1. Introduction

In 1922, Banach [1] introduced a remarkable principle, namely Banach contraction principle which asserts that every contraction self-mapping on a complete metric space has a unique fixed point. This principle plays a leading role in the development of fixed point theory. Banach's contraction principle has been generalized and extended in different directions. On his work, Edelstein [2] introduced the notion of contractive mapping and generalized Banach contraction principle. In 1973, Geraghty [3] generalized Banach's contraction principle by replacing the contraction constant by a function having certain specified properties. In 2008, Suzuki [4] introduced a new type of mapping and presented a generalization of the Banach contraction principle in which the completeness can also be characterized by the existence of a fixed point of these mappings. All these generalizations are only applicable for self-mappings.

In recent years, best proximity point theory attracted the attention of several authors. The purpose of best proximity point theory is to address a problem of finding the distance between two closed sets by using non self-mappings from one set to the other. The problem is known as the proximity point problem. Best proximity point theory analyzes the existence of an approximate solution that is optimal.

Let \(A\) and \(B\) be two non-empty subsets of a metric space \((X,d)\) and \(T:A\to B\) is a mapping, then \(d(x,Tx)\geq d(A,B)\) for all \(x\in A\). In general, for non self-mapping \(T : A \to B\), the fixed point equation \(Tx = x \) may not have a solution. In this case, it is focused on the possibility of finding an element \( x \in A\) that is an approximate solution such that the error \(d(x, Tx)\) is minimum, possibly \(d(x, Tx) = d(A,B).\)

A best proximity point becomes a fixed point if the underlying mapping is a self-mapping. Therefore, it can be concluded that best proximity point theorems generalize fixed point theorems in a natural way. In recent years, the existence and convergence of best proximity points is an interesting aspect of optimization theory which attracted the attention of many authors [5,6,7,8,9].

We recall the following notations and definitions: Let \((X,d)\) be a metric space and let \(A\) and \(B\) be non-empty subsets of \(X\).

\begin{align*}A_{0} &:= \{x \in A : d(x, y) = d(A,B)\  for \  some \  y \in B\},\\ B_{0} &:= \{y \in B : d(x, y) = d(A,B) \  for \  some \  x\in A\}.\end{align*} We denote by \(\mathcal{F}\) the class of all functions \(\beta :[0,\infty)\to [0,1)\) satisfying the following condition: \[ \beta(t_{n})\to 1 \implies t_{n}\to 0.\] We denote by \(\Phi\) the class of all functions \(\phi :[0,\infty)\to [0,\infty)\) satisfying the following conditions:

• 1. \(\phi\) is continuous,
• 2. \(\phi\) is non-decreasing, and
• 3. \(\phi(t)=0 \iff t=0\).

Definition 1. [10] Let \(A\) and \(B\) be two non-empty subsets of a metric space \((X,d)\) and \(\alpha :A\times A\to [0,\infty)\) be a function. We say that a non self-mapping \(T:A\to B\) is \(\alpha-\)proximal admissible if, for all \( x, y,u,v\in A,\) \begin{align*} & \left\{\begin{array}{ll} \alpha(x,y)\geq 1~\\ d(u,Tx)=d(A,B)\\ d(v,Ty)=d(A,B) \end{array}\right.\implies \alpha(u,v)\geq 1. \end{align*}

Definition 2. [11] Let \(A\) and \(B\) be two non-empty subsets of a metric space \((X,d)\) and \(\alpha :A\times A\to [0,\infty)\) be a function. We say that a non self-mapping \(T:A\to B\) is triangular \(\alpha-\)proximal admissible if, for all \( x, y,z, x_{1},x_{2}, u_{1}, u_{2}\in A,\) \begin{align*} &(T_{1}) \left\{\begin{array}{ll} \alpha(x_{1},x_{2})\geq 1~\\ d(u_{1},Tx_{1})=d(A,B)\\ d(u_{2},Tx_{2})=d(A,B) \end{array}\right.\implies \alpha(u_{1},u_{2})\geq 1,\\& (T_{2}) \left\{\begin{array}{ll} \alpha(x,z)\geq 1\\\alpha(z,y)\geq 1 \end{array}\right.\implies\alpha(x,y)\geq 1. \end{align*}

Definition 3. [12] Let \(A\) and \(B\) be two non-empty subsets of a metric space \((X,d)\) and \(A_0\ne \emptyset\), we say that the pair \((A,B)\) has weak \(P\)-property if and only if \begin{align*} \left\{\begin{array}{ll} d(x_{1},y_{1})=d(A,B)\\ d(x_{2},y_{2})=d(A,B) \end{array}\right.\implies d(x_{1},x_{2})\leq d(y_{1},y_{2}), \end{align*} for all \(x_{1},x_{2}~\in A\) and \( y_{1}, y_{2}\in B.\)

Definition 4. [10] Let \(A\) and \(B\) be two non-empty subsets of a metric space \((X,d)\) and \(\alpha, \eta :A\times A\to [0,\infty)\) be functions. We say that a non self-mapping \(T:A\to B\) is \(\alpha-\)proximal admissible with respect to \(\eta \) if, for all \( x, y,u,v,z,w\in A,\) \begin{align*} & \left\{\begin{array}{ll} \alpha(x,y)\geq \eta(x,y)~\\ d(u,Tx)=d(A,B)\\ d(v,Ty)=d(A,B) \end{array}\right.\implies \alpha(u,v)\geq \eta(u,v). \end{align*}

Definition 5. [13] Let \(A\) and \(B\) be two non-empty subsets of a metric space \((X,d)\) and \(T :A\to B\) be a mapping. We say that T has the RJ-property if for any sequence \(\{x_{n}\}\subseteq A,\) \begin{align*} & \left\{\begin{array}{ll} \displaystyle\lim_{n\to\infty}d(x_{n+1},Tx_{n})=d(A,B)\\ \displaystyle\lim_{n\to\infty}x_{n}=x \end{array}\right.\implies x\in A_{0}.& \end{align*}

Remark 1. [13] Any continuous mapping \(T:A\to B\) has the RJ-property provided that \(A\) and \(B\) are non-empty closed subsets of a metric space \((X,d)\). If \(A\) and \(B\) are not closed subsets of \(X\), then \(T\) may not have RJ-property.

Example 1. [13] Let \(X=\mathbb{R}\), \(A=(0,1)\) and \(B=(2,3)\). We define \(d:X\times X\to[0,\infty)\) and \(T:A\to B\) by \(d(x,y)=|x-y|\) and \(Tx=3-x\). Let \(\{x_{n}\}=\{1-\frac{1}{n}\}\subseteq A\), then \[\lim_{n\to\infty}x_{n}=1 \  and \  \lim_{n\to\infty}d(x_{n+1},Tx_{n})=\lim_{n\to\infty}d(1-\frac{1}{n+1},2+\frac{1}{n})=1=d(A,B),\] but \(1\notin A_{0}\). Hence \(T\) does not satisfy the RJ-property.

In 2016, Hamzehnejadi and Lashkaripour [13] proved best proximity point results for non self-map satisfying the RJ-property.

Definition 6. [13] Let \(A\) and \(B\) be two non-empty subsets of a metric space \((X,d)\) and \(\alpha:X\times X\to [0,\infty)\) be a function. A mapping \(T :A\to B\) is said to be a generalized \(\alpha-\phi-\)Geraghty proximal contraction if there exists \(\beta \in \mathcal{F} \) such that for all \(x,y,u,v\in A\), \begin{align*} \left\{\begin{array}{ll} d(u,Tx)=d(A,B)\\ d(v,Ty)=d(A,B) \end{array}\right.\implies \alpha(x,y)\phi(d(u,v))\leq\beta\big(\phi(M(x,y,u,v))\big)\phi(M(x,y,u,v)), \end{align*} where \(M(x,y,u,v)=\) max \(\{d(x,y),d(x,u),d(y,v)\}\) and \(\phi\in \Phi\).

Theorem 1. [13] Let \((X,d)\) be a complete metric space, \(A\) and \(B\) be non-empty subsets of \(X \), \(\alpha: X\times X\to [0,\infty)\) be a function and \(T: A\to B\) be a mapping. If the following conditions are satisfied:

• 1. \(T\) is a generalized \(\alpha-\phi-\)Geraghty proximal contraction type mapping,
• 2. \( T(A_{0}) \subseteq B_{0}\) and \(T\) is triangular \(\alpha-\)proximal admissible,
• 3. \(T\) has the RJ-property,
• 4. if \(\{x_{n}\}\) is a sequence in \(A\) such that \(\alpha(x_{n},x_{n+1})\geq 1\) for all \(n\) and \(x_{n}\to x\in A\) as \(n\to \infty\), then there exists a subsequence \(\{x_{n_{k}}\}\) of \(\{x_{n}\}\) such that \(\alpha(x_{n_{k}},x)\geq 1\) for all \(k\),
• 5. there exist \(x_{0},x_{1}\in A\) such that \(d(x_{1},Tx_{0})=d(A,B)\) and \(\alpha(x_{0},x_{1})\geq1\),

then there exists \(x^*\in A_{0}\) such that \(d(x^*,Tx^*)=d(A,B)\). Moreover, if \(\alpha(x,y)\geq 1\) for all \(x,y\in P_{T}(A)\), where \(P_{T}(A)\) denotes the set of best proximity points of \(T\), then \(x^*\) is a unique best proximity point of \(T\).

In this paper, we denote by \(\Phi_{\varphi}\) the class of functions \(\varphi: [0,\infty)\to [0,\infty)\) satisfying the following property:

\[\varphi(t)\leq \frac{1}{2}t \  for \  all \  t\geq 0.\] We denote by \(\Psi\) the set of non-decreasing functions \(\psi: [0,\infty)\to[0,\infty)\) such that \[\lim\limits_{n\to\infty}\psi^n(t)=0 \  for \  all \  t\geq 0.\] Recently, Hussain et al., [14] proved the existence of best proximity point for modified Suzuki-Edelstein \(\alpha\)-proximal contraction.

Definition 7. [14] Suppose \(A\) and \(B\) are two non-empty subsets of a metric space \((X,d)\). A non self-mapping \(T:A\to B\) is said to be modified Suzuki-Edelstein Proximal contraction if \[\varphi(d(x,Tx))-2d(A,B)\leq \alpha(x,y)d(x,y)\implies \alpha(x,y)d(Tx,Ty)\leq \psi(d(x,y)),\] for all \(x,y\in A\), where \(\varphi\in\Phi_{\varphi}\), \(\psi\in \Psi\) and \(\alpha: A\times A\to [0,\infty)\).

Theorem 2. [14] Suppose \(A\) and \(B\) are two non-empty closed subsets of a complete metric space \((X,d)\) with \(A_0\) is non-empty and let \(T: A\to B\) with \(T(A_0)\subseteq B_0\) be continuous modified Suzuzi-Edelstein proximal admissible mapping with respect to \(\eta(x,y)=2\) and the pair \((A,B)\) satisfies the weak \(P\)-property. If, the elements \(x_0\) and \(x_1\) with \(d(x_1,Tx_0)=d(A,B)\) satisfies \(\alpha(x_0,x_1)\geq 2\), then \(T\) has a unique best proximity point.

Lemma 1. [15] Suppose \((X,d)\) is a metric space and \(\{x_n\}\) be a sequence in \(X\) such that \( d(x_n, x_{n + 1} )\rightarrow 0 \) as \( n\rightarrow\infty\). If \( \{x_{n}\} \) is not a Cauchy sequence, then there exist an \(\epsilon > 0\) and sequences of positive integers \(\{m_{k}\}\) and \(\{n_{k}\}\) with \(m_{k} > n_{k} > k\) such that \(d(x_{m_{k}}, x_{n_{k}})\geq\epsilon, d(x_{m_{k}-1}, x_{n_{k}})< \epsilon\) and \[\hspace{-.1cm}(i) \lim_{k\to\infty} d(x_{m_{k}-1} , x_{n_{k}+1} ) = \epsilon\hspace{4.1cm}(ii)~ \lim_{k\to\infty} d(x_{m_{k}} , x_{n_{k}} ) = \epsilon\] \[ (iii) \lim_{k\to\infty} d(x_{m_{k}-1} , x_{n_{k}}) = \epsilon\hspace{4.cm}(iv) \lim_{k\to\infty} d(x_{m_{k}} , x_{n_{k}+1} ) = \epsilon.\] Motivated by the work of Suzuki, Edelstein and Geraghty, we introduce the notion of modified Suzuki-Edelstein-Geraghty proximal contraction and prove the existence and uniqueness of best proximity point for such mappings.

2. Main results

Definition 8. Let \(A\) and \(B\) be two non-empty subsets of metric space \((X,d)\). Let \( T :A\to B\) be non self-mapping and \(\alpha: A\times A\to [0,\infty)\) be a function. \(T\) is said to be a modified Suzuki-Edelstein-Geraghty proximal contraction if there exist \(\beta \in \mathcal{F} \) and \(\phi\in\Phi\) such that for all \(x,y\in A\),

where \(M(x,y)= \max \{d(x,y),d(x,Tx),d(y,Ty)\}\),   \(m(x,y)=\max\{d(x,Tx),d(y,Ty)\}\) and \(\varphi\in \Phi_{\varphi}\).

Theorem 3. Let \((X,d)\) be a complete metric space and \(A\) and \(B\) be non-empty closed subsets of \(X \) with \(A_0\) is non-empty. If \(T: A\to B\) be a modified Suzuki-Edelstein-Geraghty proximal contraction mapping such that the following conditions hold:

• 1. \(T(A_0)\subseteq B_0\) and the pair \((A,B)\) satisfies the weak \(P\)-property,
• 2. \(T\) is triangular \(\alpha\)-proximal admissible with respect to \(\eta(x,y)= 2\),
• 3. \(T\) is continuous,
• 4. there exist \(x_{0},x_{1}\in A\) such that \(d(x_{1},Tx_{0})=d(A,B)\) and \(\alpha(x_0,x_1)\geq2\),

then \(T\) has a unique best proximity point in \(A_0\).

Proof. By assumption (\(iv\)), there exist \(x_{0},~x_{1}\in A\) such that

Since \(Tx_{0}\in B\), by the definition of \(A_{0}\), from (2), we have \(x_{1}\in A_{0}\). Since \(T(A_{0})\subseteq B_{0},\) we have \(Tx_{1} \in B_{0}\). Hence by definition of \(B_{0}\), there exists an element \(x_{2}\in A\) such that Since \(T\) is \(\alpha\)-proximal admissible with respect to \(\eta(x,y)=2\), we obtain \(\alpha(x_1,x_2)\ge 2\). On continuing this process, we have for all \(n\in\mathbb{N}\).

Now,

From (5), we have \[\varphi(d(x_{n-1},Tx_{n-1}))-2d(A,B)\leq 2d(x_{n-1},x_{n})\leq\alpha(x_{n-1},x_{n})d(x_{n-1},x_{n}).\] By (1), it follows that where \( M(x_{n-1},x_{n})= {max}\{d(x_{n-1},x_{n}),d(x_{n-1},Tx_{n-1}),d(x_{n},Tx_{n})\}= {max}\{d(x_{n-1},x_{n}),d(x_{n-1},x_{n}),d(x_{n},x_{n+1})\},\) and \( m(x_{n-1},x_{n})={max}\{d(x_{n-1},Tx_{n-1}),d(x_{n},Tx_{n})\}={max}\{d(x_{n-1},x_{n}),d(x_{n},x_{n+1})\}.\)

Suppose \(x_{n_{0}}=x_{n_{0}+1}\) for some \(n_{0}\in \mathbb{N}\). Assume that \(x_{n_{0}+1}\neq x_{n_{0}+2}\), then by (6), it follows that

\begin{align*} \phi(d(x_{n_{0}+1},x_{n_{0}+2}))&\leq\beta\big(\phi(M(x_{n_{0}},x_{n_{0}+1}))\big)\phi({max}\{m(x_{n_0},x_{n_{0}+1})-d(A,B)\})\\ &<\phi(\{{max}\{m(x_{n_0},x_{n_{0}+1})-d(A,B)\})\\&=\phi({max}\{d(x_{n_{0}},x_{n_{0}+1}),d(x_{n_{0}+1},x_{n_{0}+2})\}-d(A,B)\})\\ &\leq\phi(\{d(x_{n_{0}+1},x_{n_{0}+2})+d(A,B)\}-d(A,B))\\&=\phi(d(x_{n_{0}+1},x_{n_{0}+2})), \end{align*} a contradiction. Therefore \(x_{n_{0}+1}= x_{n_{0}+2}\), hence \(x_{n_{0}}=x_{n_{0}+1}= x_{n_{0}+2}\), so from (4), it follows that \( d(x_{n_{0}},Tx_{n_{0}})=d(x_{n_{0}+1},Tx_{n_{0}})=d(A,B),\) i.e., \(x_{n_{0}}\) is a best proximity point of \(T\), which is the desired result. Therefore, we assume that \(x_n\neq x_{n+1}\) for all \(n\in\mathbb{N}\cup \{0\}\). From (6), we obtain If \( {max}\{d(x_{n-1},x_{n}),d(x_{n},x_{n+1})\}=d(x_{n},x_{n+1}),\) then \(\phi(d(x_{n},x_{n+1}))< \phi (d(x_{n},x_{n+1})),\) a contradiction. Hence \( {max}\{d(x_{n-1},x_{n}),d(x_{n},x_{n+1})\}=d(x_{n-1},x_{n})\), so, by (7), we have \(\phi(d(x_{n},x_{n+1}))< \phi(d(x_{n-1},x_{n})).\) By the non-decreasing property of \(\phi\), it follows that \( d(x_{n},x_{n+1})\leq d(x_{n-1},x_{n})\), for all \(n\geq 1.\) Hence we deduce that \(\{d(x_{n},x_{n+1})\}\) is a decreasing sequence of non-negative real numbers. So, there exists \(r\geq 0\) such that Suppose that \(r>0\). From (7), we have where On letting \(n\to \infty\) in (9) and using (10), we obtain \(\displaystyle\lim_{n\to\infty}\beta(\phi(d(x_{n-1},x_{n})))=1.\) Since \(\beta\in\mathcal{F},\) it follows that \(\displaystyle\lim_{n\to\infty}\phi(d(x_{n-1},x_{n}))=0.\) By continuity of \(\phi\), we get i.e., \(\phi(r)=0\), so that \(r=0\). i.e., Now, we show that \(\{x_{n}\}\) is a Cauchy sequence. Suppose \(\{x_{n}\}\) is not a Cauchy sequence. Then there exists an \(\epsilon>0\) for which we can find sequences of positive integers \(\{m_{k}\}\) and \(\{n_{k}\}\) with \(m_{k}>n_{k}>k\) such that Since \(T\) is triangular \(\alpha\)-proximal admissible with respect to \(\eta (x,y)=2\), we can show that \(\alpha(x_n, x_m)\ge 2\) for all \(n,m\in \mathbb{N}\) with \(n< m\). If \(n=m+1\), we have Suppose that \(\alpha(x_n, x_m)\ge 2\) for all \(n,m\in \mathbb{N}\) with \(n< m\). To show this we shall prove that \(\alpha(x_n,x_{m+1})\geq 2\) with \(n< m\). From (4), we have Also since \(T\) is triangular \(\alpha\)-proximal admissible with respect to \(\eta (x,y)=2\), then from (14) and (15), \(\alpha(x_n,x_{m+1})\ge 2\) for all \(n,m\in \mathbb{N}\) with \(n< m\). Hence for any \(m_k, n_k\in\mathbb{N}\) with \(n_k < m_k\), we get \(\alpha(x_{m_{k}},x_{n_{k}})\ge 2\). From (12) and (13), we can choose a positive integer \(k_1\in\mathbb{N}\) such that From (16), we have that \( \varphi(d(x_{m_{k}},Tx_{m_{k}}))-2d(A,B)\leq 2d(x_{m_{k}},x_{n_{k}})\leq\alpha(x_{m_{k}},x_{n_{k}})d(x_{m_{k}},x_{n_{k}}). \) Therefore, By (1), we get where \begin{align*} {max}\{d(x_{m_k},x_{n_k}),& {max}~\{m(x_{m_k},x_{n_k})\}-d(A,B)\}\\&= {max}\{d(x_{m_k},x_{n_k}), {max}\{d(x_{m_k},Tx_{m_k}),d(x_{n_k},Tx_{n_k})\}-d(A,B)\}\\&= {max}\{d(x_{m_k},x_{n_k}), {max}\{d(x_{m_k},x_{m_{k+1}}),d(x_{n_k},x_{n_{k+1}})\}-d(A,B)\}\\&\le {max}\{d(x_{m_k},x_{n_k}), {max}\{d(x_{m_k},x_{m_{k+1}})+d(A,B),d(x_{n_k},x_{n_{k+1}})+d(A,B)\}-d(A,B)\}\\&= {max}\{d(x_{m_k},x_{n_k}),d(x_{m_k},x_{m_{k+1}}),d(x_{n_k},x_{n_{k+1}})\} . \end{align*} Hence by applying Lemma 1 \begin{align*} \lim\limits_{k\to \infty} {max}\{d(x_{m_k},x_{n_k}), {max}~\{m(x_{m_k},x_{n_k})\}-d(A,B)\}\le\lim\limits_{k\to \infty} {max}\{d(x_{m_k},x_{n_k}),d(x_{m_k},x_{m_{k+1}}),d(x_{n_k},x_{n_{k+1}})\}=\epsilon. \end{align*} On letting \(k\to\infty\) in (18), it follows that \[0< \frac{\phi(\epsilon)}{\phi(\epsilon)}\leq\displaystyle\lim\limits_{k\to\infty}\beta(\phi(M(x_{m_{k}},x_{n_{k}})))\leq 1,\] thus, \[\displaystyle\lim\limits_{k\to\infty}\beta(\phi(M(x_{m_{k}},x_{n_{k}})))= 1.\] Since \(\beta\in\mathcal{F},\) we have \[\displaystyle\lim\limits_{k\to\infty}\phi( {max}\{d(x_{n_k},x_{m_k}),d(x_{m_k},x_{m_{k+1}}),d(x_{n_k},x_{n_{k+1}})\})=0.\] This yields, by continuity of \(\phi\), that \[\phi(\displaystyle\lim_{k\to\infty}( {max}\{d(x_{n_k},x_{m_k}),d(x_{m_k},x_{m_{k+1}}),d(x_{n_k},x_{n_{k+1}})\}))=0,\] i.e., \(\phi(\epsilon)=0\) and hence \(\epsilon=0\), a contradiction. Hence \(\{x_{n}\}\) is a Cauchy sequence. By the completeness of \(X\) and closed property of \(A\), there exists \(x^*\in A\) such that \(\displaystyle\lim_{n\to\infty}x_{n}=x^*\). Since \(T\) is continuous, from (4), we obtain \begin{equation*} d(A,B)= \lim\limits_{n\to}d(x_{n+1},Tx_{n})=d(x^*,Tx^*), \end{equation*} hence \(x^*\) is the best proximity point of \(T\).

We now show the uniqueness of best proximity point. Suppose that \(u\) and \(v\) are the two distinct best proximity points of \(T\). Since \(d(u,Tu)=d(A,B)=d(v,Tv)\), by weak \(P\)-property of the pair \((A,B),\) we get

Now, we consider From (20), we have that \[\varphi(d(u,Tu))-2d(A,B)\leq 0\leq\alpha(u,v)d(u,v).\] By (1) and (19), it follows that \begin{align*} \phi(d(u,v))&\leq \phi(d(Tu,Tv))\leq\alpha(u,v)\phi(d(Tu,Tv))\\&\leq\beta\big(\phi(M(u,v))\big)\phi( {max}\{d(u,v), {max}\{m(u,v)\}-d(A,B)\})\\&<\phi( {max}\{d(u,v), {max}\{m(u,v)\}-d(A,B)\})\\&=\phi( {max}\{d(u,v), {max}\{d(u,Tu),d(v,Tv)\}-d(A,B)\})\\&=\phi(d(u,v)), \end{align*} a contradiction. Hence \(u=v\).

Example 2. Let \(X=\mathbb{R}^2,~A=[0,\infty)\times\{1\},~B=[0,\infty)\times\{0\}, A_{0}=[0,1]\times\{1\} \) and \(B_{0}=[0,1]\times\{0\}\). We define \(d\) by \(d((x_{1},x_{2}),(y_{1},y_{2}))=\sqrt{(x_{1}-y_{1})^2+(x_{2}-y_{2})^2}\) for all \((x_{1},x_{2}),(y_{1},y_{2})\in X\) and a map \(T:A\to B\) by \begin{align*} &T(x,1) = \left\{\begin{array}{ll} \big(\frac{1}{2}x,0\big) & \mbox{   if  \(x\in[0,1]\)}\\ (\frac{3}{2}x-1,0)&\mbox{   if   \(x\geq1\)}. \end{array} \right. \end{align*} We also define functions \(\alpha: A\times A\to [0,\infty),~\beta: [0,\infty)\to [0,1), and \  \phi:[0,\infty)\to[0,\infty)\) by \begin{align*} \alpha((p,q),(r,s)) &= \left\{\begin{array}{ll} 2&\mbox{   if   \((p,q),(r,s)\in [0,1]\times\{1\}\)}.\\ 0&\mbox{   otherwise,} \end{array} \right.\\ \beta(t) &= \left\{\begin{array}{ll} 0& \mbox{   if   \(t=0\)},\\ \frac{1+t}{1+2t} & \mbox{   if \(t>0\), } \end{array} \right. \end{align*} and \begin{align*}\phi(t) &= \left\{\begin{array}{ll} \frac{1}{2}t&\mbox{   if   \(t\in[0,1]\)},\\ \frac{t}{1+t}&\mbox{   if   \(t\geq1\)}. \end{array} \right. \end{align*} Clearly, \(~T(A_{0})\subseteq B_{0}\), \(d(A,B)=1\) and \(T\) is continuous. We choose \(x_{0}=(\frac{1}{2},1)\) and \(x_{1}=(\frac{1}{4},1)\) such that \(d(x_{1},Tx_{0})=d(A,B)\) and \(\alpha(x_{0}, x_{1})\geq 2.\) Now, let \((x,1),~(y,1),~(u,1),~(v,1)\in A\) such that

From (21), we obtain \(x,y\in [0,1],~u=\frac{1}{2}x\in[0,\frac{1}{2}]\), \(v=\frac{1}{2}y\in[0,\frac{1}{2}]\) and hence \(d((u,1),(v,1))\leq d(T(x,1),T(y,1))\). Therefore, the pair \((A,B)\) satisfies the weak \(P\)-property. Also, \(\alpha((u,1),(v,1))\geq 2\). Hence \(T\) is \(\alpha-\)proximal admissible. Clearly, \(T\) is a triangular \(\alpha-\)proximal admissible with respect to \(\eta ((x,1),(y,1))= 2\).

Now, we show that \(T\) is a modified Suzuki-Edelstein-Geraghty proximal contraction mapping. Let \((x,1),~(y,1)\in [0,1]\times\{1\}\), then \(\alpha((x,1),~(y,1))=2\). Without loss of generality assume that \(x>y\). Now, we consider

\begin{align*} {max}&\left\{d\left(\left(x,1\right),\left(y,1\right)\right),{max}\left\{m\left(\left(x,1\right),\left(y,1\right)\right)\right\}-d\left(A,B\right)\right\} \\&={max}\left\{d\left(\left(x,1\right),\left(y,1\right)\right),{max}~\left\{d\left(\left(x,1\right),T\left(x,1\right)\right),d\left(\left(y,1\right),T\left(y,1\right)\right)\right\}-d\left(A,B\right)\right\}\\&={max}\left\{d\left(\left(x,1\right),\left(y,1\right)\right),{max}~\left\{d\left(\left(x,1\right),\left(\frac{1}{2}x,0\right)\right),d\left(\left(y,1\right),\left(\frac{1}{2}y,0\right)\right)\right\}-1\right\}\\&={max}\left\{d\left(\left(x,1\right),\left(y,1\right)\right),~\left\{d\left(\left(x,1\right),\left(\frac{1}{2}x,0\right)\right)\right\}-1\right\}\\&\leq{max}\left\{d\left(\left(x,1\right),\left(y,1\right)\right),~\left\{d\left(\left(x,1\right),\left(\frac{1}{2}x,0\right)\right)\right\}\right\} \\&=d\left(\left(x,1\right),\left(\frac{1}{2}x,0\right)\right). \end{align*} Also we consider \begin{align*} \varphi\left(d\left(\left(x,1\right),T\left(x,1\right)\right)\right)&\leq\frac{1}{2}d\left(\left(x,1\right),T\left(x,1\right)\right)=\frac{1}{2}d\left(\left(x,1\right),\left(\frac{x}{2},0\right)\right)\leq 2d\left(\left(x,1\right),\left(\frac{x}{2},0\right)\right)\\&\leq 2\left(d\left(\left(x,1\right),\left(y,1\right)\right) +d\left(\left(y,1\right),\left(\frac{x}{2},0\right)\right)\right)\leq 2\left(d\left(\left(x,1\right),\left(y,1\right)\right) +d\left(\left(\frac{x}{2},1\right),\left(\frac{x}{2},0\right)\right)\right)\\&=2d\left(\left(x,1\right),\left(y,1\right)\right) + 2d\left(A,B\right). \end{align*} Hence \(\varphi(d((x,1),T(x,1)))-2d(A,B)\leq2d((x,1),(y,1))=\alpha ((x,1),(y,1))d((x,1),(y,1)).\) By (1), it follows that \begin{align*} \frac{1}{4}\sqrt{(x-y)^2}&=\phi(d((T(x,1),T(y,1))))\leq \alpha ((x,1),(y,1))\phi(d((T(x,1),T(y,1))))\\ &\leq\beta\big(\phi(M((x,1),(y,1)))\big)\phi({max}\{d((x,1),(y,1)),{max}\{m((x,1),(y,1))\}-d(A,B)\})\\ &<\phi({max}\{d((x,1),(y,1)),{max}\{m((x,1),(y,1))\}-d(A,B)\})\\ &=\phi(d((x,1),(\frac{1}{2}x,0)))=\frac{\sqrt{\frac{1}{4}x^2+1}}{1+\sqrt{\frac{1}{4}x^2+1}}. \end{align*} Therefore, \(T\) is a modified Suzuki-Edelstein-Geraghty proximal contraction mapping. The point \((0,1)\in A_0\) is the unique best proximity point of \(T\).

If \(\phi(t)=t\) for all \(t\geq 0\), in Theorem 3, we have the following corollary.

Corollary 1. Suppose that \(A\) and \(B\) are two non-empty closed subsets of a complete metric space \((X,d)\) with \(A_0\) is non-empty. Assume that \(\alpha: A\times A\to [0,\infty)\) is a function and there exists \(\beta \in \mathcal{F} \). If \(T:A\to B\) be a non self-mapping such that for all \(x,y\in A\), \( \varphi(d(x,Tx))-2d(A,B)\leq \alpha(x,y)d(x,y)\) implies \(\alpha(x,y)d(Tx,Ty) \leq\beta\big(M(x,y)\big){max}\{d(x,y),m(x,y)-d(A,B)\}, \) where \(M(x,y)=\max \{d(x,y),d(x,Tx),d(y,Ty)\},\) \(m(x,y)={max}\{d(x,Tx),d(y,Ty)\}\) and \(\varphi\in \Phi_{\varphi}\) with the following conditions hold:

• 1. \(T(A_0)\subseteq B_0\) and the pair \((A,B)\) satisfies the weak \(P\)-property,
• 2. \(T\) is triangular \(\alpha\)-proximal admissible with respect to \(\eta(x,y)= 2\),
• 3. \(T\) is continuous,
• 4. there exist \(x_{0},x_{1}\in A\) such that \(d(x_{1},Tx_{0})=d(A,B)\) and \(\alpha(x_0,x_1)\geq2\),

then \(T\) has a unique best proximity point in \(A_0\).

We can prove the existence and uniqueness of best proximity point by replacing the continuity assumption with RJ-property in Theorem 3.

Theorem 4. Let \((X,d)\) be a complete metric space and \(A\) and \(B\) be non-empty closed subsets of \(X \) with \(A_0\) is non-empty. If \(T: A\to B\) be a modified Suzuki-Edelstein-Geraghty proximal contraction mapping such that the following conditions hold:

• 1. \(T(A_0)\subseteq B_0\) and the pair \((A,B)\) satisfies the weak \(P\)-property,
• 2. \(T\) is triangular \(\alpha\)-proximal admissible with respect to \(\eta(x,y)= 2\),
• 3. \(T\) has the RJ-property and assume that \(\alpha(x,y)\geq 2\) for all \(x,y\in A\),
• 4. there exist \(x_{0},x_{1}\in A\) such that \(d(x_{1},Tx_{0})=d(A,B)\) and \(\alpha(x_0,x_1)\geq2\),

then \(T\) has a unique best proximity point in \(A_0\).

Proof. From the proof of Theorem 3, \(\{x_n\}\) is cauchy such that \(x_n\to x*\in A\) as \(n\to\infty\). Since \(T\) has the RJ-property \(x^*\in A_0\). We shall prove that \(d(x^*, Tx^*) = d(A,B).\) From the proof of Theorem 3, we have \(d(x_{n+1},x_{n+2})\leq d(x_{n},x_{n+1})\) for all \(n\in\mathbb{N}\cup \{0\}\).

Suppose \(d(x_n,x^*)< \frac{1}{2}d(x_n,x_{n+1})\) and \(d(x_{n+1},x^*)< \frac{1}{2}d(x_{n+1},x_{n+2}),\) for some \(n\in\mathbb{N}.\) Therefore,

\begin{align*} d(x_n,x_{n+1})&\leq d(x_n,x^*)+d(x^*,x_{n+1})< \frac{1}{2}d(x_n,x_{n+1})+\frac{1}{2}d(x_{n+1},x_{n+2})\\ &\leq\frac{1}{2}d(x_n,x_{n+1})+\frac{1}{2}d(x_{n},x_{n+1})=d(x_{n},x_{n+1}), \end{align*} a contradiction. Hence \(d(x_n,x^*)\ge\frac{1}{2}d(x_n,x_{n+1})\) and \(d(x_{n+1},x^*)\ge\frac{1}{2}d(x_{n+1},x_{n+2}),\) for all \(n\in\mathbb{N}.\)

Now, we consider

\begin{align*} \varphi(d(x_n,Tx_n))&\le\frac{1}{2}d(x_n,Tx_n)\le d(x_n,Tx_n)\le d(x_n,x_{n+1})+d(x_{n+1},Tx_n)\\&\leq 2d(x_n,x^*)+d(A,B)\leq 2d(x_n,x^*)+2d(A,B). \end{align*} From the above inequality, we obtain \[\varphi(d(x_n,Tx_n))-2d(A,B)\leq 2d(x_n,x^*)\leq \alpha(x_n,x^*)d(x_n,x^*).\] Since \(x_n\in A ~\forall n\in \mathbb{N}\) and \(x^*\in A\), we obtain \(\alpha(x_n,x^*)\geq 2\). Thus from (1), it follows that Letting \(n\to\infty\) in (22), we obtain Since \(d(x^*,Tx^*)\leq d(x^*,x_{n+1})+d(x_{n+1},Tx_n)+d(Tx_n,Tx^*),\) we get \begin{equation*}d(x^*,Tx^*)-d(x_{n+1},Tx_n)\leq d(x^*,x_{n+1})+d(Tx_n,Tx^*).\end{equation*} From the property of \(\phi\), it follows that \begin{equation*}\phi\big(d(x^*,Tx^*)-d(x_{n+1},Tx_n)\big)\leq\phi\big(d(x^*,x_{n+1})+d(Tx_n,Tx^*)\big).\end{equation*} Hence By (23) and (24), we have \begin{equation*} \phi\big(d(x^*,Tx^*)-d(A,B)\big)\leq\phi\big(\lim\limits_{n\to\infty}d(Tx_n,Tx^*)\big)< \phi\big(d(x^*,Tx^*)-d(A,B)\big),\end{equation*} a contradiction. Hence \(d(x^*,Tx^*)=d(A,B)\). Therefore, \(x^*\) is the best proximity point of \(T\). Uniqueness follows from the proof of Theorem 3.

If we take \(\phi(t)=t\) for all \(t\geq 0\), in Theorem 4, we have the following corollary.

Corollary 2. Suppose that \(A\) and \(B\) are two non-empty closed subsets of a complete metric space \((X,d)\) with \(A_0\) is non-empty. Further suppose that \(\alpha: A\times A\to [0,\infty)\) is a function and there exists \(\beta \in \mathcal{F} \). If \(T:A\to B\) be a non self-mapping such that for all \(x,y\in A\), we have \( \varphi(d(x,Tx))-2d(A,B)\leq \alpha(x,y)d(x,y)\) implies \(\alpha(x,y)d(Tx,Ty) \leq\beta\big(M(x,y)\big){max}\{d(x,y),m(x,y)-d(A,B)\}, \) where \(M(x,y)=\max \{d(x,y),d(x,Tx),d(y,Ty)\}\), \(m(x,y)={max}\{d(x,Tx),d(y,Ty)\}\) and \(\varphi\in \Phi_{\varphi}\), with the following conditions hold:

• 1. \(T(A_0)\subseteq B_0\) and the pair \((A,B)\) satisfies the weak \(P\)-property,
• 2. \(T\) is triangular \(\alpha\)-proximal admissible with respect to \(\eta(x,y)= 2\),
• 3. \(T\) has the RJ-property and assume that \(\alpha(x,y)\geq 2\) for all \(x,y\in A\),
• 4. there exist \(x_{0},x_{1}\in A\) such that \(d(x_{1},Tx_{0})=d(A,B)\) and \(\alpha(x_0,x_1)\geq2\),

then \(T\) has a unique best proximity point in \(A_0\).

Conflicts of Interest

The author declares no conflict of interest.

1. Introduction and Preliminaries

A novel Coronavirus or COVID-19 is an infectious ailment brought about by a recently distinguished infection which is known to be just transmitted through a set of all animals yet as of late influenced people too. Since December 2019, various instances of "obscure viral pneumonia" identified with a nearby Seafood Wholesale Market was accounted for in Wuhan City, China [1]. A Novel coronavirus fit for tainting people was officially affirmed on January 6, 2020 [2]. As indicated by nature, the spread of coronavirus ailment 2019 (COVID-19) is getting relentless and has just arrived at the important epidemiological measures for it to be announced a pandemic [3]. COVID-19 is an intense settled disease however it can likewise be lethal, with a 2% case casualty rate [4]. Like other coronaviral pneumonia, for example, a serious intense respiratory disorder brought about by coronavirus, COVID-19 can likewise prompt intense respiratory trouble condition [2].

There is a dire requirement for viable treatment. Current spotlight has been on the improvement of novel therapeutics, including antivirals and antibodies. Gathering proof recommends that a subgroup of patients with serious COVID-19 may have a cytokine storm condition [5]. The most widely recognized test method as of now utilized for COVID-19 determination is an ongoing converse interpretation polymerase chain response (RT-PCR) [6]. COVID-19 can cause intense heart injury. In the vast majority of the cases, the patients who have co-morbidity like diabetes, circulatory strain, coronary illness [7].

The side effects of these sicknesses resemble whatever another ordinary influenza which is a disadvantage of distinguishing the genuine influenced ones. The side effects can be demonstrated roughly in the middle of 14 days. As this COVID-19 is another infection for the clinical network, so still explicit treatment with respect to COVID-19 is difficult. There are some recognized side effects in regards to COVID-19, proposed by the World Health Organization (WHO). For example, high fever or mellow fever, hack, breathing problem, exhaustion, muscle or body throbs, migraine, loss of taste or smell, sore throat, clog or runny nose, spewing, diarrhoea. It straightforwardly influences the lung. X-Ray based images can assist us with knowing the lung condition so we can discover more COVID-19 cases as per the lung report. CT scan reports likewise can be utilized [8]. In spite of the fact that by far most of patients just have a typical, gentle type of sickness, around 15-20% of the patients fall into the serious gathering, which means they require helped oxygenation as a major aspect of treatment [9].

While it is about images-based problems, deep convolutional neural network can comprehend this all the more effectively these days. Deep neural-based frameworks can group images or related issues all the more precisely and productively by its condition of workmanship algorithmic strength. These are some enormous algorithms have been presented by deep learning researchers.

In this examination, author assesses the viability of cutting edge pre-trained convolutional neural systems proposed by established researchers, with respect to their mastery in the programmed analysis of COVID-19 from thoracic X-ray images. Author utilized pre-trained models, for example, DenseNet-121, ResNet-50, EfficientNet-B4 and base convolutional model. Our assessment is dependent on AUC.

Further part of this paper-situated as related works, dataset, methodology, results, discussions and conclusion.

2. Related works

The novel coronavirus is another new disease in the field of the clinical network. Clinical researchers, just as deep learning specialists, are attempting to determine this issue. The fundamental test is to distinguish COVID-19 cases in a less measure of time and minimal cost. So the AI research network has come up to handle this test all the more proficiently. Less measure of works has been done as such far. In this segment, we go over some past and effective works with respect to this challenge from AI specialists.

Mangla et al., [10] have attempted to tackle COVID-19 case identification utilizing pre-prepared deep convolutional neural systems. Their model contains pre-prepared CheXNet, with a 121-layer Dense Convolutional Network (DenseNet) spine, trailed by a completely associated layer. They supplant CheXNet's last classifier of 14 classes with our characterization layer of 4 classes, each with a sigmoid actuation to deliver the last yield. They wound up with a consequence of AUROC 0.9994 and precision of 87.2% in 4 class grouping. They named their model as CovidAID. El Asnaou et al., [11] have attempted to discover a few inquiries in regards to COVID-19 early recognition utilizing deep learning methods. They executed a few generally utilized deep learning structures, for example, VGG16, VGG19, MobileNet V2, Resnet50, DenseNet201, Inception ResNet V2 and Inception V3 in X-Ray just as Ct-Scan images, where they infer that Inception ResNet V2 has performed superior to different architectures with a 92.18% accuray. Apostolopoulos et al., [12] have utilized pre-prepared deep learning models in their test. They tested in a dataset which contains 1427 X-Ray images, where 700 images are typical pneumonia, 224 images with affirmed Covid-19 cases and 504 images of ordinary conditions. They utilized MobileNet v2, VGG19, Inception, Xception and Inception ResNet v2 designs. Where VGG 19 has given the best yield 98.75% accuracy in 2-class order.

Abiyev et al., [13] conventional convolutional neural system to distinguish chest related ailment. They spoke to a correlation between the convolutional neural system, supervised back-propagation neural system and competitive neural system utilizing chest X-Ray images. Where the convolutional neural system has performed superior to different models. Abbas et al., [14] have actualized a tuned and altered deep neural system in X-ray images to distinguish COVID-19 cases all the more productively. They re-manufacture their model and named as DeTraC which contains 3 periods of layers. They built up this by utilizing ResNet-18 in backend and gets an accuracy of 95.12% in the X-Ray dataset. Rahimzadeh et al., [15] have actualized a connected of Xception and ResNet50V2 design to distinguish COVID-19 cases. In their trial, they utilized unbalanced X-Ray dataset. They observed numerous deep learning models look at the best result. The altered model which is a blend of Xception and ResNet50V2 has accomplished 91.40% accuracy on average. Naurin et al., [16] have executed convolutional neural systems, for example, Inception V3, Inception ResNetV3 and ResNet50 for the identification of COVID-19 cases by X-Ray images. They saw around 98% accuracy in pre-prepared ResNet50 model, which is higher than Inception V3 model.

Considering all references, author attempts to handle this continuous COVID-19 detection issue by various deep learning procedures. We executed EfficientNet-B4, ResNet-50, DenseNet-121 and base CNN model to legitimize which one performs better in this analysis.

3. Dataset

In this investigation, author has utilized and retrieved another arrangement of a dataset for the COVID-19 detection framework. It is accessible for theresearch community to battle against COVID-19 and quicken the exploration results. Later on, this dataset has been presented by Kaggle as an ongoing competition. The dataset contains a total of 5907 X-Ray images, where it has 5283 images for train purpose and 624 images for test purpose. It additionally has two classifications, for example, normal class and pneumonia class. Pneumonia class has four division, for example, SARS, COVID-19, ARDS and Streptococcus. The dataset can be downloaded from https://github.com/ieee8023/covid-chestxray-dataset. The sample of the dataset has appeared in Figure 1.

4. Methodology

In the dataset, 5907 images are utilized to investigate the examination, where the dataset has a few arrangements of images, for example, Normal, SARS, COVID-19, ARDS and Streptococcus. Be that as it may, in this analysis our primary goal to identify COVID cases. To identify all the more proficiently, in this manner author is going to build a model which separates between normal case, Pneumonia and COVID-19 cases. Author additionally lessen the classification number to two. Later on, author applied different pre-trained deep learning models, for instance, Base convolutional neural network, DenseNet-121, ResNet-50 and EfficientNet-B4 to distinguish COVID-19 cases and to locate the best exact outcome as indicated by the individual exhibitions.

4.1. Data augmentation

Data augmentation is a strategy that enables practitioners to significantly increase the diversity of data available for training models, without actually collecting new data. Data augmentation techniques such as cropping, padding, and horizontal flipping are commonly used to train large neural networks.

4.2. Base convolutional neural network

Convolutional neural networks are practically equivalent to conventional artificial neural networks in that they are included neurons that self-streamline through learning [17]. It fundamentally center around the premise that the info will be involved images. This centers the engineering to be set up in a manner to best suit the requirement for managing the particular sort of data. There are a few functionalities to explain convolutional neural network more briefly. As found in various kinds of artificial neural network, the information layer will hold the pixel estimations of the image. Convolutional layers will choose the yield of neurons of which are related to close-by regions of the commitment through the check of the scalar the thing between their heaps and the area related to the data volume.

Take where, \(X\) is the representation of input volume size (height \(x\) weight \(x\) depth), \(Y\) stands for the receptive field size, \(A\) stands for the size of zero padding and \(K\) stands for stride.

The reviewed straight unit hopes to apply an elementwise inception capacity or initiation work, for instance, sigmoid to the yield of the sanctioning made by the past layer. Pooling layers will by then simply perform down-testing along the spatial the dimensionality of the given information, further reducing the amount of limits inside that incitation. completely associated layers will by then play out comparative commitments found in standard artificial neural networks and try to convey class scores from the institutions, to be used for portrayal. It is moreover suggested that ReLU may be used between these layers, as to improve execution. Zero-padding is the basic procedure of cushioning the outskirt of the information and is a compelling strategy to give further control with regards to the dimensionality of the yield volumes. Boundary sharing chips away at the supposition that in the event that one area highlight is helpful to register at a set spatial area, at that point, it is probably going to be valuable in another locale. In the event that we compel every individual initiation map inside the yield volume to similar loads and predisposition, at that point we will see a huge decrease in the quantity of boundaries being created by the convolutional layer [17].

4.3. Densely connected convolutional network

Densely connected convolutional network (DenseNet) interfaces each layer to each other layer in a feed-forward design [18], while convolutional neural systems with \(N\) layers have \(N\) associations, one between each layer and its ensuing layer. DenseNets have a few convincing points of interest: they lighten the vanishing gradient issue, fortify element spread, empower include reuse, and significantly diminish the quantity of boundaries. It has better accuracy than ResNet in object recognition [18]. DenseNets are worked from thick squares and pooling activities, where each thick square is an iterative connection of past element maps. This design can be viewed as an augmentation of ResNets [19], which performs an iterative summation of past component maps. In any case, this little change makes them intrigue suggestions such as, boundary proficiency, DenseNets are more efficient in the boundary use. Understood profound oversight, DenseNets perform profound management on account of short ways to all component maps in the design and highlight reuse, all layers can without much of a stretch access their first layers making it simple to reuse the data from recently figured element maps. The attributes of DenseNets make them an awesome fit for the semantic division as they normally actuate skip associations and multi-scale management. Fully connected DenseNets are worked from a downsampling way, an upsampling way and skip associations. Skip associations help the upsampling way recoup spatially point by point data from the downsampling way, by reusing highlights maps. The objective of our model is to additionally misuse the component reuse by broadening the more refined DenseNet engineering while at the same time maintaining a strategic distance from the element blast at the upsampling way of the system [18]. In this investigation, author has utilized pre-trained DenseNet-121 design to actualize in our dataset.

4.4. Residual network

Residual network has been created and acquainted by Microsoft Research with handle image recognition all the more without any hurdle. ResNet has about 3.57% less error than VGGNet [19]. It has around 152 layers top to bottom, which is eight multiple times higher than VGGNet design. Its architecture has been inspired by VGGNets architecture. We are meaning the mapping as \(H(a)\), another non-direct mapping can be communicated as \(F(a) = H(a)- a\), the primary mapping can be communicated as \(F(a) + a\). We receive lingering figuring out how to each couple of stacked layers. The structure square can be defined as [19]: where, \(a\) and \(b\) can be considered as the vectors. The function \(F(a ,{Wi})\) is considered as figuring out how to outline. The elements of \(a\) and \(F\) must be the equivalent. On the off chance that we can not figure out how to do as such, we can include a projection vector into the detailing. The projection must be straight and, for example, \(W_s\). Yet, we will appear by tests that the character mapping is adequate for tending to the debasement issue and is efficient, and accordingly, \(W_s\) is possibly utilized when coordinating measurements. It has two plan rule and scarcely followed by VGGNet. The principles are: it has multiple times 3 convolutional layers, for a similar yield highlight map size, the layers have a similar number of channels and if the component map size is split, the quantity of channels is multiplied in order to save the time multifaceted nature per layer [19]. The downsampling stride must be 2. In this analysis, author utilized pre-trained ResNet-50 architecture.

4.5. EficientNet

EfficientNet has been created and presented by Mingxing Tan, staff software engineer at Google. EfficientNet is a systematical model scaling and distinguishes that cautiously adjusting system profundity, width, and goals can prompt better performance [20]. It is propelled by ResNet and MobileNet, and scaling up or down to legitimize better exactness. This is a compound demonstrating framework. There are numerous approaches to scale a ConvNet for various asset limitations: ResNet [19] can be downsized for example ResNet-18 or up e.g., ResNet-200 by altering system profundity or the quantities of layers. A convolutional neural layer can be detailed as \(B_i\) = \(F_i(A_i)\), where, \(B_i\) is the yield tensor, \(A_i\) is the information tensor and \(F_i\) is the employable capacity. Tensor shape \((X_i, Y_i, Z_i)\) where \(X_i\) and \(Y_i\) are spatial measurements and \(Z_i\) is the channel measurement. There are three significant boundaries to consider for scaling reason, for example, profundity, width and goals. Scaling system profundity is the most well-known way utilized by numerous convolutional systems. The instinct is that more profound convolutional systems can catch more extravagant and progressively complex highlights, and sum up well on new errands. Notwithstanding, more profound systems are additionally progressively hard to prepare because of the disappearing inclination issue. The genuine errand of this model is to scaling the profundity, width and goals all the more productively to change the assignment prerequisite, goals with a lot of fixed scaling coefficients. For instance, on the off chance that we need to utilize \(2^n\) times progressively computational assets, at that point we can just build the system profundity by \(\alpha^n\), width by \(\beta^n\), and image size by \(\gamma^n\), where \(\alpha, \beta, \gamma\) are consistent coefficients controlled by a little lattice search on the first little model. In this analysis, author has utilized pre-trained EfficientNet-B4 model for COVID-19 for detecting purpose.

4.6. Activation function

Actuation functions are numerical conditions that decide the yield of a neural system. The capacity is appended to every neuron in the system and decides if it ought to be actuated or not, founded on whether every neuron's input is significant for the model's expected output. In this investigation, we utilized two individual activation functions, for example, Sigmoid [21] and ReLU [22] activation function. We utilized Sigmoid in Base CNN model and ReLU in rest of different models as initiation work.

• Sigmoid: The sigmoid activation function is here and there alluded to as the strategic capacity or crushing capacity in some literatures [21]. The Sigmoid is a non-direct enactment work utilized for the most part in feedforward neural systems.
  \begin{equation} \text{Sigmoid} = [\frac{1}{(1+ exp ^-x)}]^-1.4\,. \end{equation}

  (3)
  The sigmoid capacity shows up in the yield layers of the deep learning structures, and they are utilized for anticipating likelihood-based yield and has been applied effectively in binary characterization issues, demonstrating strategic relapse undertakings just as other neural system areas [21].
• ReLU: Rectified Linear Units (ReLU) as the arrangement work in a deep neural system [22]. Customarily, ReLU is utilized as an actuation work in deep neural systems, with Softmax work as their arrangement work. It works by thresholding values at 0, model \(f (a) = max(0, a)\). Basically, it yields 0 when \(a < 0\), and then again, it yields a straight capacity when \(a 0\). ReLU is not just as an activation function in each concealed layer of a neural system yet additionally as the grouping capacity at the last layer of a system. Thus, the anticipated class for ReLU classifier would be \(\hat{b}\) [22],
  \begin{equation} \hat{b} = \text{arg max}_{i=1,.....,N} (0,0). \end{equation}

  (4)

4.7. Optimizer

The optimizer is a robust algorithm that helps to reduce the loss of a deep neural system by changing some attributes such as learning rate and changing weight and enhance the overall performance of the system. An optimizer can improve the performance of a neural system. It is essential to use an optimizer to reduce loss functions. In this experiment, author used two extensively used optimizers such as Stochastic Gradient Descent and Adam. We used Adam in EfficientNet-B4 algorithm and ResNet-50, on the other hand, author used Stochastic Gradient Descent (SGD) in DenseNet-121 and Base CNN model.

• Stochastic gradient descent: Stochastic Gradient Descent is widely utilized optimizer, much of the time, it is utilized in traditional CNN model to streamline [23]. It is an updated form of Batch SGD. SGD gets rid of this repetition by performing each update in turn. It is subsequently generally a lot quicker and can likewise be utilized to learn on the web. SGD performs visit refreshes with a high change that prompt the target capacity to change intensely.
• Adam: Adam is an extensively utilized optimizer, which is 1st order gradient-based optimizer. It is a strategy for proficient stochastic advancement that just requires first-request slopes with little memory prerequisite. The technique registers individual versatile taking in rates for various boundaries from evaluations of first and second snapshots of the angles, the name Adam is gotten from versatile second estimation [24]. This can be formulated as:
  \begin{equation} x_1 = (1 - \beta_1) \sum_{i=0}^t \beta_1^{t-1} y_1. \end{equation}

  (5)

5. Performance matrix

We evaluated our models by AUC, accuracy, precision, specificity and sensitivity. Here, \(XP\) and \(XN\) denote as true positive and true negative, \(YP\) and \(YN\) denote as false positive and false negative respectively.

6. Results and discussions

In this analysis, author utilized a few convolutional neural system models to empower better outcome. Author executed EfficientNet-B4, ResNet-50, DenseNet-121 and Base convolutional neural Network model to identify COVID-19 cases all the more proficiently. There are a few works have been done before in an exceptionally brief timeframe to handle this obstacle by colossal scientists. Different specialists have created gathering algorithms for identifying COVID-19 cases.

In this examination, EfficientNet-B4 has performed better. It has 98.86% accuracy and 0.996 AUC. Different models, for example, ResNet-50, DenseNet-121 and Base CNN have additionally performed well. Author set the epochs to 20 in every examination. Be that as it may, Base CNN has the most reduced accuracy of 84.50% where ResNet-50 has 97.31% and DenseNet-121 has an accuracy of 96.50%. Author executed the sigmoid activation function in Base CNN model and ReLU activation function in the remainder of the models. Then again, Author utilized stochastic gradient descent (SGD) optimizer in Base CNN model and DenseNet-121 model. Author also likewise utilized Adam optimizer in EfficientNet-B4 and ResNet-50 model. Table 1 has demonstrated the presentation examination of each algorithm. Figure 5, has shown performances of every algorithm with AUC, training loss, validation loss and validation AUC, and Figure 6, we have shown EfficientNet-B4's model accuracy by increasing epochs and model loss in order to increase epochs.

Table 1. Performance analysis of every algorithm (%).

Algorithm	AUC	Accuracy	Specificity	Sensitivity
Base CNN Model	0.762	84.50	81.43	88.29
DenseNet-121	0.874	96.50	92.66	93.28
ResNet-50	0.967	97.31	97.78	96.12
EfficientNet-B4	0.997	98.87	99.46	98.77

Finally, because of excellent outcome in chest X-Ray images, we propose and concoct the choice that EfficientNet-B4 model can be utilized for additional identifying limit with regards to clinical network to handle this hardest time for the world.

7. Conclusion

Conclusion of COVID-19 is basic to follow the influenced individuals and limit the transmission as it is a viral disease. RT-PCR method is costly and needs more time to detect COVID-19 cases. These days medical images preparing is one of the fundamental assignments for the scientists to foresee or recognize infections all the more productively. So as referenced, we attempted to handle COVID-19 discovery issue by utilizing medical images, all the more explicitly chest X-Ray images. In this investigation, author has actualized a few pre-tranied deep convolutional neural networks, for example, Base CNN, DenseNet-121, ResNet-50 and EfficientNet-B4 in chest X-Ray dataset. Convolutional neural systems can think of an effective and powerful result than any conceivable way. In this investigation, EfficientNet-B4 has performed superior to different models with an accuracy of 98.86% just as containing higher AUC of 0.996. Then again, ResNet-50 has likewise performed well with accuracy and AUC of 97.31% and 0.967 separately. Different models, for example, DenseNet-121 and Base CNN have an accuracy of 96.50% and 84.50%. So in rundown, author would recommend and proposed EfficientNet-B4 for additional headway of distinguishing COVID-19 cases by utilizing X-Ray images. For future improvement, a more sophisticated dataset is needed with more amount of images to train our model for better outcomes.

Conflicts of interest

The author declares no conflict of interest.

1. Introduction

The classical calculus of derivatives and integrals which involves integer orders is extended with fractional orders that belong to the real numbers. In last few decades, the fractional calculus theory receives more attention due to its significant applications in several scopes such as physics, fluid dynamics, computer networking, image processing, biology, signal processing, control theory and other scopes. Because of the importance of fractional calculus, many researchers have shown their intense interest. One of the prevalent approaches among researchers is the use of fractional derivatives and integral operators. As a consequence, several distinct kinds of fractional integrals and derivatives operators have been realized, such as the Liouville, Riemann-Liouville, Katugampola, Weyl types, Hadamard and some other types which can be found in Kilbas et al., [1].

Hilfer [2] in (2000), through his contribution to improve the fractional calculus, established a new fractional derivative operator for any real order \(\delta \), which gives the Caputo derivative and the Riemann-Liouville fractional operator. The primary concept and properties and more information of \(\psi \)-Riemann-Liouville fractional derivative and integral can be found in [1]. In (2017), Almeida [3], introduced \(% \psi \)-Caputo fractional derivative and investigated its significant properties. Recently, in 2018, Sousa and Oliveira [4], introduced a generalization of many existing fractional derivative operators called \(\psi \)-Hilfer derivative.

The mathematical inequalities play a very reliable role in classical integral and differential equations as well as in the past few years, many of useful mathematical inequalities have been originated by many authors, see [5,6,7,8]. One of the most significant integral inequalities is that discovered by Hermite [9] and Hadamard [10] for convex function \(f\) as follows

If \(f\) is a concave function then both inequalities in (1) are held in a reversed direction. For some historical of Hermite-Hadamard inequalities [11] and the references therein. In the last few decades, these inequalities have been received a considerable attention by many authors and several articles have appeared in the literature, see [12,13,14]. In 2010, Dahmani [15], studied the Hermite-Hadamard type inequalities for concave functions by means of Riemann-Liouville fractional integral. Sarikaya et al., in 2013 [16], gave the Hermite-Hadamard type inequalities for convex function using Riemann-Liouville fractional integral. In 2014, Set et al., established Hermite-Hadamard type inequalities for s-convex functions in the second sense proved by Dragomir et al., [17] and \(m\)-convex functions via fractional integrals. In (2015), Noor et al., [18], derived some quantum estimates for Hermite-Hadamard inequalities for \(q\)-differentiable quasi convex functions and \(q\)-differentiable convex functions. Liu et al., [19] in 2016, introduced some inequalities of Hermite-Hadamard type for MT-convex functions using classical integrals and Riemann-Liouville fractional integrals. In 2017, Agarwal et al., [20], obtained some Hermite-Hadamard type inequalities for convex functions via \((k,s)-\)Riemann-Liouville fractional integrals. Muhammad A. Khan [21] in 2018, proved new Hermite-Hadamard inequalities for convex function, \(s-\)convex and coordinate convex functions by using conformable fractional integrals. Recently in 2019, a lots of researchers studied Hermite-Hadamard inequalities for several kinds of convexity of the functions, for more details we refer the readers to see [22,23]. Very recently, in 2020, Chudziak and Ołdak introduced notion of a co-ordinated \((F,G)\)-convex function defined on an interval in \( \mathbb{R} ^{2}.\)

The main objective of this paper is to establish some new fractional integral Hermite-Hadamard inequalities for concave functions by using \(\psi - \)Riemann-Liouville fractional integral operator. Moreover, we introduce some new fractional integral inequalities related to the Hermite-Hadamard inequalities via \(\psi -\)Riemann-Liouville fractional integral operator. The paper is organized as follows: In Section 2, we collect some notations, definitions, results and preliminary facts which are used throughout this paper. In Section 3, we present the reverse Hermite-Hadamard's inequalities for concave functions. In Section 4, we give some other related results of Hermite-Hadamard type inequalities which involving \(\psi -\)Riemann-Liouville fractional integral operator.

2. Basic definitions and tools

This section is dedicated for some basic definitions and properties of fractional integrals used to obtain and discuss our new results. We also outline some basic results related to this work.

Let \(\delta >0,\) \(m\in \mathbb{N} ,\) with \(\Upsilon =\left[ a,b\right] \) \(\left( -\infty \leq a< t< b\leq \infty \right) ,\) be a finite or infinite interval. Assume that \(f\) be an integrable function defined on \(\Upsilon \) and \(\psi :\Upsilon \rightarrow \mathbb{R} \) be an increasing function for all \(t\in \Upsilon ,\) which belong to \( C^{1}\left( \Upsilon , \mathbb{R} \right) \) with condition that \(\psi ^{\prime }\left( t\right) \) must be nonzero along the interval \(\Upsilon \). The \(\psi -\)Riemann-Liouville fractional derivative of order \(\delta \) of a function \(f\) are defined by [1,24]:

and

Definition 1. Let \(\delta >0\) and \(f\) be an integrable function defined on \(\Upsilon \) and \(\psi \left( t\right) \in C^{1}\left( \Upsilon , \mathbb{R} \right) \) be an increasing function such that \(\psi ^{\prime }\left( t\right) \neq 0\) for all \(t\in \Upsilon .\) The left and right \(\psi -\) Riemann-Liouville fractional integral of order \(\delta \) with respect to the function \(\psi \) of a function \(f\) are respectively defined by [1,24]:

and

Lemma 1. [1] Let \(\delta >0\) and \(\mu >0.\) If \(f\left( t\right) =\left[ \psi \left( t\right) -\psi \left( \xi \right) \right] ^{\mu -1},\) then \begin{equation*} \mathcal{I}_{a^{+}}^{\delta ;\psi }f\left( t\right) =\frac{\Gamma \left( \mu \right) }{\Gamma \left( \delta +\mu \right) }\left[ \psi \left( t\right) -\psi \left( \xi \right) \right] ^{\delta +\mu -1}. \end{equation*}

Definition 2. The function \(f\) \(:\left( \Lambda \subseteq %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion \right) \longrightarrow %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion \) is said to be concave function if the following inequality holds

for all \(x,y\in \Lambda \) and \(\lambda \in \left[ 0,1\right] .\) We say that \( f\) is convex if the inequality (6) is reversed.

Theorem 3. [16] Let \(f:\left[ a,b\right] \longrightarrow %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion \) be a convex positive function on \(\left[ a,b\right] \) with \(0\leq a< b,\) then for all \(\delta >0,\) the following inequality holds:

Lemma 2. [15] Let \(h:\left[ a,b\right] \longrightarrow %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion \) be a concave. Then the following inequality holds: \begin{equation*} h\left( a\right) +h\left( b\right) \leq h\left[ a+b-t\right] +h\left( t\right) \leq 2h\left( \frac{a+b}{2}\right) . \end{equation*}

Theorem 4. [15] Let \(f\) and \(g\) be two positive functions on \(\left[ 0,\infty \right) \). If \(f\) and \(g\) are a concave functions on \(\left[ 0,\infty \right) \), then for all \(p>1\), \(q>1\) and \(\delta >0,\) the following inequality holds: \begin{eqnarray*} 2^{-p-q}\left[ f\left( 0\right) +f\left( x\right) \right] ^{p}\left[ g\left( 0\right) +g\left( x\right) \right] ^{q}\left( \mathcal{I}^{\delta }x^{\delta -1}\right) ^{2} \left. \leq \right. \mathcal{I}^{\delta }\left[ x^{\delta -1}f^{p}\left( x\right) \right] \mathcal{I}^{\delta }\left( x^{\delta -1}g^{q}\left( x\right) \right) . \end{eqnarray*}

Theorem 5. [15] Let \(f\) and \(g\) be two positive functions on \(\left[ 0,\infty \right) \). If \(f\) and \(g\) are a concave functions on \(\left[ 0,\infty \right) \), then for all \(p>1\), \(q>1\) and \(\delta >0,\) \(\sigma >0\) the following inequality holds: \begin{eqnarray*} &&2^{2-p-q}\left[ f\left( 0\right) +f\left( x\right) \right] ^{p}\left[ g\left( 0\right) +g\left( x\right) \right] ^{q}\left( \mathcal{I}\left( ^{\delta }x^{\sigma -1}\right) \right) ^{2} \\ &&\left. \leq \right. \frac{\Gamma \left( \sigma \right) }{\Gamma \left( \delta \right) }\mathcal{I}^{\sigma }\left[ x^{\sigma -1}f^{p}\left( x\right) \right] +\mathcal{I}^{\delta }\left[ x^{\sigma -1}f^{p}\left( x\right) \right] \frac{\Gamma \left( \sigma \right) }{\Gamma \left( \delta \right) }\mathcal{I}^{\sigma }\left[ x^{\sigma -1}g^{q}\left( x\right) \right] +\mathcal{I}^{\delta }\left[ x^{\sigma -1}g^{q}\left( x\right) \right] . \end{eqnarray*}

3. The reverse Hermite-Hadamard's inequalities for fractional integral

Now, we give the reverse Hermite-Hadamard's inequalities involving concave functions for \(\psi \)-Riemann-Liouville fractional integral operators.

Theorem 6. Let \(\psi :\left[ a,b\right] \longrightarrow \Lambda \subseteq %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion \) with \(0\leq a< b\) be an increasing and bijective function having a continuous derivative \(\psi ^{\prime }\left( x\right) \neq 0\,\) \(\forall \) \(% x\in \left[ a,b\right] ,\) \(\psi \left( 0\right) =0,\) \(\psi \left( 1\right) =1\) and \(f:\Lambda \longrightarrow %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion \) be an increasing and differentiable function on \(\Lambda ^{\circ }\) such that \(\left( f\circ \psi \right) :\left[ a,b\right] \longrightarrow %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion \) be an integrable mapping on \(\left[ a,b\right] \). If \(\psi \) and \(f\) are concave and positive functions, then the following inequality holds:

Proof. For any \(x,y\in \left[ a,b\right] ,\) using the concavity of \(f\) and \(\psi ,\) we have

Putting \(\lambda =\frac{1}{2}\) and using (9), we can write Let and where \(x,y\) are variables containing \(t.\) By substituting (11) and (12) in (10), we get Now, multiplying both sides of (13) by \(\psi ^{\prime }\left( t\right) \psi ^{\delta -1}\left( t\right) ,\) then integrating the resulting inequality with respect to \(t\) over \(\left[ 0,1\right] ,\) we obtain From (11), we have \begin{equation*} \frac{d}{dt}\psi \left( x\right) =\frac{d}{dt}\left[ \psi \left( t\right) \psi \left( a\right) +\left[ 1-\psi \left( t\right) \right] \psi \left( b\right) \right] \Longrightarrow \frac{\psi ^{\prime }\left( x\right) dx}{% \psi \left( a\right) -\psi \left( b\right) }=\psi ^{\prime }\left( t\right) dt, \end{equation*} and \begin{equation*} \psi \left( t\right) =\frac{\psi \left( x\right) -\psi \left( b\right) }{% \psi \left( a\right) -\psi \left( b\right) }. \end{equation*} So, we have Also from (12), we have \begin{equation*} \frac{d}{dt}\psi \left( y\right) =\frac{d}{dt}\left[ \left[ 1-\psi \left( t\right) \right] \psi \left( a\right) +\psi \left( t\right) \psi \left( b\right) \right] \Longrightarrow \frac{\psi ^{\prime }\left( y\right) dy}{% \left[ \psi \left( b\right) -\psi \left( a\right) \right] }=\psi ^{\prime }\left( t\right) dt, \end{equation*} and \begin{equation*} \psi \left( t\right) =\frac{\psi \left( y\right) -\psi \left( a\right) }{% \psi \left( b\right) -\psi \left( a\right) }. \end{equation*} So, we have Using (14), (15) and (16), we get which is the first inequality in (8). To prove the second inequality and using the concavity of \(f\) and \(\psi \), we can write for \( \lambda \in \left[ 0,1\right]\) and Adding (18) and (19), we obtain Now, multiplying both sides of (20) by \(\psi ^{\prime }\left( t\right) \psi ^{\delta -1}\left( t\right) ,\) then integrating the resulting inequality with respect to \(t\) over \(\left[ 0,1\right] ,\) we obtain Hence, by combining the inequalities (17) and (20), we get the desired inequality (8).

Remark 1.

• (i) If we put \(\psi \left( x\right) =x,\) \(\forall x\in \left[ a,b\right] ,\) for \(f\) a convex function, then both inequalities (8) reversed and Theorem 6 reduce to Theorem 3 obtained by Sarikaya et al., in [16] for classical Riemann-Liouville fractional integral.
• (ii) Applying inequalities (8) for \(\psi \left( x\right) =x\), \(% \forall x\in \left[ a,b\right] \) and \(\delta =1,\) for \(f\) be a convex function, we obtain the classical Hermite-Hadamard inequalities (1).

4. Hermite-Hadamard type inequalities for fractional integral

In this section, we generalize some Hermite-Hadamard type inequalities involving concave functions introduced by Dahmani [15] using the Riemann-Liouville fractional integral with respect to other monotone and bijective function. In present part, we use only the left-sided fractional integrals (4). Moreover, we consider \(a=0\) to obtain and discuss our results. We first prove the following lemma:

Lemma 3. Let \(\psi :\left[ 0,\infty \right) \longrightarrow \Lambda \) be an increasing and bijective function having a continuous derivative \(\psi ^{\prime }\left( t\right) \neq 0\,\) \(\forall \) \(t\in \left[ 0,\infty \right) , \) \(\psi \left( 0\right) =0,\) \(\psi \left( 1\right) =1\) and \(h:\Lambda \longrightarrow %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion \) be an increasing and differentiable function on \(\Lambda ^{\circ }\) such that \(\left( h\circ \psi \right) :\left[ 0,\infty \right) \longrightarrow %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion \) be an integrable mapping on \(\left[ 0,\infty \right) \). If \(h\) is a concave functions on \(\Lambda \), then we have

Proof. Since \(h\) be a concave function on \(\Lambda ,\) so for any \(c,d\in \left[ 0,\infty \right) ,\) we can write

If we choose \(\psi \left( t\right) =\lambda \psi \left( c\right) +\left[ 1-\lambda \right] \psi \left( d\right) ,\) then we have \begin{eqnarray*} &&\frac{1}{2}\left[ h\left[ \psi \left( c\right) +\psi \left( d\right) -\lambda \psi \left( c\right) -\left[ 1-\lambda \right] \psi \left( d\right) % \right] +h\left[ \lambda \psi \left( c\right) +\left[ 1-\lambda \right] \psi \left( d\right) \right] \right] \\ &&\left. =\frac{1}{2}\left[ h\left[ \lambda \psi \left( d\right) +\left[ 1-\lambda \right] \psi \left( c\right) \right] +h\left[ \lambda \psi \left( c\right) +\left[ 1-\lambda \right] \psi \left( d\right) \right] \right] \right. . \end{eqnarray*} Using the concavity of \(h\), we obtain By (23) and (24), we get \begin{eqnarray*} \left( h\circ \psi \right) \left( c\right) +\left( h\circ \psi \right) \left( d\right) \leq h\left[ \psi \left( c\right) +\psi \left( d\right) -\psi \left( t\right) \right] +\left( h\circ \psi \right) \left( t\right) \leq 2h\left( \frac{\psi \left( c\right) +\psi \left( d\right) }{2}\right) , \end{eqnarray*} which is the required inequality (22).

Theorem 7. Let \(\psi :\left[ 0,\infty \right) \longrightarrow \Lambda \) be an increasing positive and bijective function having a continuous derivative \(\psi ^{\prime }\left( x\right) \neq 0\,\) \(\forall \) \(x\in \left[ 0,\infty \right) ,\) \(\psi \left( 0\right) =0,\) \(\psi \left( 1\right) =1\) and \(% f,g:\Lambda \longrightarrow %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion \) be an increasing and differentiable functions on \(\Lambda ^{\circ }\) such that \(\left( f\circ \psi \right) ,\left( g\circ \psi \right) :\left[ 0,\infty \right) \longrightarrow %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion \) are two integrable mappings on \(\left[ 0,\infty \right) \). If \(f\) and \(g\) are a concave functions on \(\Lambda \). Then for all \(p>1\), \(q>1\) and \(\delta >0,\) the following inequality holds:

Proof. Since \(f^{p}\) and \(g^{q}\) are a concave functions on \(\Lambda ,\) so by Lemma (3), for any \(x,y>0,\) we have

and Multiplying both sides of (26) and (27) by \(\frac{\psi ^{\prime }\left( y\right) }{\Gamma \left( \delta \right) }\left[ \psi \left( x\right) -\psi \left( y\right) \right] ^{\delta -1}\psi ^{\delta -1}\left( y\right) ,\) \(y\in \left( 0,x\right) \) and integrating the resulting inequalities with respect to \(y\) over \(\left( 0,x\right) ,\) we obtain and Using the change of variable \(\psi \left( u\right) =\psi \left( x\right) -\psi \left( y\right) ,\) where \(u\in \left[ 0,\infty \right) \) is a variable containing \(y,\) we have \begin{equation*} \frac{d}{dy}\left[ \psi \left( u\right) \right] =\frac{d}{dy}\left[ \psi \left( x\right) -\psi \left( y\right) \right] \Longrightarrow \psi ^{\prime }\left( u\right) du=-\psi ^{\prime }\left( y\right) dy. \end{equation*} Then, we can write and Now, by using (28) and (30), we get and using (29) and (31), we get The inequalities (32) and (33) yields On the other hand, we have \(f\) and \(g\) are positive functions and \(\psi \) is increasing function on \(\left[ 0,\infty \right) \). Then for any \(x>0,\) \( p\geq 1,\) \(q\geq 1\), we can write and Multiplying both sides of (35) and (36) by \(\frac{\psi ^{\prime }\left( y\right) }{\Gamma \left( \delta \right) }\left[ \psi \left( x\right) -\psi \left( y\right) \right] ^{\delta -1}\psi ^{\delta -1}\left( y\right) ,\) \(y\in \left( 0,x\right) \), then integrating the resulting inequalities with respect to \(y\) over \(\left( 0,x\right) ,\) we get and The inequalities (37) and (38) yields Combining the inequalities (34) and (39), we obtain the desired inequality (25).

Remark 2.

• (i) If we put \(\psi \left( x\right) =x\) for all \(x\in \left[ 0,\infty \right) ,\) then Lemma 3 reduce to Lemma 2 and Theorem 7 reduce to Theorem 4 obtained by Dahmani in [15].
• (ii) Applying Theorem 7 for \(\psi \left( x\right) =x\) for all \(x\in \left[ 0,\infty \right) ,\) \(\delta =1,\) we obtain Theorem 5 obtained by Set et al., in [25].

No, we give the following version of Theorem 7 with two parameters for \(\psi \)-Riemann-Liouville fractional integral operator.

Theorem 8. Let \(\psi :\left[ 0,\infty \right) \longrightarrow \Lambda \) be an increasing positive and bijective function having a continuous derivative \(\psi ^{\prime }\left( x\right) \neq 0\,\) \(\forall \) \(x\in \left[ 0,\infty \right) ,\) \(\psi \left( 0\right) =0,\) \(\psi \left( 1\right) =1\) and \(% f,g:\Lambda \longrightarrow %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion \) be an increasing and differentiable functions on \(\Lambda ^{\circ }\) such that \(\left( f\circ \psi \right) ,\left( g\circ \psi \right) :\left[ 0,\infty \right) \longrightarrow %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion \) are two integrable mappings on \(\left[ 0,\infty \right) \). If \(f\) and \(g\) are a concave functions on \(\Lambda \). Then for all \(p>1\), \(q>1\) and \(\delta >0,\) \(\sigma >0,\) the following inequality holds:

Proof. By using Lemma 3 and as \(f^{p}\) and \(g^{q}\) are concave functions on \(\Lambda ,\) then we have for any \(x,y>0\)

and Now, multiplying both sides of (41) and (42) by \(\frac{\psi ^{\prime }\left( y\right) }{\Gamma \left( \delta \right) }\left[ \psi \left( x\right) -\psi \left( y\right) \right] ^{\delta -1}\psi ^{\sigma -1}\left( y\right) ,\) \(y\in \left( 0,x\right) \), then integrating the resulting inequalities with respect to \(y\) over \(\left( 0,x\right) ,\) we obtain and Using the change of variable \(\psi \left( u\right) =\psi \left( x\right) -\psi \left( y\right) ,\) where \(u\in \left[ 0,\infty \right) \) is a variable containing \(y,\) we have and By using (43) and (45), we obtain and using (44) and (46), we get The inequalities (47) and (48) imply that Similarly as before, we have \(f\) and \(g\) are positive functions and \(\psi \) is increasing function on \(\left[ 0,\infty \right) \). Then for any \(x>0,\) \(% p\geq 1,\) \(q\geq 1\), we can write and The inequalities (50) and (51) imply that Combining the inequalities (49) and (52), we obtain the desired inequality (40).

Remark 3. If we put \(\psi \left( x\right) =x\) for all \(x\in \left[ 0,\infty \right) ,\) then Theorem 7 reduce to Theorem 5 obtained by Dahmani in [15].

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflicts of interest

The authors declare no conflict of interest.

1. Introduction and Preliminaries

For a convex mapping \(\prod:I\rightarrow \mathbb{R}\) on a real interval, for all \(f_1,f_2\in I\) and \(t\in[0,1]\), the inequality

is known as the Hermite-Hadamard inequality [1]. The inequality (1) has been established for several generalized convex functions [2,3,4,5,6,7,8,9]. Dragomir [10] and Sarikaya [11] calculated Hermite-Hadamard inequality for co-ordinated convex functions. They define co-ordinated convex function as:

Definition 1. [10] A function \(\prod:\Delta=[f_1,f_2]\times [g_1,g_2]\subseteq \mathbb{R}^{2} \rightarrow \mathbb{R}\) is called co-ordinate convex on \(\Delta\) with \(f_1 < f_2\) and \(g_1 < g_2\), if the partial functions \begin{equation} \prod_{y}:[f_1,f_2] \rightarrow \mathbb{R}, \prod_{y}(u)=\prod(u,y), and \prod_{x}:[g_1,g_2] \rightarrow \mathbb{R}, \prod_{x}(v)=\prod(x,v), \end{equation} are convex for all \(x\in[f_1,f_2]\) and \(y\in [g_1,g_2]\).

Sarikaya [11] define the co-ordinated convex function as:

Definition 2. [11] A function \(\prod:\Delta=[f_1,f_2]\times [g_1,g_2]\subseteq \mathbb{R}^{2}\rightarrow \mathbb{R}\) is called coordinate convex on \(\Delta\) with \(f_1< f_2\) and \(g_1< g_2\), if \begin{align*} \begin{split} &\prod(t_1x+(1-t_1)z,t_2 y+(1-t_2)w) \\ &\leq t_1t_2\prod(x,y)+t_1(1-t_2)\prod(x,w)+(1-t_1)t_2 \prod(z,y)+(1-t_1)(1-t_2)\prod(z,w), \end{split} \end{align*} holds for all \(t_1,t_2\in [0,1]\) and \((x,y),(z,w)\in\Delta\).

Every convex function is co-ordinated convex but not conversely [10].

Theorem 3. [10] Let \(\prod:\Delta=[f_1,f_2]\times [g_1,g_2]\subseteq \mathbb{R}^{2} \rightarrow \mathbb{R}\) be convex on \(\Delta\) with \(f_1< f_2\) and \(g_1< g_2\). Then

Definition 4. [12] A function \(\prod:\Delta=[f_1,f_2]\times [g_1,g_2]\subseteq \mathbb{R}^{2} \rightarrow \mathbb{R}\) is called harmonically convex on \(\Delta\) with \(f_1< f_2\) and \(g_1< g_2\), if \begin{equation*} \prod\left(\frac{xz}{t_1x+(1-t_1)z},\frac{yw}{t_2 y+(1-t_2)w}\right) \leq t_1t_2 \prod(x,y)+(1-t_1)(1-t_2)\prod(z,w), \end{equation*} holds for all \(t_1,t_2\in [0,1]\) and \((x,y),(z,w)\in\Delta\).

Definition 5. [12] A function \(\prod:\Delta=[f_1,f_2]\times [g_1,g_2]\subseteq (0,\infty)\times (0,\infty) \rightarrow \mathbb{R}\) is called coordinated harmonically convex on \(\Delta\) with \(f_1< f_2\) and \(g_1< g_2\), if \begin{align*} \begin{split} &\prod\left(\frac{xz}{t_1x+(1-t_1)z},\frac{yw}{t_2 y+(1-t_2)w}\right) \\ &\leq t_1t_2 \prod(x,y)+t_1(1-t_2)\prod(x,w)+(1-t_1)t_2 \prod(z,y)+(1-t_1)(1-t_2)\prod(z,w), \end{split} \end{align*} holds for all \(t_1,t_2\in [0,1]\) and \((x,y),(z,w)\in\Delta\).

Note that, a function \(\prod:\Delta=[f_1,f_2]\times [g_1,g_2]\subseteq (0,\infty)\times (0,\infty) \rightarrow \mathbb{R}\) is called coordinated harmonically convex on \(\Delta\) with \(f_1< f_2\) and \(g_1< g_2\), if the partial functions

\begin{equation} \prod_{y}:[f_1,f_2] \rightarrow \mathbb{R}, \prod_{y}(u)=\prod(u,y), \prod_{x}:[g_1,g_2] \rightarrow \mathbb{R}, \prod_{x}(v)=\prod(x,v), \end{equation} are harmonically convex for all \(x\in[f_1,f_2]\) and \(y\in [g_1,g_2]\), (for more detail, see [9,12]).

Theorem 6. [12] Let \(\prod:\Delta=[f_1,f_2]\times [g_1,g_2]\subseteq (0,\infty)\times (0,\infty) \rightarrow \mathbb{R}\) be co-ordinated harmonically convex on \(\Delta\) with \(f_1< f_2\) and \(g_1< g_2\). Then

Definition 7. [13] Let \(\prod \in L[f_1,f_2]\). The right-hand side and left-hand side Riemann- Liouville fractional integrals \(J^{\alpha}_{f_1+}\prod\) and \(J^{\alpha}_{f_2-}\prod\) of order \(\alpha > 0\) with \(f_2 > f_1\geq 0\) are defined by \begin{equation*} J^{\alpha}_{f_1+}\prod(x)=\frac{1}{\Gamma(\alpha)}\int_{f_1}^{x}(x-t)^{\alpha-1}\prod(t)dt,\ x>f_1, \end{equation*} and \begin{equation*} J^{\alpha}_{f_2-}\prod(x)=\frac{1}{\Gamma(\alpha)}\int_{x}^{f_2}(t-x)^{\alpha-1}\prod(t)dt,\ x< f_2, \end{equation*} respectively, where \(\Gamma(\alpha)\) is the Gamma function defined by \(\Gamma(\alpha)=\int_{0}^{\infty}e^{-t}t^{\alpha-1}dt\).

Theorem 8. [14] Let \(\prod:I\subseteq (0,\infty)\rightarrow \mathbb{R}\) be a function such that \(\prod\in L_1(f_1,f_2)\) where \(f_1,f_2\in I\) with \(f_1< f_2\). If \(\prod\) is harmonocally convex function on \([f_1,f_2]\), then following inequality for fractional integral hold:

where \(\alpha>0\) and \(\Omega(x)=\frac{1}{x}\).

Definition 9. [11] Let \(\prod\in L_{1}([f_1,f_2]\times [g_1,g_2])\). The Riemann-Liouville integrals \(J^{\alpha,\beta}_{f_1+,g_1+}\), \(J^{\alpha,\beta}_{f_1+,g_2-}\), \(J^{\alpha,\beta}_{f_2-,g_1+}\) and \(J^{\alpha,\beta}_{f_2-,g_2-}\) of order \(\alpha,\beta>0\) with \(f_1,g_1\geq 0\) are defined by \begin{equation*} J^{\alpha,\beta}_{f_1+,g_1+}\prod(x,y)=\frac{1}{\Gamma(\alpha)\Gamma(\beta)}\int_{f_1}^{x}\int_{g_1}^{y}(x-t)^{\alpha-1}(y-s)^{\beta-1}\prod(t,s)dsdt,\ x>f_1 \ y>g_1, \end{equation*} \begin{equation*} J^{\alpha,\beta}_{f_1+,g_2-}\prod(x,y)=\frac{1}{\Gamma(\alpha)\Gamma(\beta)}\int_{f_1}^{x}\int_{y}^{g_2}(x-t)^{\alpha-1}(y-s)^{\beta-1}\prod(t,s)dsdt,\ x>f_1 \ y< g_2, \end{equation*} \begin{equation*} J^{\alpha,\beta}_{f_2-,g_1+}\prod(x,y)=\frac{1}{\Gamma(\alpha)\Gamma(\beta)}\int_{x}^{f_2}\int_{g_1}^{y}(x-t)^{\alpha-1}(y-s)^{\beta-1}\prod(t,s)dsdt,\ x< f_2 \ y>g_1, \end{equation*} and \begin{equation*} J^{\alpha,\beta}_{f_2-,g_2-}\prod(x,y)=\frac{1}{\Gamma(\alpha)\Gamma(\beta)}\int_{x}^{f_2}\int_{y}^{g_2}(x-t)^{\alpha-1}(y-s)^{\beta-1}\prod(t,s)dsdt,\ x< f_2 \ y< g_2, \end{equation*} respectively. Here \(\Gamma\) is the Gamma function.

Theorem 10. [11] Let \(\prod:\Delta=[f_1,f_2]\times [g_1,g_2]\subseteq \mathbb{R}^{2} \rightarrow \mathbb{R}\) be convex on \(\Delta\) with \(f_1< f_2\) and \(g_1< g_2\) and \(\prod\in L_{1}(\Delta)\). Then

In this paper, we gave integral results for co-ordinated harmonically convex functions via fractional integrals.

2. Main Results

In this section, our aim is to prove some Hermite-Hadamard type ineqalities for co-ordinated harmonically convex functions in fractional integrals.

Theorem 11. Let \(\prod:\Delta=[f_1,f_2]\times [g_1,g_2]\subseteq (0,\infty)\times (0,\infty) \rightarrow \mathbb{R}\) be harmonically convex on \(\Delta\) with \(f_1< f_2\) and \(g_1< g_2\) and \(\prod\in L_{1}(\Delta)\). Then

where \(\Omega(x,y)=\left(\frac{1}{x},\frac{1}{y} \right) \) for all \((x,y)\in ([\frac{1}{f_2},\frac{1}{f_1}],[\frac{1}{g_2},\frac{1}{g_1}])\).

Proof. Let \((x,y),(z,w)\in \Delta\) and \(t_1,t_2 \in [0,1]\). Since \(\prod\) is co-ordinated harmonically convex on \(\Delta\), we have

By taking \(x=\frac{f_1f_2}{t_1f_1+(1-t_1)f_2}\), \(z=\frac{f_1f_2}{t_1f_2+(1-t_1)f_1}\), \(y=\frac{g_1g_2}{t_2 g_1+(1-t_2)g_2}\), \(w=\frac{g_1g_2}{t_2 g_2+(1-t_2)g_1}\) and \(t_1=t_2=\frac{1}{2}\) in (7), we get Multiplying both sides of (8) by \(t_1^{\alpha-1}t_2^{\beta-1}\) and then integrating with respect to \((t_1,t_2)\) over \([0,1]\times [0,1]\), we get Applying change of variable, we find Then by multiplying and dividing by \(\Gamma(\alpha)\Gamma(\beta)\) on right hand side of inequality (10), we get the first inequality of (6). For the second inequality of (6) we use the co-ordinated harmonically convexity of \(\prod\) as: \begin{align*} \begin{split} &\prod\left( \frac{f_1f_2}{t_1f_1+(1-t_1)f_2},\frac{g_1g_2}{t_2 g_1+(1-t_2)g_2}\right) \\ &\leq t_1t_2 \prod(f_1,g_1)+t_1(1-t_2)\prod(f_1,g_2)+(1-t_1)t_2 \prod(f_2,g_1)+(1-t_1)(1-t_2)\prod(f_2,g_2), \end{split} \end{align*} \begin{align*} \begin{split} &\prod\left(\frac{f_1f_2}{t_1f_1+(1-t_1)f_2},\frac{g_1g_2}{t_2 g_2+(1-t_2)g_1}\right) \\ &\leq t_1t_2 \prod(f_1,g_2)+t_1(1-t_2)\prod(f_1,g_1)+(1-t_1)t_2 \prod(f_2,g_2)+(1-t_1)(1-t_2)\prod(f_2,g_1), \end{split} \end{align*} \begin{align*} \begin{split} &\prod\left(\frac{f_1f_2}{t_1f_2+(1-t_1)f_1},\frac{g_1g_2}{t_2 g_1+(1-t_2)g_2}\right) \\ &\leq t_1t_2 \prod(f_2,g_1)+t_1(1-t_2)\prod(f_2,g_2)+(1-t_1)t_2 \prod(f_1,g_1)+(1-t_1)(1-t_2)\prod(f_1,g_2), \end{split} \end{align*} and \begin{align*} \begin{split} &\prod\left(\frac{f_1f_2}{t_1f_2+(1-t_1)f_1},\frac{g_1g_2}{t_2 g_2+(1-t_2)g_1}\right) \\ &\leq t_1t_2 \prod(f_2,g_2)+t_1(1-t_2)\prod(f_2,g_1)+(1-t_1)t_2 \prod(f_1,g_2)+(1-t_1)(1-t_2)\prod(f_1,g_1). \end{split} \end{align*} Then by adding above inequalities, we get Thus by multiplying (11) by \(t_1^{\alpha-1}t_2^{\beta-1}\) and then integrating with respect to \((t_1,t_2)\) over \([0,1]\times [0,1]\), we get the second inequality of (6). Hence the proof is completed.

Remark 1. In Theorem 11, if one takes \(\alpha=\beta=1\) and using change of variable \(u=1/x\) and \(v=1/y\), then one has Theorem in [12].

Theorem 12. Let \(\prod:\Delta=[f_1,f_2]\times [g_1,g_2] \subseteq (0,\infty)\times (0,\infty)\rightarrow \mathbb{R}\) be harmonically convex on \(\Delta\) with \(f_1< f_2\) and \(g_1< g_2\) and \(\varPsi\in L_{1}(\Delta)\). Then

where \(\Omega(x,y)=\left(\frac{1}{x},\frac{1}{y} \right) \), \(\Omega_{1}(x,y)=\left(\frac{1}{x},y \right) \) and \(\Omega_{2}(x,y)=\left(x,\frac{1}{y} \right) \) for all \((x,y)\in \left( [\frac{1}{f_2},\frac{1}{f_1}],[\frac{1}{g_2},\frac{1}{g_1}]\right) \).

Proof. Since \(\prod\) is co-ordinated harmonically convex on \(\Delta\) then we have \(\prod_{\frac{1}{x}}:[f_1,f_2]\rightarrow \mathbb{R}\), \(\prod_{\frac{1}{x}}(y)=\prod(\frac{1}{x},y)\), is harmonically convex on \([g_1,g_2]\) for all \(x\in \left[ \frac{1}{f_2},\frac{1}{f_1}\right] \). Then from inequality (4), we have

In other words, for all \(x\in\left[ \frac{1}{f_2},\frac{1}{f_1}\right] \). Now by multiplying (14) by \(\frac{\alpha(x-1/f_2)^{\alpha-1}}{2}\left( \frac{f_1f_2}{f_2-f_1}\right) ^{\alpha}\) and \(\frac{\alpha(1/f_1-x)^{\alpha-1}}{2}\left( \frac{f_1f_2}{f_2-f_1}\right) ^{\alpha}\), and then integrating with respect to \(x\) over \([1/f_2,1/f_1]\), respectively, we find and Again by similar arguments for \(\prod_{\frac{1}{y}}:[f_1,f_2]\rightarrow \mathbb{R}\), \(\prod_{\frac{1}{y}}(x)=\prod(x,\frac{1}{y})\), we get and By adding inequalities (15)-(18), we have This completes the second and third inequality of (12). Now again using (4), we have Adding (20) and (21), we get This completes the first inequality of (12). For the last inequality by using (4), we have \begin{align*} \begin{split} &\frac{\alpha}{2}\left( \frac{f_1f_2}{f_2-f_1}\right) ^{\alpha}\left[\int_{1/f_2}^{1/f_1}\left( \frac{1}{f_1}-x\right) ^{\alpha-1}\prod\left( \frac{1}{x},g_1\right) {\text d}x+\int_{1/f_2}^{1/f_1}\left( x-\frac{1}{f_2}\right) ^{\beta-1}\prod\left( \frac{1}{x},g_1\right) {\text d}x\right] \\ &\leq \frac{\prod(f_1,g_1)+\prod(f_2,g_1)}{2},\\ &\frac{\alpha}{2}\left( \frac{f_1f_2}{f_2-f_1}\right) ^{\alpha}\left[\int_{1/f_2}^{1/f_1}\left( \frac{1}{f_1}-x\right) ^{\alpha-1}\prod\left( \frac{1}{x},g_2\right) {\text d}x+\int_{1/f_2}^{1/f_1}\left( x-\frac{1}{f_2}\right) ^{\beta-1}\prod\left( \frac{1}{x},g_2\right) {\text d}x\right] \\ &\leq \frac{\prod(f_1,g_2)+\prod(f_2,g_2)}{2},\\ &\frac{\beta}{2}\left( \frac{g_1g_2}{g_2-g_1}\right) ^{\beta}\left[\int_{1/g_2}^{1/g_1}\left( \frac{1}{g_1}-y\right) ^{\beta-1}\prod\left( f_1,\frac{1}{y}\right) {\text d}y+\int_{1/g_2}^{1/g_1}\left( y-\frac{1}{g_2}\right) ^{\beta-1}\prod\left( f_1,\frac{1}{y}\right) {\text d}y\right] \\ &\leq \frac{\prod(f_1,g_1)+\prod(f_1,g_2)}{2},\\ &\frac{\beta}{2}\left( \frac{g_1g_2}{g_2-g_1}\right) ^{\beta}\left[\int_{1/g_2}^{1/g_1}\left( \frac{1}{g_1}-y\right) ^{\beta-1}\prod\left( f_2,\frac{1}{y}\right) {\text d}y+\int_{1/g_2}^{1/g_1}\left( y-\frac{1}{g_2}\right) ^{\beta-1}\prod\left( f_2,\frac{1}{y}\right) {\text d}y\right] \\ &\leq \frac{\prod(f_2,g_1)+\prod(f_2,g_2)}{2}. \end{split} \end{align*} Thus by adding all above inequalities, we get the last inequality of (12). Hence the proof is completed.

Lemma 1. Let \(\prod:\Delta=[f_1,f_2]\times [g_1,g_2]\subseteq (0,\infty)\times (0,\infty) \rightarrow \mathbb{R}\) be a partial differentiable mapping on \(\Delta\) with \(0< f_1 < f_2\) and \(0< g_1< g_2\). If \(\partial^{2} \prod/\partial t_1\partial t_2\in L_1(\Delta)\), then following holds:

where and \(A_{t_1}=t_1f_1+(1-t_1)f_2\), \(B_{t_2}=t_2 c+(1-t_2)d\). Also, \(g(x,y)=(\frac{1}{x},\frac{1}{y})\), \(g_{1}(x,y)=(\frac{1}{x},y)\), and \(g_{2}(x,y)=(x,\frac{1}{y})\) for all \((x,y)\in \Delta\).

Proof. By integration by parts and using the change of variable \(x=\frac{A_{t_1}}{f_1f_2}\) and \(y=\frac{B_{t_2}}{g_1g_2}\), we find that

Similarly, we can have Thus from equalities (25)-(28), we have Multiplying both sides of equality (29) by \(\frac{f_1f_2g_1g_2(f_2-f_1)(g_2-g_1)}{4}\), we get the desired equality (23).

Theorem 13. Let \(\prod:\Delta=[f_1,f_2]\times [g_1,g_2]\subseteq (0,\infty)\times (0,\infty) \rightarrow \mathbb{R}\) be a partial differentiable mapping on \(\Delta\) with \(0< f_1< f_2\) and \(0< g_1< g_2\). If \(\left| \partial^{2} \prod/\partial t_1\partial t_2\right| \) is a harmonically convex on the co-ordinates on \(\Delta\), then following holds:

where

Proof. Using Lemma 1, we have

Now using co-ordinated harmonically convexity of \(\left| \frac{\partial^{2}\prod}{\partial t_1\partial t_2}\right| \), we get After calculating above integrations, we get the required result.

Theorem 14. Let \(\prod:\Delta=[f_1,f_2]\times [g_1,g_2]\subseteq (0,\infty)\times (0,\infty) \rightarrow \mathbb{R}\) be a partial differentiable mapping on \(\Delta\) with \(0< f_1< f_2\) and \(0< g_1< g_2\). If \(\left| \partial^{2} \prod/\partial t_1\partial t_2\right|^{q} \), \(q>1\), is a harmonically convex on the co-ordinates on \(\Delta\), then following holds:

where

Proof. Applying the Holder's inequality for double integrals in (35), we get

Using co-ordinated harmonically convexity of \(\left| \frac{\partial^{2}\prod}{\partial t_1\partial t_2}\right|^{q} \), we get By calculating all integrals, we get the required result (37).

3. Conclusion

In Theorem 11 and 12, we have proved some new Hermite-Hadamard type inequalities for co-ordinated harmonically convex on a rectangle via Riemann-Liouville fractional integrals. In Lemma 1, we have proved a fractional integral identity and then with the help of this Lemma 1 we proved some fractional Hermite-Hadamard type inequalities on the co-ordinates.

Acknowledgments

The present investigation is supported by National University of Science and Technology(NUST), Islamabad, Pakistan.

Authors Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflicts of interest

The authors declare no conflict of interest.

1. Introduction and Preliminaries

It is proposed to advance the mechanics of binding energies of vibration of interest for dipole energetics in 1) Ionized water systems and 2) Nonlinear or chaotic computational regimes. Complex water molecule structures of various states of water, similar to microtubulin proteins, or to silicon-based photovoltaic panels, have a morphology of a naturally-occurring micromachine, with the potential to be engineered to make quantum computers, e.g. chemical neural nets, as similarly microtubulins are the intelligent scaffolding for neural computation [1].

The hydrogen bond is an attractive interaction between a hydrogen atom from a molecule or a molecular fragment X-H in which X is more electronegative than H, and an atom or a group of atoms in the same or a different molecule, in which there is evidence of bond formation [2].

2. Proposal and study

Quantum can be defined at \(h=6.62607015 \times 10^{-34}J.s\) for propagation physics of subatomic decoherence and superposition linkeages of electromagnetic wavelengths across space and time axes. Entanglement can be defined as any physical network of related, asymmetric action as interrelated, therefore systemic mechanisms such as solutions [3]. A one-cell-buffer of monadic value as a finite-state machine or floating-point unit can be represented [4].

Entanglement in the physical explanation can be understood as a process involving either Quantum-scale or Newtonian-scale physical formula and algorithms. A given water molecule \(H_2O\), and hydrogen bonds, as polar compounds, bonding at \(104.48^{o}\) [5] can essentially be understood as a computational mechanism, in process, but completely physical in nature [6]. This physical mechanism is based upon Bell's Theorem from Shannon entropy where

\begin{equation*} H(x)=-\sum_{i=1}^{n}P(x_{i})logP(x_{i}). \end{equation*} Demonstrations of quantum tunneling behavior of water have been noted for physical isomorphisms in Chronoelectromagnetogravidynamic [-CEMGD] hypothesis, therefore to define the computational quantum-physical process and mechanics for the hydrogen bond [7], it is necessary to hypothesize that these models as either a kind and type of:

• 1. Automata computing machine composed of algorithmic strings, therefore quantum-entangled, and/or a kind and type of
• 2. Automorphic, or monoidal algorithmic string that is solely or atomically quantum-entangled, vis-a-vis the corresponding system. For the intelligent scaffolding hypothesis, Albrecht-Buehler has demonstrated such cellular infrared detection and mechanics to originate in the centrosome and centrioles [8].

3. Materials and methods

Now it is demonstrated in gedanken-experiment that the entire molecule of \(H_2O\) and subsequent hydrogen bonds being physical and computational, but without specific analysis of kind of computation proposed or invoked, is based on advances from the simple Radon transform where \begin{eqnarray*} \notag Rf(\alpha, s)&=&\int_{-\infty}^{\infty}f(x(z),y(z))dz\\ &=&\int_{-\infty}^{\infty}f((z sin \alpha+s cos \alpha),(-zcos \alpha+ssin \alpha))dz. \end{eqnarray*} It is more dimensionally-exact to posit that water molecules are the result of basal quantum entanglement of forces and energetics occurring at Planck scale and then forming energy from energy into local entropic maximals, sending and receiving forms of energetic phases, e.g. mechanical, chemical or electromagnetic in nature at the newtonian, \(+nm\) scale. Regarding demonstration of direct testing of these hypotheses, a simple system is proposed for scaling of entanglement processes. There may be weak examples of entanglement at room-temperature at the atomic to molecular scale, and it is proposed at the Planck scale between separable states A and B [Wightman] in a given density matrix for the proposed given the first order of Quantum Entanglement, \(E_1\): \begin{equation*} P([\psi],[\phi])=\frac{|\langle \psi,\phi \rangle|^2}{||\psi||^2||\phi||^2}, \end{equation*} and per Local Quantum Field Theory [LQFT] therefore \begin{equation*} \frac{\delta \rho}{\delta t}+\nabla. (\rho v)=0, \end{equation*} and regarding the continuum mechanics inherent in quantum entanglement to activate the conjectured Second Order Of Entanglement, \(E_2\): \begin{equation*} > >v(>>r,t)=\frac{\nabla S(>>r,t)}{m}, \end{equation*} c.f. Schrodinger's equation for a single relativistic particle \begin{eqnarray*} i\hbar \partial \partial t\psi(r,t)&=&[-\hbar22m\nabla2+V(r,t)]\psi(r,t)i\hbar \frac{\partial}{\partial t}\psi(r,t)\\ &=&\left[\frac{-\hbar^{2}}{2m}\nabla^{2}+V(r,t)\right]\psi(r,t) \end{eqnarray*}

4. Results

Macroscopic energy systemics of polarization of simple compound molecules of air at approxi-mately 78% N and 20% O demonstrate future potential technologies e.g. air-based tractor beams [9], pointing to a type of limited macroscale entanglement. These should be defined from \begin{equation*} | \beta(x,y)\langle=\left(\frac{|0,y\rangle +(-1)^{x}|1,Y\rangle}{}\right), \end{equation*} and extended formulas for the Bell States.

Mathematical physics, Topological Quantum Field Theory [TQFT], Gauge theory, and Quantum mechanics are utilized as system of computation to understand any relevant linkeages between quantum to classical systems, e.g. the hydrophysical cellular systems proposed by the Author [Jonah Lissner] as either quasi-ice [crystalline structure] or quasi-vapor [gas structure] states for quantum computation regimes of qubits. There is conjectured a theoretical, historical similarity between Wave-Particle duality debate and the subsequent, Copenhagen-stochastic versus De Broglie-Bohm-Hiley-Bekenstein Holographic Principle [DBBHBHP] quantum field mechanics-discrete, systemic interpretations. It is therefore proposed there be a scale-boundary for clopen wavefunction collapse. This action may in effect be the origin of modes of quantum entanglement in an entropic system or subsys-tem, and the composition of the ionic bonding values of water.

This is a postulate estimated considering the factor of entropy in the given system and subsystem for the wavefunction collapse at a maximal distribution method from Boltzmann's constant \(k\), at every scale of time-gravity, for the given wavefunction frequency, especially for multibody problems. Based on metrics and formula from the Quantum Vacuum State/Vacuum Expectation Value [QVS/VEV] we can estimate at relativistic speeds of \(1\) electronvolt

\begin{equation*} p=1GeV/c=(1/times 10^{9}).(1.60217646 \times 10^{-19}C).V/(2.99792457 \times 10^{8}m/s)=5.344285 \times 10^{-19}kg.m/s\,. \end{equation*} There is a great finite amount of electro-magnetic string landscape energy. In a given water molecule \(H_2O\), there are requirements for efficient energy usage of the entire volume of the water structure. For the requirements to change would point to chaotic regions of physical constants which are not classical, but quantum. Electrical Resistivity Tomography [ERT] has demonstrated advances for the discovery of isomorphisms in Chronoelectromagnetogravidynamic [-CEMGD] regions of the earth releasing energy from seismoelectric and seismogravitic activity [10].

Quantum gravity and quantum chronodynamics are directly correlated to measurable entropy of systems. De Broglie-Bohm-Hiley-Bekenstein Holographic Principle [DBBHBHP] and Wheeler-Feynman have strong points in this debate; strict Copenhagen is supposed as a basis for heuristics and stochastic methods as Von Neumann advanced. Here the Principle of Least Action can be invoked for further study of quantum mechanics of cellular systems, specifically hydrological regimes where

\begin{equation*} L(q,q^{.},t)=K(q^{.})-U(q,t)\,. \end{equation*} Orders of magnitude, in given system domains of power contain variations or automata of physical systems: The quantum scale contains orders of entanglement and probability translated up-ward into constant-entangled systems at scales and parameters of entropy of physical systems, e.g. from Planckscale atoms to the cellular \(+nm\) scale being past molecular scale, into molecular chains \([H_2O]\).

These therefore form two major systems strata having internal and external components in continual interrelationship over space-and-time expending and absorbing energy in an exegetics environment. The quantum cellular automata theory can be understood as algorithms of powers or factors of physical constants applied to given systems as heuristics, correct to the extent of such heuristics which being mechanical are part of the mathematical solutions required to understand the actions of quantum probability of systems.

Although there is proof for room-temperature quantum entanglement of photonic streams, it is not scale-free in that the specific pilot-wave has collapsed at a given drop or point of entropy and dispersed its energy at a given system scale of maximum entropy [Point B]. The minimum scale of entropy is the point of diffraction [Point A]. We can see this in the continual reference to such experiments as the Wave dispersement double-slit experiment and requisite string scattering amplitudes for plasmonic field confinement [11]. The mechanisms of scalar changes in entanglement demonstrate that there are too many points of energetic discourse to have a completely scale-free system. The keys of the scale changes are the points of energetic absorption in a given hydrogen bond structures. They are routed cell-to-cell and bond-to-bond across such networked regions.

It is therefore hypothesized to measure the quantum processes of any physical system going past current magnetic resonance imaging. It is given there are many kinds of physical entanglements or energetic interactions occurring at any given time and location utilizing Dynamic Global Workspace Theory-Intelligent Computational System Organization [DGWT-ICSO] of the inherent environmental quantum pathways, given the path of least resistance.

5. Discussion

Now it can be proposed to determine how a water molecule is a quantum-entangled object in this instance. The water molecule \(H_2O\) is at the outer physical layer of the given quantum processes occurring around, across, through and beneath it in 4 axes of the spin isomers of H - orthohydrogen [parallel], and parahydrogen [antiparallel]. Therefore the water-cellular model is demonstrated the result of these processes and acts and re-acts in the two-state Newtonian system, which somehow mechanizes from the quantum mechanical states in the greater physical system.

A theory of topological geometry of matrices in topological quantum field theory, is a proposed quantum cellular automata decomplexifying at greater and greater scales toward a temporary local maximum entropy of classical biophysical processes.

\begin{equation*} H(i,0)=0, \ \ \ 0\leq i \leq m, \end{equation*} \begin{equation*} H(j,0)=0, \ \ \ 0\leq j \leq n, \end{equation*} \begin{equation*} H(i,j)-max\{H(i-1,j-1)+s(a-i,b)j)\text{Match}/\text{Mismatch}\}, \ \ \ 1\leq i \leq m, \ \ \ 1\leq j \leq n, \end{equation*} \begin{equation*} max \ \ k\geq1 \{H(i-j,k)+W_{k}\} \ \ \ \text{Deletion}, \end{equation*} \begin{equation*} max \ \ i\geq1 \{H(i,j-1)+W_{l}\} \ \ \ \text{Insertion}. \end{equation*} This model can be demonstrated in Adenosine triphosphate [ATP] cycles as the rule automata are completed, as in Smith-Waterman algorithm.

Marx generators have successfully used N for the spark gap and \(H_2O\) as water capacitor to modulate voltage dynamics in the capacitor portion [12]. Chronoelectromagnetogravidynamics [-CEMGD] of water adsorption is proposed as a function of Adenosine triphosphate [ATP] cellular physiology and cell energetics of \(H_2O\) adsorption, from electron field transduction mechanisms. Some unified macroscale and quantum physical methods to separate water molecule chains in given water mediums use superconductivity from magnetism or sonic vibration, e.g. Macrosonics, or Supercavitation, to desalinate water, than by forward osmosis or reverse osmosis.

Nikola Tesla studied induced harmonics utilizing the advanced magnifying transmitter with an array of grounding and systems of ionospheric plasma induction, to guide Quantum Vacuum State/Vacuum Expectation Value [QVS/VEV] or Quantum-scale electromagneto-gravidynamics at specific wavelengths from power generation to relay station, as an example of applied geo-physics of lightning storms.

The vacuum state, or exclusion zone, has the properties of propagation as the ionosphere and the subcrust geology, as noted in Tesla's engineering for advanced magnifying transmis-sion [13]> Regarding Nikola Tesla's work with Hydrodynamics, heat and mechanics of turbine systems and water adsorption zones, models of exclusion zones can be found in ionospheric, and geological regimes.

6. Conclusion

Regarding water molecule physics, it can be understood that transduction of environmental entropy is slowed down by each molecular bond [14,15,16]. These mimetics are defined as engineer-ing of differences in energetic values using Boundary Layer Effect utilizing the Von Karman momentum integral, where \begin{equation*} \frac{\tau_{w}}{\rho U^2}=\frac{1}{U^2}\frac{\partial}{\partial t}(U \delta_{1})+\frac{\partial \delta_{2}}{\partial x}+\frac{2 \delta_{2}+\delta_{1}}{U}+\frac{\partial U}{\partial x}+\frac{v_{w}}{U}, \end{equation*} for \begin{equation*} \tau_{w}=\mu\left(\frac{\partial u}{\partial y}\right)_{y=0}, \ \ \ v_{w}=v(x,0,t), \ \ \ \delta_{1}=\int_{0}^{\infty}\left(1+\frac{u}{U}\right)dy, \ \ \ \delta_{2}=\int_{0}^{\infty}\frac{u}{U}\left(1+\frac{u}{U}\right)dy\,. \end{equation*} In the case of the water-adsorption zone, the complete water-gas phase is engineerable for water-batteries, water-conduits, water-circuitry, and \(H_2O\) molecule-based computation engines with prototypes using engineered materials for transduction. In summary, the gestalt effect of the transduced energy does not overrule the standard theory, but advances the previous theories of nanophysics of Water, and how electrons are transmitted across nm-scale ionic bonds, in given quantum computational regimes.

Conflict of Interests

''The author declares no conflict of interest.''

1. Introduction and Preliminaries

Free convection flow over vertical surfaces has been widely applied in different engineering areas such as mechanical forming processes, glass-fibre production processes, extrusion, food processing, melt spinning etc. Consequently, this has aroused lots of research interests on the flow phenomena following the experimental investigations of Schmidt and Beckmann [1] and the pioneering theoretical work of Ostrach et al. [2]. An extended work was done by Sparrow and Gregg [3] on the free convection under the influence of a uniform surface heat flux. In another work, Cianfrini et al. [4] studied the flow phenomena over a non-isothermal vertical plate. Lefevre [5] examined the free convection of an inviscid flow under low Prandtl-numbers while Stewartson and Jones [6] as well as Eshghy [7] analyzed the flow over a heated vertical plate at high Prandtl number. Although, Roy [8] also studied the free convection flow process under a large Prandtl number, the effects of uniform surface heat flux on the flow phenomena was also investigated. Kuiken [9,10] and Kuiken and Rotem [11] examined the free convection flow over the vertical plates at both low and high Prandtl numbers. In recent times, different analytical and numerical methods have been used to examine the laminar free convection [12,13,14,15,16,17,18]. Also, various parametric studies on the nonlinear models and fluid flow problems have been presented in literature [19,20,21,22,23,24,25,26,27].

Different types of fluid such as tomato sauce, honey, printing inks, blood, concentrated fruit juices and Jelly have been classified as the non-Newtonian fluids called Casson fluids [28]. These fluids exhibit shear thinning nature with an assumed infinite viscosity at zero rate of shear, a yield stress below which no flow occurs, and a zero viscosity at an infinite rate of shear [29]. Its important areas of applications have provoked some studies [30,31]. Also, the effects of thermal radiation, magnetic field and nanoparticles on the fluid flow processes have been extensively studied [32,33,34,35,36,37,38,39,40]. However, most of these studies which are based on time invariant, focused on the free convection currents caused by the temperature difference. It should be stated that the flow is also affected by the differences in concentration on material constitution such as seen in atmospheric flows, chemical processing, formation and dispersion of fog, distributive temperature and moisture over agricultural fields. Also, the transient behaviours of the fluids flow before a steady state is reached should be well investigated. Hence, the study of the transient heat and mass transfer of the fluid over the vertical plate is very much important. Moreover, to the best of the authors knowledge, a study on transient behaviours of free convection boundary-layer flow, heat and mass transfer of Casson nanofluids over a vertical plate under the influences of thermal radiation, flow medium porosity and nanoparticles has not been presented in literature. Therefore, the transient magnetohydrodynamics free convection heat and mass transfer of Casson nanofluid past an isothermal vertical flat plate embedded in a porous media subjected to thermal radiation is investigated numerically using implicit finite difference scheme of Crank-Nicolson type. The numerical solutions are used to carry out parametric studies. The results are presented graphically and discussed.

2. Problem formulation and mathematical analysis

Consider a two-dimensional unsteady free-convection flow, heat and mass transfer of a Casson nanofluid over a vertical plate embedded in a porous media and parallel to the direction of the generating body force as shown in Figure 1.

In order to set up the flow, heat and mass transfer of the Casson nanofluid, the following assumptions are made

• 1. The flow is incompressible and laminar.
• 2. The heat transfer from the plate to the fluid is proportional to the local surface temperature.
• 3. Pressure is uniform across the boundary layer and Boussinesq approximation is used.
• 4. The thermal diffusion and diffusion thermal effects which are called the Soret and Dufour effects, respectively are insignificant and they are therefore negligible.
• 5. The effect of viscous dissipation on the fluid flow process is negligible.
• 6. There is no chemical reaction taking place in the mass of the fluid.

Taken \(x\)-coordinate to be directed upward along the plate in the flow direction and \(y\)-coordinate is taken normal to the plate. Then under the stated assumptions, the governing equation of the flow, heat and mass transfer could be written as In this work, we adopt a conditions that the plate and the fluid are initially at the same concentration and temperature level that is the same in the fluid everywhere. Then at time \(t > 0\), the plate temperature is suddenly raised to \(T_{\infty}\), and the concentration level near the plate is also raised to \(C_{\infty}\), which are thereafter maintained constant. Therefore, the initial condition is given as and the appropriate boundary conditions under no slip conditions are given as The thermal radiation term in Equation (3) could be linearized using Rosseland's approximation as follows Substituting Equation (10) into Equation (3), we the governing equations of the flow, heat and mass transfer as where the various physical and thermal properties in Equations (13) and (14) are given as where \(m\) in the above Hamilton Crosser's model in Equation (19) is the shape factor which numerical values for different shapes are given in Table 1.

Table 1. The values of different shapes of nanoparticles [41, 42].

Sr. No.	Name	Shape factor (m)	Sphericity($\psi$)
1	Sphere	3.0	1.000
2	Brick	3.7	0.811
3	Cylinder	4.8	0.625
4	Platelet	5.7	0.526
5	Lamina	16.2	0.185

We now introduce the following non-dimensional quantities

We have the dimensionless forms of the governing equations as The appropriate initial and boundary conditions are given as

3. Method of solution: finite difference method

Indisputably, the finite difference method is one of the efficient methods of obtaining numerical solutions to differential equations. The difference scheme could be applied using explicit or implicit schemes. However, the applications of the finite difference scheme to the practical situations depict that the stability and convergence conditions for explicit finite-difference methods become extremely complicated coupled with high computational cost. In order to avoid obviate the inherent problems in the explicit schemes, implicit schemes are recommended. Such schemes permit large time-steps to be used with unconditional stability. Also, the use and the accuracy of finite difference method for the analysis of nonlinear problems has earlier been pointed out by Han et al. [43]. Therefore, in this work, implicit finite difference method of Crank-Nicolson type is used to discretize the systems of coupled non-linear ordinary differential equations combined with the initial and boundary conditions in governing Equations (21)-(24) as well as Equations (25)-(29), respectively. The finite difference forms for the systems of coupled non-linear ordinary differential equations, the initial and boundary conditions are given as The appropriate initial and boundary conditions in finite difference forms are given as where, \(\triangle x_{j}=x_{i}-x_{i-1}\), \(\triangle y_{j}=y_{i}-y_{i-1}\), \(\triangle y_{j}^{+}=y_{i+1}-y_{i}\), \(\overline{\triangle y_{j}}=y_{i+1}-y_{i-1}\), \(\triangle y_{j}^{-}=y_{i}-y_{i-1}\).

The implicit finite difference scheme is used for the Cartesian coordinate. The rectangular region of the integration (Figure 2) considered varies from 0 to 1 and y varying from 0 to 15. It is taken in this study that \(y_{max}\) lies well outside the momentums, energy and concentration boundary layers. It should be pointed out that after some preliminary investigations, the maximum of \(y\) was chosen as 15. This value is arrived at in order to make the boundary conditions in Equations (25) and (26) to be satisfied within the tolerance limit of \(10^{-5}\). For the rectangular region considered, the \(x\)- and \(y\)-directions are divided into \(M\) and \(L\) grid spacings, respectively as shown in Figure 2. In order to reduce the computational time, variable mesh sizes are used in the \(x\)- and \(y\)-directions as follows

It should be stated that the subscripts \(i\) designates the grid points with \(x\)-direction while \(j\) designates the grid points \(y\)-direction as shown in Figure 2. The superscript \(n\) represents a value of time in \(t\)-direction. The velocity, temperature and concentration distributions at all interior nodal points are computed by successive applications of the above finite difference equations (30)-(32) along with the initial and boundary conditions (33) - (37). The process of the computations is explained below.

From the initial conditions (33) - (37), the initial values (at \(t = 0\)) of \(u\), \(v\), \(T\) and \(C\) are known at all grid points. For the subsequent computations, it should be stated that during any one-time step, the coefficients appearing in the finite difference equations are treated as constants. Using the known values at previous time level \((n)\), the computations of \(u,\ v,\ T,\) and \(C\) at time level \((n + 1)\) are calculated as follows:

At every internal nodal point on a particular \(i-\) level, the finite difference equation (32) constitutes a tri-diagonal system of equations which is solved by Thomas algorithm as explained by Carnahan et al. [44]. Therefore, the values of \(C\) are determined at every nodal point on a particular \(i-\)level at \((n+ 1)th\) time level. In the same way, the values of \(T\) are computed using the finite difference Equation (31). Substituting the values of \(C\) and \(T\) at \((n +1)th\) time level in the finite difference Equation (30), the values of \(u\) at (\(n+ 1)th\) time level are found. Having determined the values of \(C\), \(T\) and \(u\) on a particular \(i-\)level, with the aid of the finite difference Equation (30) at every nodal point on a particular \(i-\)level at \((n +1)th\) time level, the values of \(v\) are calculated explicitly. This process is repeated for various \(i-\)levels. Thus, the values of \(C\), \(T\), \(u\) and \(v\) are known at all grid points in the rectangular region at \((n+ 1)th\) time level. The computations are carried out till the steady state is reached. In this present study, it is assumed that the steady state solution is reached, when the absolute difference between the values of \(u\), \(T\) and \(C\) at two consecutive time steps are less than \(10-5\) at all grid points.

4. Flow, heat and mass transfer parameters of engineering interests

In addition to the determination of the velocity, temperature and concentration distributions, it is often desirable to compute other physically important quantities (such as shear stress, drag, heat transfer and mass transfer rates) associated with the free convection flow, heat transfer and mass transfer problems. Consequently, flow, heat and mass transfer parameters are computed.

4.1. Fluid flow parameter

The local skin-friction are derived as

4.2. Heat transfer parameter

The local Nusselt number are given as

4.3. Mass transfer parameter

The Sherwood are developed as The derivatives involved in the Equations (41), (42) and (43) are evaluated using five-point approximation formula while the integrals are evaluated using Newton-Cotes closed integration formula.

5. Results and discussion

The results of the simulations are presented in this section and the effects of the various model parameters are presented.

5.1. Effects of Casson parameter on the fluid velocity, temperature and concentration distributions

The effects of Casson parameter on the flow velocity, temperature and concentrations profiles of the nanofluid are shown in Figures 3, 4 and 5, respectively. The figures depict that the flow velocity of the nanofluid near the plate decreases as the Casson parameter increases as illustrated in Figure 3 The trend in the figure could explained that, physically, increasing values of Casson parameter develop the viscous forces which in consequent retards the flow of the and thereby reduced the flow velocity. It could be established from the results that the temperature as well as the concentration of the fluid increase as the Casson fluid parameter increase as shown in Figures 4 and 5.

5.2. Effects of radiation parameter on the fluid velocity, temperature and concentration distributions

Figures 6, 7 and 8 show that the viscous, thermal and concentration boundary layers increase with the increase of radiation parameter, \(R\). It is shown that increase in radiation parameter causes the velocity of the fluid to increase. This is because as the radiation parameter is increased, the absorption of radiated heat from the heated plate releases more heat energy released to the fluid and the resulting temperature increases the buoyancy forces in the boundary layer which also increases the fluid motion and the momentum boundary layer thickness accelerates. This is expected, because the considered radiation effect within the boundary layer increases the motion of the fluid which increases the surface frictions.

5.3. Effects of nanoparticle shape on the fluid velocity, temperature and concentration

The use of nanoparticles in the fluids exhibited better properties relating to the heat transfer of fluid than heat transfer enhancement through the use of suspended millimeter- or micrometer-sized particles which potentially cause some severe problems, such as abrasion, clogging, high pressure drop, and sedimentation of particles. The very low concentrations applications and nanometer sizes properties of nanoparticles in basefluid prevent the sedimentation in the flow that may clog the channel. It should be added that the theoretical prediction of enhanced thermal conductivity of the basefluid and prevention of clogging, abrasion, high pressure drop and sedimentation through the addition of nanoparticles in basefluid have been supported with experimental evidences in literature.

Figures 9, 10 and 11 show the influence of the shape of nanoparticle on the flow velocity, temperature and concentrations profiles of the nanofluid. It is observed that lamina shaped nanoparticle carries maximum velocity whereas spherical shaped nanoparticle has better enhancement on heat transfer than other nanoparticle shapes. In fact, it is in accordance with the physical expectation since it is well known that the lamina nanoparticle has greater shape factor than other nanoparticles of different shapes, therefore, the lamina nanoparticle comparatively gains maximum temperature than others. The velocity decrease is maximum in spherical nanoparticles when compared with other shapes. The enhancement observed at lower volume fractions for non-spherical particles is attributed to the percolation chain formation, which perturbs the boundary layer and thereby increases the local Nusselt number values. The results show that the maximum decrease in velocity and maximum increase in temperature are caused by lamina, platelets, cylinder and sphere, respectively. It is also observed that irreversibility process can be reduced by using nanoparticles, especially the spherical particles. This can potentially result in higher enhancement in the thermal conductivity of a nanofluid containing elongated particles compared to the one containing spherical nanoparticle, as exhibited by the experimental data in the literature. It is therefore required that that proper choice of nanoparticles should made as this will be helpful in controlling fluid flow, heat and mass transfer processes.

5.4. Effect of Prandtl number on the fluid velocity, temperature and concentration distributions

The figures also show the effects of Prandtl number (Pr) on the velocity and temperature profiles are shown in Figures 12, 13 respectively. It is indicated that the velocity of the Casson nanofluid decreases as the Pr increases but the temperature of the nanofluid increases as the Pr increases. This is because the nanofluid with higher Prandtl number has a relatively low thermal conductivity, which reduces conduction, and thereby reduces the thermal boundary-layer thickness, and as a consequence, increases the heat transfer rate at the surface. For the case of the fluid velocity that decreases with the increase of \(Pr\), the reason is that fluid of the higher Prandtl number means more viscous fluid, which increases the boundary-layer thickness and thus, reduces the shear stress and consequently, retards the flow of the nanofluid. Also, it can be seen that the velocity distribution for small value of Prandtl number consist of two distinct regions. A thin region near the wall of the plate where there are large velocity gradients due to viscous effects and a region where the velocity gradients are small compared with those near the wall. In the later region, the viscous effects are negligible and the flow of fluid in the region can be considered to be inviscid. Also, such region tends to create uniform accelerated flow at the surface of the plate.

5.5. Effect of Schmidt and Grashof numbers on fluid velocity and temperature distributions

Figures 14 and 15 shows the effects of Schmidt number (Sc) on the velocity and concentration profiles of the Casson nanofluid, respectively. Figure 16 shows that the as Grashof number increases, the velocity of the fluid increases. However, as in the case of the effect of Prandtl number on the velocity and temperature distribution, it is depicted in the figures that the velocity of the nanofluid decreases as the Sc increases but the temperature of the nanofluid increases as the Sc increases. This is because the nanofluid with higher Schmidt number has a relatively low diffusion coefficient, which reduces mass diffusion thereby reduces the concentration boundary-layer thickness, and as a consequence, increases the mass transfer rate at the surface. In Figure 14, where the fluid velocity decreases with the increase of Sc, this is because the fluid of the higher Schmidt number means more viscous fluid, which increases the boundary-layer thickness and thus, reduces the shear stress and consequently, retards the flow of the nanofluid. It is also observed that the species concentration decreases with increasing Schmidt number as shown in Figure 15. It was also found that the temperature increases with increasing Schmidt number. A further investigation revealed that an increase in the Schmidt number leads to a decrease in Grashof number \(Gr\).

5.6. Effects of magnetic field and flow medium Porosity on Casson nanofluid velocity distributions

Figure 17 shows the effect of magnetic field on the flow velocity of the fluid. It is revealed that there is a diminution in the velocity field occurs for increasing value of the magnetic field number, Hartmann number (Ha). This confirms the general physical behavior of the magnetic field that say that the fluid velocity depreciates for improved values of Ha. The magnetic field produces Lorentz force which is drag-like force that produces more resistance to the flow and reduces the fluid velocity. So large Ha values implies that the Lorentz force increases and the resistance to the flow increases, and consequently, the velocity of the fluid decreases. Practically, the Lorentz force has a resistive nature which opposes motion of the fluid and as a result heat is produced which increases thermal boundary layer thickness and fluid temperature. The magnetic field tends to make the boundary layer thinner, thereby increasing the wall friction. Consequently, the boundary layer thickness is a decreasing function of Ha. i.e. presence of magnetic field slows fluid motion at boundary layer and hence retards the velocity field.

A porous medium studies is very important in a number of engineering applications such as geophysics, die filling, metal processing, agricultural and industrial water distribution, oil recovery techniques, and injection molding. Therefore, Figure 18 shows the effect of flow medium porosity on the fluid velocity. As it is illustrated, the fluid velocity increases as the flow medium porosity, Darcy number increases. This is because, as the Darcy number increases, there is less resistance to fluid flow through the flow medium.

5.7. Effect of Buoyancy ratio parameter on the fluid velocity, temperature and concentration distributions

Figures 19, 20 and 21 show the impacts of buoyancy ratio parameter (N) on the velocity, temperature and concentration profiles. Figure 19 depicts that the as buoyancy ratio parameter increases, the velocity of the fluid increases. However, an increase in buoyancy ratio parameter leads to a decrease in the fluid temperature and concentration as shown in Figures 20 and 21.

5.8. Effect of flow time on the fluid velocity, temperature and concentration distributions

The effects of Schmidt number on local skin friction, Nusselt and Sherwood numbers are shown in Figures 25, 26 and 27, respectively. The figures reveal that the local skin friction, Nusselt and Sherwood numbers decrease as the Schmidt number increases. An opposite trend was recorded when the impact of the buoyancy ratio parameter on Nusselt number was investigated. In the investigation, it was found that as the local Nusselt number increases as the buoyancy ratio parameter increases. It was shown that at small values of x (near the leading edge of the plate), the local Nusselt number is not affected by both buoyancy ratio parameter and Schmidt number due to the pure diffusion and conduction at the location

6. Conclusion

In this present study, the transient free convection heat and mass transfer of Casson nanofluid past an isothermal vertical flat plate embedded in a porous media under the influences of thermal radiation and magnetic field have been investigated. The governing systems of nonlinear partial differential equations of the flow, heat and mass transfer processes are solved using implicit finite difference scheme of Crank-Nicolson type. The numerical solutions are used to carry out parametric studies and the follow results were established:

• 1. The temperature and the concentration of the fluid increase as the Casson fluid and radiation parameters as well as Prandtl and Schmidt numbers increase.
• 2. The increase in the Grashof number, radiation, buoyancy ratio and flow medium porosity parameters causes the velocity of the fluid to increase. However, the Casson fluid parameter, buoyancy ratio parameter, the Hartmann (magnetic field parameter), Schmidt and Prandtl numbers decrease as the velocity of the flow increases.
• 3. The time to reach the steady state concentration, the transient velocity, Nusselt number and the local skin-friction decrease as the buoyancy ratio parameter and Schmidt number increase.
• 4. The steady-state temperature and velocity decrease as the buoyancy ratio parameter and Schmidt number increase.
• 5. The local skin friction, Nusselt and Sherwood numbers decrease as the Schmidt number increases. However, the local Nusselt number increases as the buoyancy ratio parameter increases.

Based on the results in this work, it is believed that the present study will greatly assist in various areas of industrial and engineering applications of the flow problems.

Nomenclature

\(\mathbf{B_{o}}\)   electromagnetic induction

\(\mathbf{c_{p}}\)   specific heat capacity

\(\mathbf{C}\)   species concentration

\(\mathbf{D}\)   species diffusion coefficient

\(\mathbf{g}\)   acceleration due to gravity

\(\mathbf{Gr}\)   Grashof number

\(\mathbf{Ha}\)   Hartmann number/magnetic field parameter

\(\mathbf{k}\)   thermal conductivity

\(\mathbf{K}\)   the absorption coefficient

\(\mathbf{m}\)   shape factor

\(\mathbf{N}\)   buoyancy ratio parameter

\(\mathbf{p}\)   pressure

\(\mathbf{Pr}\)   Prandtl number

\(\mathbf{\overline{p}}\)   pressure

\(\mathbf{R}\)   Radiation number

\(\mathbf{Sc}\)   Schmidt number

\(\mathbf{t}\)   time

\(\mathbf{\overline{T}}\)   temperature of the fluid

\(\mathbf{\overline{u}}\)   velocity component in x-direction

\(\mathbf{\overline{v}}\)   velocity component in y-direction

\(\mathbf{U_{w}}\)   fluid inflow velocity at the wall

\(\mathbf{\overline{x}}\)   coordinate axis parallel to the plate

\(\mathbf{\overline{y}}\)   coordinate axis perpendicular to the plate.

Symbols

\(\beta\)   volumetric extension coefficients

\(\rho_{nf}\)   density of the nanofluid

\(\rho_{f}\)   density of the base fluid

\(\mu_{nf}\)   dynamic viscosity of the nanofluid

\(\rho_{s}\)   density of the solid/nanoparticles

\(\phi\)   fraction of nanoparticles in the nanofluid

\(\gamma\)   Casson parameter

\(\tau\)   shear stress

\(\tau_{o}\)   Casson yield stress,

\(\mu\)   dynamic viscosity

\(\sigma\)   shear rate

Subscript

\(\mathbf{f}\)   fluid

\(\mathbf{s}\)   solid

\(\mathbf{nf}\)   nanofluid

\(\mathbf{w}\)   wall.

Acknowledgments

The author expresses sincere appreciation to University of Lagos, Nigeria for providing material supports and good environment for this work.

Conflict of Interests

''The author declares no conflict of interest.''

1. Introduction

Analyzing the rheological behavior of Eyring-Powell fluid models (at low and high shear rates), which can be easily deduced from the molecular theory of rarefied gases [1], is very important for pseudoplastic systems nowadays. Until recently, only such fluid flow problems, whether ``steady'' or ``unsteady'', have been concerned with finding the velocity and temperature distribution through the use of some particular problem-solving methodologies [2,3,4,5,6,7,8,9,10,11]. However, from a mathematical viewpoint, it can be desirable to have a series expansion which converges in the semi-infinite intervals faster than a series expansion with a smaller interval of convergence. In this way, Khoshrouye Ghiasi and Saleh [12] employed a rather convergent feature of homotopy analysis method (HAM) by adding to the truncation technique, namely, tHAM, for solving the Falkner-Skan boundary value problem (FBVP) and spotting many errors. In fact, they could give rigorous proof of their observations showing how the P-time can be minimized without any loss of accuracy. Furthermore, it is to be mentioned here that some other types of technical problems [13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29] can be solved quite simply through this starting point.

Unlike many side benefits of the truncation technique, the lack of tHAM for solving many flow problems is pronounced yet. To this aim, an efficient tHAM for studying the laminar flow velocity distribution in an Eyring-Powell fluid model subjected to inclined magnetic field is developed here. The obtained results concerning the steady case are compared and validated with those simulations available in the open literature. It is shown that the tHAM can be considered as a powerful tool for deriving high-accuracy approximations.

2. Problem formulation

Consider a two-dimensional unsteady laminar flow past a continuous stretching sheet of velocity \(U_{w}=\frac{ax}{1-bt}\) where \(a>0\) and \(b\geq0\) are the initial stretching rate and unsteadiness coefficient, respectively. Here an inclined magnetic field strength \(\mathcal{B}\) is applied across the conducting fluid at an acute angle \(\theta\) below the free surface.

According to the Eyring-Powell fluid model, the differential equations of mass and linear momentum conservation for the assumed flow pattern can be expressed as,

\begin{equation*} u_{,x}+v_{,y}=0, \end{equation*} \begin{equation*} \dot{u}+uu_{,x}+vv_{,y}=\left(v+\frac{1}{\rho\Gamma\Lambda}\right)u_{,yy}-\frac{1}{2\rho\Gamma\Lambda^3}u^{2}_{,y}u_{,yy}-\frac{\sigma \mathcal{B}^2sin^{2}\theta}{\rho}u, \end{equation*} with the boundary conditions \begin{equation*} u=U_{w}(x,t)=\frac{ax}{1-bt} \ \ \ \ , v=V(x,t) \ \ \ \ \text{at} \ \ \ \ y=0, \end{equation*} \begin{equation*} \ \ \ \ \ \ \ u\rightarrow 0, \ \ \ \ \ v\rightarrow 0, \ \ \ \ \ \ \ \text{as} \ \ \ \ \ \ y\rightarrow \infty, \end{equation*} where \(v\) is the kinematic viscosity, \(\rho\) is the density, \(\Gamma\) and \(\Lambda\) are the material constants, \(\sigma\) is the electrical conductivity and \(V(x,t)\) is the mass transfer rate. It is to be noted that the comma sign and dot-superscript followed by independent variables signify the partial derivative involving \(\frac{\partial}{\partial x}\)(or \(\frac{\partial}{\partial x}\) ) and differentiation with respect to the time , respectively.

Introducing these variables \(\tau=y=\sqrt{\frac{a}{v(1-bt)}}\), \(u=\frac{ax}{1-bt}\varphi_{,\tau}\) and \(v=-\frac{av}{1-bt}\varphi\), the governing equations and associated boundary conditions are given by

\begin{equation*} (1+\lambda)\varphi_{,\tau\tau\tau}-\beta(\varphi_{,\tau}+\frac{1}{2}\tau\phi_{,\tau\tau})+\varphi\varphi_{,\tau\tau}-\varphi^{2}_{,\tau}-\lambda \delta \varphi^{2}_{,\tau\tau}\varphi_{,\tau\tau\tau}-\mathrm{M}^2sin^{2}\theta \varphi_{,\tau}=0 \end{equation*} \begin{equation*} \varphi=0, \ \ \ \ \varphi_{,\tau}=1, \ \ \ \ \text{at} \ \ \ \ \tau =0, \end{equation*} \begin{equation*} \varphi_{,\tau}=0 \ \ \ \ \ \ \ \ \ \ \ \ \text{as} \ \ \ \ \tau\rightarrow \infty, \end{equation*} where \(\lambda=\frac{1}{\rho\Gamma\Lambda}\) and \(\delta=\frac{a^3x^2}{2v\Lambda^2}\), are the Eyring-Powell fluid parameters, \(\beta=\frac{b}{a}\) is the unsteadiness parameter and \(\mathrm{M}=\sqrt{\frac{\sigma B^2}{\rho a}}\) is the magnetic field parameter.

Here the dimensionless quantity skin friction coefficient \(C_{f}\) is defined as,

\begin{equation*} \sqrt{Re_{x}}C_{f}=[(1+\lambda)\varphi_{,\tau\tau}(0)-\frac{1}{3}\lambda\delta\varphi^{3}_{,\tau\tau}(0)], \end{equation*} where \(Re_{x}=\frac{U_{w}x}{v}\) is the local Reynolds number.

3. Solution methodology

Let us denote the following nonlinear equation by, \begin{equation*} \mathcal{N}[\varphi(\tau)]=0, \end{equation*} where \(\mathcal{N}\) is a nonlinear operator. Using \(q\in[0,1 ]\) as an embedding parameter, the homotopy function \(\mathcal{H}\) is constructed as [30] where \(\hat{\varphi}\) is an unknown function, \(\hbar\neq 0\) is an auxiliary parameter, \(\mathcal{L}\neq 0\) is an auxiliary linear operator and \(\varphi_{0}\) is an initial guess of \(\varphi\). It is to be noted that in the limit as approaches 0 and 1, \(\hat{\varphi}(\tau,q)\) varies from the initial guess to the solution of \(\varphi(\tau )\). In fact, as follows from Equation (1), \(\hat{\varphi}(\tau; 0)=\varphi_{0}(\tau)\) and \(\hat{\varphi}(\tau; 1)=\varphi(\tau)\) are the solution of \(\mathcal{H}(\hat{\varphi;q,h})|_{q=0}=0\) and \(\mathcal{H}(\hat{\varphi;q,h})|_{q=1}=0\), respectively. By expanding \(\hat{\varphi}(\tau;q)\) n the Taylor's series with respect to, one would get \begin{equation*} \hat{\varphi}(\tau;q)=\hat{\varphi}(\tau;0)+\sum^{\infty}_{j=1}\frac{1}{j!}\hat{\varphi}^{(j)}_{,q}(\tau;q)|_{q=0}=0=\varphi_{0}(\tau)+\sum^{\infty}_{j=1}\varphi_{j} (\tau)q^{j}, \end{equation*} where \(\varphi_{j}\) is the jth-order deformation derivative. By setting \(\mathcal{H}(\hat{\varphi;q,h})\) and \(q\) equal to zero, the zeroth-order deformation equation is obtained as [30]; \begin{equation*} \mathcal{L}[\hat{\varphi}(\tau;0)-\varphi_{0}(\tau)]=0. \end{equation*} By differentiating \(\mathcal{H}(\hat{\varphi;q,h})\), \(j\) times with respect to \(q\), setting \(q=0\) and dividing it by \(j!\), after dropping the hats, the th-order deformation equation is constructed as \begin{equation*} \mathcal{L}[\varphi_{j}(\tau)-x_{j}\varphi_{j-1}(\tau)]+\frac{1}{(j-1)!}\hbar \mathcal{N}^{j-1}_{,q}[\varphi(\tau;q)]|_{q=0}=0, \end{equation*} where \begin{equation*} X_{j} = \left\{ \begin{array}{rl} 0 &\text{if }j\leq 1, \\ 1 &\text{if }j> 1. \\ \end{array} \right. \end{equation*} Here the initial guess and auxiliary linear operator are taken to be \begin{equation*} \varphi_{0}(\tau)=1-e^{-\tau}, \end{equation*} \begin{equation*} \mathcal{L}[\varphi(\tau;q)]=\varphi_{,\tau\tau\tau}(\tau,q)-\varphi_{,q}(\tau,q), \end{equation*} with the property \begin{equation*} \mathcal{L}[\alpha_{1}+\alpha_{2}e^{\tau}+\alpha_{3}e^{-\tau}]=0, \end{equation*} where \(\alpha_{1}-\alpha_{2}\) are the integration constants. Expanding \(\alpha(\tau;q)\) in the Taylor's series gives the following series expansion, \begin{equation*} \alpha(\tau;q)=\varphi_{0}(t)+q\varphi_{1}(t)+q^2\varphi_{2}(t)+\cdot\cdot\cdot. \end{equation*} The nonlinear operator in this case is given by \begin{eqnarray*} \mathcal{N}[\alpha(t;q)]&=&(1+\lambda)\varphi_{,\tau\tau\tau}(\tau,q)-\beta(\alpha_{,\tau}(\tau;q)+\frac{1}{2}\tau\varphi_{,\tau\tau}(\tau,q))-\varphi^{2}(\tau;q) \notag\\&&-\lambda\delta\varphi^{2}_{,\tau\tau}(\tau,q)\varphi_{,\tau\tau\tau}(\tau;q)-\mathcal{M}^2sin^2\theta\varphi_{,\tau}(\tau;q). \end{eqnarray*} The zeroth-order deformation equation and associated boundary conditions are written in the form \begin{equation*} \varphi_{,\tau\tau\tau}(\tau)-\varphi_{0,\tau}(\tau)=0, \end{equation*} \begin{equation*} \varphi(\tau;q)=0, \ \ \ \ \varphi_{,\tau}(\tau;q)=1\ \ \ \ \ at \ \ \ \ \tau=0, \end{equation*} \begin{equation*} \varphi_{,\tau}(\tau;q)\rightarrow \infty. \end{equation*} The th-order deformation equation is generated as which goes to zero boundary conditions.

To find a more explicit way of representing the th-order deformation Equation (2), it is required to express \(\frac{1}{j-1!}\hbar\mathcal{N}^{j-1}_{,q} [\varphi(\tau,q)]|_{q=0}\) by a linear combination of independent functions. With an inner product of any two independent functions in such a way that \(\psi_{m},\psi_{n}=\int^{\infty}_{0} (\tau)\psi_{m}(\tau)\psi_{n}(\tau)d\tau\) \(m\geq1\) and \(n\leq r\) [30] where \(k\) and \(\tau\) are the weight function and number of truncation, respectively. The Schmidt-Gram procedure [31] as well as the Kronecker delta functions can be applied to calculate \(r\). Therefore, \(\frac{1}{j-1!}\hbar\mathcal{N}^{j-1}_{,q} [\varphi(\tau,q)]|_{q=0}\) will be approximated by an orthonormal set of bases \(e_{1}, e_{2}, ... e_{r}\) at each point \(\tau\). It is to be noted here that after making this substitution and then solving the \(p-\)th order deformation Equation (2), the th-order approximate solution is achieved by

\begin{equation*} \varphi_{p}=\sum^{p}_{j=0}\varphi_{j}(\tau). \end{equation*} In theory, the square residual error at the th-order of approximation can be defined as [32]; \begin{equation*} \Delta_{p}=\frac{1}{d+1}\sum^{d}_{l=0}\left(\mathcal{N}\left[\sum^{p}_{z=0}\varphi_{z}(\tau)\right]\right)^2. \end{equation*}

4. Results and discussion

As it was mentioned earlier, employing the tHAM for those problems lie in the semi-infinite intervals is sufficient not just for a much faster rate of convergence, but also for the less obtained P-time. To do this, the geometric and physical properties discussed in Section 2, unless stated otherwise, are given in Table 1. It is to be noted here that if the order of approximation is selected as \(p=8\), the auxiliary parameter and square residual error take the same values as before. This fact is presented in Table 2.

Table 1. Geometric and physical properties of the fluid.

\(\lambda\)	\(\beta\)	\(\delta\)	\(\mathcal{M}\)	\(\theta\)
\(0.6\)	\(-0.5\)	\(0.1\)	\(0.2\)	\(45^{o}\) [1ex]

Table 2. Convergence and uniqueness of the series expansion by selecting an appropriate auxiliary parameter, when the amount of P-time is rounded up to two digits.

\(p\)	\(\hbar\)	\(\triangle_{p}\)	\(P-times(s)\)
\(5\)	\(-0.864\)	\(6.403 \times 10^{-9}\)	\(7.41\)
\(6\)	\(-0.871\)	\(5.179 \times 10^{-10}\)	\(15.66\)
\(7\)	\(-0.875\)	\(2.436 \times 10^{-11}\)	\(32.90\)
\(8\)	\(-0.875\)	\(2.436 \times 10^{-11}\)	\(73.75\)
\(9\)	\(-0.875\)	\(2.436 \times 10^{-11}\)	\(169.15\)
\(10\)	\(-0.875\)	\(2.436 \times 10^{-11}\)	\(406.24\)

In view of the results given in Table 2, the 8th-order tHAM converges rapidly for \(\hbar=-0.875\). However, due to the loop-like behavior of the tHAM, the P-time is greatly enhanced by increasing the order of approximation. According to the case reported by Zhao et al. [33], the main idea behind the truncation technique is to reduce the computation of series expansion and P-time made of (two or more) independent functions with a given \(k\). Since the number of truncation can simply be calculated as \(k=40\) [12] therefore, the square residual error will be minimized.

Table 3. Skin friction coefficient vs. those geometric and physical properties.

\(\lambda\)	\(\beta\)	\(\delta\)	\(\mathcal{M}\)	\(\theta\)	\([(1+\lambda)\varphi_{,\tau\tau}(0)-\frac{1}{3}\lambda\delta\varphi^{3}_{,\tau\tau}(0)]\)
\(0.6\)	\(-0.5\)	\(0.1\)	\(0.2\)	\(30^{o}\)	\(1.0191\)
\(0.7\)					\(0.9591\)
\(0.8\)					\(0.9262\)
\(0.8\)	\(-0.4\)				\(0.9280\)
	\(-0.3\)				\(0.9516\)
	\(-0.2\)				\(0.9733\)
\(0.8\)	\(-0.2\)	\(0.2\)			\(1.1170\)
		\(0.3\)			\(1.1046\)
		\(0.4\)			\(1.1776\)
\(0.8\)	\(-0.2\)	\(0.4\)	\(0.3\)		\(1.1046\)
			\(0.4\)		\(1.1053\)
			\(0.5\)		\(1.1061\)
\(0.8\)	\(-0.2\)	\(0.4\)	\(0.5\)	\(45^{o}\)	\(0.9970\)
				\(60^{o}\)	\(0.9811\)
				\(90^{o}\)	\(0.9684\)

A comparison of the local velocity distribution obtained by different solution methodologies with geometric and physical properties \(\lambda=0.3\) and \(\theta=90^{o}\) is shown in Figure 1. According to this Figure 1, the 8th-order tHAM gives accurate answers compared to those findings reported by Hayat et al. [34]; however, the order of approximation in the case analyzed by Hayat et al. [21] is radically different. Furthermore, the local velocity distribution for the same system without using the truncation is consistent with the tHAM results, but instead takes more P-time to consistency (i.e., 996.43s). It is worth noting that, based on the results reported in Table 2 and Figure 1, although the square residual error should be taken to optimize the value of auxiliary parameter, the only way that the P-time can reduce appears in the combination of truncation technique with the HAM.

Table 3 investigates the variation of skin friction coefficient versus different geometric and physical properties presented in Section 2. It is seen that the skin friction coefficient enhances when \(\beta\), \(\delta\) and \(\mathcal{M}\) are increased in any case. In contrast to the previous observation, the skin friction coefficient is a diminishing function of \(\theta\) and \(\lambda\). Hence, one can conclude that such findings involved in Table 3 are particularly useful for developing thermodynamic characteristics of an Eyring-Powell fluid model with slip velocity at the wall.

5. Conclusions

Utilizing the tHAM for analyzing the unsteady Eyring-Powell fluid embedded in a shrinking wall under inclined magnetic field was the main purpose of this article. Furthermore, the square residual error at each order of approximation was minimized to accelerate the rate of convergence for the present system. It was found that the 8th-order tHAM together with \(\hbar0.875\) and \(k=40\) can give high accurate results compared to any other solution methodology simply because the P-time in this case is taken as \(73.75s\).

6. Nomenclature

• \(u,v\) ; Velocity components along \(x\) and \(y\) axes, respectively \([ms^{-1}]\)
• \(\beta\) ; Magnetic field strength \([kgs^{-2}A^{-1}]\)
• \(U_{w}\) ; Velocity at the wall \([ms^{-1}]\)
• \(t\) ; time \([s]\)
• \(a\) ; Initial stretching rate \([a]\)
• \(b\) ; Unsteadiness coefficient \([b]\)
• \(V\) ; Mass transfer rate \([s^{-1}]\)
• \(\mathcal{M}\) ; Magnetic field parameter
• \(C_{f}\) ; Skin fiction coefficient
• \(Re_{x}\) ; Reynolds number

Greek symbols

• \(v\) ; Kinematic viscosity \([m^2s^{-1}]\)
• \(\rho\) ; Density \([kgm^{3}]\)
• \(\Gamma, \Lambda\) ; Material constants
• \(\sigma\) ; Electrical conductivity \([Sm^{-1}]\)
• \(\theta\) ; Inclined angle of magnetic field
• \(\eta\) ; Similarity variable
• \(\varphi\) ; Similarity function
• \(\lambda, \delta\) ; Eyring-Powell fluid parameter,
• \(\beta\) ; Unsteadiness parameter

Authorcontributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflictofinterests

The authors declare no conflict of interest.

1. Introduction

The calculus of time scales was initially developed by Stefan Hilger (see [1]). A time scale is an arbitrary nonempty closed subset of the real numbers. The three commonly known examples of calculus on time scales are differential calculus, difference calculus, and quantum calculus, i.e., when \(\mathbb{T}=\mathbb{R}\), \(\mathbb{T}=\mathbb{N}\) and \(\mathbb{T}=q^{\mathbb{N}_{0}}=\{q^{t}:t\in\mathbb{N}_{0}\}\) where \(q>1\). The time scales calculus is divided into delta calculus, nabla calculus and diamond--alpha calculus. During the last two decades, many researchers have established several dynamic inequalities (see [2,3,4,5,6,7,8,9,10]). The fundamental work on dynamic inequalities is done by Ravi Agarwal, George Anastassiou, Martin Bohner, Allan Peterson, Donal O'Regan, Samir Saker and many other researchers.

There have been recent developments and refinements of the theory and applications of dynamic inequalities on time scales. From the theoretical perspective, the work provides a coalition and amplification of conventional differential, difference and quantum equations. Moreover, it is a key mechanism in many mathematical, computational, biological, economical and numerical applications.

In this research article, it is accepted that all considerable integrals exist and are finite and \(\mathbb{T}\) denotes as usual the time scale, \(a,b\in\mathbb{T}\) with \(a< b\) and an interval \([a,b]_{\mathbb{T}}\) means the intersection of a real interval with the given time scale.

2. Preliminaries

We need here basic concepts of delta calculus. The results of delta calculus are adopted from monographs [6,11]. For \(t\in \mathbb{T}\), the forward jump operator \(\sigma:\mathbb{T} \rightarrow \mathbb{T}\) is defined by \begin{equation*}\sigma(t):=\inf\{s\in\mathbb{T}:s>t\}.\end{equation*} The mapping \(\mu:\mathbb{T}\rightarrow \mathbb{R}^{+}_{0}=[0,+\infty)\) such that \(\mu(t):=\sigma(t)-t\) is called the forward graininess function. The backward jump operator \(\rho:\mathbb{T} \rightarrow\mathbb{T}\) is defined by \begin{equation*}\rho(t) :=\sup \{s\in\mathbb{T}:s< t\}.\end{equation*} The mapping \(\nu:\mathbb{T} \rightarrow\mathbb{R}^{+}_{0}=[0,+\infty)\) such that \(\nu(t) :=t-\rho(t)\) is called the backward graininess function. If \(\sigma(t)>t\), we say that \(t\) is right--scattered, while if \(\rho(t)< t\), we say that \(t\) is left--scattered. Also, if \(t< \sup\mathbb{T}\) and \(\sigma(t)=t\), then \(t\) is called right--dense, and if \(t>\inf\mathbb{T}\) and \(\rho(t)=t\), then \(t\) is called left--dense. If \(\mathbb{T}\) has a left--scattered maximum \(M\), then \(\mathbb{T}^{k}=\mathbb{T}-\{M\}\), otherwise \(\mathbb{T}^{k}=\mathbb{T}\).

For a function \(f:\mathbb{T}\rightarrow \mathbb{R}\), the delta derivative \(f^{\Delta}\) is defined as follows; Let \(t\in\mathbb{T}^{k}\). If there exists \(f^{\Delta}(t)\in\mathbb{R}\) such that for all \(\epsilon>0\), there is a neighborhood \(U\) of \(t\), such that

\begin{equation*}|f(\sigma(t))-f(s)-f^{\Delta}(t)(\sigma(t)-s)|\leq\epsilon |\sigma(t)-s|,\end{equation*} for all \(s\in U\), then \(f\) is said to be delta differentiable at \(t\), and \(f^{\Delta}(t)\) is called the delta derivative of \(f\) at \(t\). A function \(f : \mathbb{T}\rightarrow\mathbb{R}\) is said to be right-dense continuous (rd-continuous), if it is continuous at each right-dense point and there exists a finite left-sided limit at every left-dense point. The set of all rd-continuous functions is denoted by \(C_{rd}(\mathbb{T},\mathbb{R})\).

The next definition is given in [6,11].

Definition 1. A function \(F : \mathbb{T}\rightarrow \mathbb{R}\) is called a delta antiderivative of \(f : \mathbb{T}\rightarrow \mathbb{R}\), provided that \(F^{\Delta}(t)=f(t)\) holds for all \(t\in \mathbb{T}^{k}\). Then the delta integral of \(f\) is defined by \begin{equation*}\int^{b}_{a} f(t)\Delta t=F(b)-F(a).\end{equation*} The following results of nabla calculus are taken from [6,11,12]. If \(\mathbb{T}\) has a right--scattered minimum \(m\), then \(\mathbb{T}_{k}=\mathbb{T}-\{m\}\), otherwise \(\mathbb{T}_{k}=\mathbb{T}\). A function \(f:\mathbb{T}_{k}\rightarrow\mathbb{R}\) is called nabla differentiable at \(t\in \mathbb{T}_{k}\), with nabla derivative \(f^{\nabla}(t)\), if there exists \(f^{\nabla}(t)\in \mathbb{R}\) such that given any \(\epsilon>0\), there is a neighborhood \(V\) of \(t\), such that \begin{equation*}|f(\rho(t))-f(s)-f^{\nabla}(t)(\rho(t)-s)|\leq \epsilon|\rho(t)-s|,\end{equation*} for all \(s\in V\).

A function \(f:\mathbb{T} \rightarrow\mathbb{R}\) is said to be left-dense continuous (ld-continuous), provided it is continuous at all left-dense points in \(\mathbb{T}\) and its right-sided limits exist (finite) at all right-dense points in \(\mathbb{T}\). The set of all ld-continuous functions is denoted by \(C_{ld}(\mathbb{T},\mathbb{R})\). The next definition is given in [6,11,12].

Definition 2. A function \(G : \mathbb{T}\rightarrow \mathbb{R}\) is called a nabla antiderivative of \(g : \mathbb{T}\rightarrow \mathbb{R}\), provided that \(G^{\nabla}(t)=g(t)\) holds for all \(t\in \mathbb{T}_{k}\). Then the nabla integral of \(g\) is defined by \begin{equation*}\int^{b}_{a} g(t)\nabla t=G(b)-G(a).\end{equation*}

The following definition is taken from [3,5].

Definition 3. For \(\alpha\geq 1\), the time scale \(\Delta\)-Riemann-Liouville type fractional integral for a function \(f\in C_{rd}\) is defined by

which is an integral on \([a,t)_{\mathbb{T}}\), see [13] and \(h_{\alpha}:\mathbb{T}\times \mathbb{T}\rightarrow \mathbb{R}\), \(\alpha\geq0\) are the coordinate wise rd--continuous functions, such that \(h_{0}(t,s)=1\), Notice that \begin{equation*} \mathcal{I}^{1}_{a}f(t)=\int^{t}_{a}f(\tau)\Delta \tau, \end{equation*} which is absolutely continuous in \(t\in [a,b]_{\mathbb{T}}\), see [13].

The following definition is taken from [4,5].

Definition 4. For \(\alpha\geq 1\), the time scale \(\nabla\)-Riemann-Liouville type fractional integral for a function \(f\in C_{ld}\) is defined by

which is an integral on \((a,t]_{\mathbb{T}}\), see [13] and \(\hat{h}_{\alpha}:\mathbb{T}\times \mathbb{T}\rightarrow \mathbb{R}\), \(\alpha\geq0\) are the coordinate wise ld--continuous functions, such that \(\hat{h}_{0}(t,s)=1\), Notice that \begin{equation*} \mathcal{J}^{1}_{a}f(t)=\int^{t}_{a}f(\tau)\nabla \tau, \end{equation*} which is absolutely continuous in \(t\in [a,b]_{\mathbb{T}}\), see [13].

3. Dynamic Young's inequality

In order to present our main results, first we give a straightforward proof for an extension of dynamic Young's inequalities by using the time scale \(\Delta\)-Riemann-Liouville type fractional integral.

Theorem 5. Let \(w,f,g\in C_{rd}\left([a,b]_{\mathbb{T}},\mathbb{R}\right)\) be \(\Delta\)-integrable functions and \(h_{\alpha-1}(.,.),\) \(h_{\beta-1}(.,.)>0\). If \(p,q>1\) with \(\frac{1}{p}+\frac{1}{q}=1\), then the following inequalities hold true for \(\alpha,\beta\geq1\):

and

Proof. For the proof of inequality (5), we set \(\psi=\frac{|f(y)|}{|g(y)|}\) and \(\omega=\frac{|f(z)|}{|g(z)|}\), \(|g(y)|,|g(z)|\neq 0\), \(y,z\in[a,x]_{\mathbb{T}}\), \(\forall x\in[a,b]_{\mathbb{T}}\), in the classical Young's inequality \(\psi\omega\leq\frac{\psi^{p}}{p}+\frac{\omega^{q}}{q}\), \(\psi,\omega\geq0\), we obtain

Multiplying inequality (8) by \(h_{\alpha-1}(x,\sigma(y))h_{\beta-1}(x,\sigma(z))|w(y)w(z)|\), \(y,z\in[a,x)_{\mathbb{T}}\), \(\forall x\in[a,b]_{\mathbb{T}}\) on both sides and double integrating over \(y\) and \(z\), respectively, from \(a\) to \(x\), we get Inequality (5) follows from inequality (9).

For the proof of inequality (6), we set \(\psi=\frac{|f(y)|}{|f(z)|}\) and \(\omega=\frac{|g(y)|}{|g(z)|}\), \(|f(z)|,|g(z)|\neq 0\), \(y,z\in[a,x]_{\mathbb{T}}\), \(\forall x\in[a,b]_{\mathbb{T}}\), in the classical Young's inequality \(\psi\omega\leq\frac{\psi^{p}}{p}+\frac{\omega^{q}}{q}\), \(\psi,\omega\geq0\) and following the same steps used in the proof of inequality (5), we obtain the desired result.

Now, for the proof of inequality (7), we set \(\psi=|f(y)g(z)|\) and \(\omega=|f(z)g(y)|\), \(y,z\in[a,x]_{\mathbb{T}}\), \(\forall x\in[a,b]_{\mathbb{T}}\), in the classical Young's inequality \(\psi\omega\leq\frac{\psi^{p}}{p}+\frac{\omega^{q}}{q}\), \(\psi,\omega\geq0\) and following the same steps used in the proof of inequality (5), we obtain the desired result. This completes the proof of Theorem 5.

Next, we give a straightforward proof for an extension of dynamic Young's inequalities by using the time scale \(\nabla\)-Riemann-Liouville type fractional integral.

Theorem 6. Let \(w,f,g\in C_{ld}\left([a,b]_{\mathbb{T}},\mathbb{R}\right)\) be \(\nabla\)-integrable functions and \(\hat{h}_{\alpha-1}(.,.),\) \(\hat{h}_{\beta-1}(.,.)>0\). If \(p,q>1\) with \(\frac{1}{p}+\frac{1}{q}=1\), then the following inequalities hold true for \(\alpha,\beta\geq1\):

and

Proof. Similar to the proof of Theorem 5.

Remark 1. Let \(\alpha=\beta=1\), \(\mathbb{T}=\mathbb{Z}\), \(a=1\), \(x=b=n+1\), \(w\equiv1\), \(f(k)=x_{k}\in[0,+\infty)\) and \(g(k)=y_{k}\in[0,+\infty)\) for \(k= 1,2,\ldots,n\). Then inequalities (5), (6) and (7) become

and We give an extension of more dynamic Young's inequalities by using the time scale \(\Delta\)-Riemann-Liouville type fractional integral.

Theorem 7. Let \(w,f,g\in C_{rd}\left([a,b]_{\mathbb{T}},\mathbb{R}\right)\) be \(\Delta\)-integrable functions and \(h_{\alpha-1}(.,.),\) \(h_{\beta-1}(.,.)>0\). If \(p,q>1\) with \(\frac{1}{p}+\frac{1}{q}=1\), then the following inequalities hold true for \(\alpha,\beta\geq1\):

and

Proof. For the proof of inequality (16), we set \(\psi=|f(y)||g(z)|^{\frac{2}{p}}\) and \(\omega=|f(z)|^{\frac{2}{q}}|g(y)|\), \(y,z\in[a,x]_{\mathbb{T}}\), \(\forall x\in[a,b]_{\mathbb{T}}\), in the classical Young's inequality \(\psi\omega\leq\frac{\psi^{p}}{p}+\frac{\omega^{q}}{q}\), \(\psi,\omega\geq0\), we obtain

Multiplying (19) by \(h_{\alpha-1}(x,\sigma(y))h_{\beta-1}(x,\sigma(z))|w(y)w(z)|\), \(y,z\in[a,x)_{\mathbb{T}}\), \(\forall x\in[a,b]_{\mathbb{T}}\) on both sides and double integrating over \(y\) and \(z\), respectively, from \(a\) to \(x\), we get Inequality (16) follows from inequality (20).

For the proof of inequality (17), we set \(\psi=\frac{|f(y)|^{\frac{2}{p}}}{|f(z)|}\) and \(\omega=\frac{|g(y)|^{\frac{2}{q}}}{|g(z)|}\), \(|f(z)|,|g(z)|\) \(\neq0\), \(y,z\in[a,x]_{\mathbb{T}}\), \(\forall x\in[a,b]_{\mathbb{T}}\), in the classical Young's inequality \(\psi\omega\leq\frac{\psi^{p}}{p}+\frac{\omega^{q}}{q}\), \(\psi,\omega\geq0\) and following the same steps used in the proof of inequality (16), we obtain the desired result.

Now, for the proof of inequality (18), we set \(\psi=|f(y)|^{\frac{2}{p}}|g(z)|\) and \(\omega=|f(z)|^{\frac{2}{q}}|g(y)|\), \(y,z\in[a,x]_{\mathbb{T}}\), \(\forall x\in[a,b]_{\mathbb{T}}\), in the classical Young's inequality \(\psi\omega\leq\frac{\psi^{p}}{p}+\frac{\omega^{q}}{q}\), \(\psi,\omega\geq0\) and following the same steps used in the proof of inequality (16), we obtain the desired result. This completes the proof of Theorem 7.

Next, we give an extension of more dynamic Young's inequalities by using the time scale \(\nabla\)-Riemann-Liouville type fractional integral.

Theorem 8. Let \(w,f,g\in C_{ld}\left([a,b]_{\mathbb{T}},\mathbb{R}\right)\) be \(\nabla\)-integrable functions and \(\hat{h}_{\alpha-1}(.,.),\) \(\hat{h}_{\beta-1}(.,.)>0\). If \(p,q>1\) with \(\frac{1}{p}+\frac{1}{q}=1\), then the following inequalities hold true for \(\alpha,\beta\geq1\):

and

Proof. Similar to the proof of Theorem 7.

Remark 2. Let \(\alpha=\beta=1\), \(\mathbb{T}=\mathbb{Z}\), \(a=1\), \(x=b=n+1\), \(w\equiv1\), \(f(k)=x_{k}\in[0,+\infty)\) and \(g(k)=y_{k}\in[0,+\infty)\) for \(k= 1,2,\ldots,n\). Then inequalities (16), (17) and (18) become

and

4. Dynamic Qi's inequality

In this section, we give an extension of dynamic Qi's inequalities by using the time scale \(\Delta\)-Riemann-Liouville type fractional integral.

Theorem 9. Let \(w,f,g,h\in C_{rd}\left([a,b]_{\mathbb{T}},\mathbb{R}-\{0\}\right)\) be \(\Delta\)-integrable functions with \(0< m\leq\frac{|f(y)|}{|g(y)|}\leq M< \infty\) on \([a,x]_{\mathbb{T}}\), \(\forall x\in[a,b]_{\mathbb{T}}\) satisfying \(|f(y)|^{\frac{1}{p}}|g(y)|^{\frac{1}{q}}|h(y)|^{\frac{1}{r}}=c\), where \(c\) is a positive real number. Assume further that \(\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=0\), \(h_{\alpha-1}(.,.)>0\) and \(\alpha\geq1\).

• (i) If \(p>0\), \(q>0\), \(r< 0\), then
  \begin{equation} \label{4.1} \left\{\mathcal{I}^{\alpha}_{a}\left(|w(x)||f(x)|\right)\right\}^{\frac{1}{q}} \left\{\mathcal{I}^{\alpha}_{a}\left(|w(x)||g(x)|\right)\right\}^{\frac{1}{p}} \left\{\mathcal{I}^{\alpha}_{a}\left(|w(x)||h(x)|\right)\right\}^{\frac{1}{r}} \geq c\left(\frac{m^{\frac{1}{q^{2}}}}{M^{\frac{1}{p^{2}}}}\right)^{-r}. \end{equation}

  (27)
• (ii) If \(p< 0\), \(q< 0\), \(r> 0\), then
  \begin{equation} \left\{\mathcal{I}^{\alpha}_{a}\left(|w(x)||f(x)|\right)\right\}^{\frac{1}{q}} \left\{\mathcal{I}^{\alpha}_{a}\left(|w(x)||g(x)|\right)\right\}^{\frac{1}{p}} \left\{\mathcal{I}^{\alpha}_{a}\left(|w(x)||h(x)|\right)\right\}^{\frac{1}{r}} \leq c\left(\frac{m^{\frac{1}{q^{2}}}}{M^{\frac{1}{p^{2}}}}\right)^{-r}. \end{equation}

  (28)

Proof. Case (i). The given condition \(\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=0\) can be rearranged to yield \(\frac{1}{\left(-\frac{p}{r}\right)}+\frac{1}{\left(-\frac{q}{r}\right)}=1\). Applying dynamic Rogers--Hölder's inequality [2] for \(-\frac{p}{r}>1\) and \(-\frac{q}{r}>1\), we get

From (29), we have that From the given condition, we obtain \begin{equation*} |f(y)|^{-\frac{r}{p}}\leq \left(M|g(y)|\right)^{-\frac{r}{p}},~~~~~~|g(y)|^{-\frac{r}{q}}\leq m^{\frac{r}{q}}|f(y)|^{-\frac{r}{q}}, \end{equation*} on the set \([a,x]_{\mathbb{T}}\), \(\forall x\in[a,b]_{\mathbb{T}}\). From the above inequality, it follows that Therefore Again, applying dynamic Rogers-Hölder's inequality on the right-hand side of inequality (32), we obtain Using the condition that \(|f(y)|^{\frac{1}{p}}|g(y)|^{\frac{1}{q}}|h(y)|^{\frac{1}{r}}=c\), where \(c\) is a positive real number and \(y\in[a,x]_{\mathbb{T}}\), \(\forall x\in[a,b]_{\mathbb{T}}\), the inequality (33) becomes Taking power \(-\frac{1}{r}>0\) on both sides of inequality (34) and replacing \(|w(y)|\) by \(h_{\alpha-1}(x,\sigma(y))|w(y)|\), \(y\in[a,x)_{\mathbb{T}}\), \(\forall x\in[a,b]_{\mathbb{T}}\), we obtain the desired inequality (27).

The proof of Case (ii) is similar to that of Case (i). This completes the proof of Theorem 9.

Next, we give an extension of dynamic Qi's inequalities by using the time scale \(\nabla\)-Riemann-Liouville type fractional integral.

Theorem 10. Let \(w,f,g,h\in C_{ld}\left([a,b]_{\mathbb{T}},\mathbb{R}-\{0\}\right)\) be \(\nabla\)-integrable functions with \(0< m\leq\frac{|f(y)|}{|g(y)|}\leq M< \infty\) on \([a,x]_{\mathbb{T}}\), \(\forall x\in[a,b]_{\mathbb{T}}\) satisfying \(|f(y)|^{\frac{1}{p}}|g(y)|^{\frac{1}{q}}|h(y)|^{\frac{1}{r}}=c\), where \(c\) is a positive real number. Assume further that \(\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=0\), \(\hat{h}_{\alpha-1}(.,.)>0\) and \(\alpha\geq1\).

• (i) If \(p>0\), \(q>0\), \(r< 0\), then
  \begin{equation} \left\{\mathcal{J}^{\alpha}_{a}\left(|w(x)||f(x)|\right)\right\}^{\frac{1}{q}} \left\{\mathcal{J}^{\alpha}_{a}\left(|w(x)||g(x)|\right)\right\}^{\frac{1}{p}} \left\{\mathcal{J}^{\alpha}_{a}\left(|w(x)||h(x)|\right)\right\}^{\frac{1}{r}} \geq c\left(\frac{m^{\frac{1}{q^{2}}}}{M^{\frac{1}{p^{2}}}}\right)^{-r}. \end{equation}

  (35)
• (ii) If \(p< 0\), \(q< 0\), \(r>0\), then
  \begin{equation} \left\{\mathcal{J}^{\alpha}_{a}\left(|w(x)||f(x)|\right)\right\}^{\frac{1}{q}} \left\{\mathcal{J}^{\alpha}_{a}\left(|w(x)||g(x)|\right)\right\}^{\frac{1}{p}} \left\{\mathcal{J}^{\alpha}_{a}\left(|w(x)||h(x)|\right)\right\}^{\frac{1}{r}} \leq c\left(\frac{m^{\frac{1}{q^{2}}}}{M^{\frac{1}{p^{2}}}}\right)^{-r}. \end{equation}

  (36)

Proof. Similar to the proof of Theorem 9.

Remark 3. Let \(\alpha=1\), \(\mathbb{T}=\mathbb{R}\), \(x=b\), \(r=-1\), \(c=1\), \(w\equiv1\) and \(f(y),g(y)\in(0,+\infty)\), \(\forall y\in[a,b]\). Then inequality (27) reduces to

The inequality (37) can be found in [14].

Remark 4. Let \(\alpha=1\), \(x=b\), \(r=-1\), \(c=1\), \(w\equiv1\) and \(f(y),g(y)\in(0,+\infty)\), \(\forall y\in[a,b]_{\mathbb{T}}\). Then inequality (27) reduces to

The inequality (38) may be found in [10].

Corollary 1. Let \(x_{k},y_{k},z_{k}\in(0,+\infty)\) with \(0 < m\leq\frac{x_{k}}{y_{k}}\leq M< \infty\) for \(k\in\{1,2,\ldots,n\}\) satisfying \(x^{\frac{1}{p}}_{k}y^{\frac{1}{q}}_{k}z^{\frac{1}{r}}_{k}=c\), where \(c\) is a positive real number. Assume further that \(\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=0\), \(p,q,r\in\mathbb{R}-\{0\}\).

• (i) If \(p>0\), \(q>0\), \(r< 0\), then
  \begin{equation} \label{4.13} \left(\sum^{n}_{k=1}x_{k}\right)^{\frac{1}{q}} \left(\sum^{n}_{k=1}y_{k}\right)^{\frac{1}{p}} \left(\sum^{n}_{k=1}z_{k}\right)^{\frac{1}{r}} \geq c\left(\frac{m^{\frac{1}{q^{2}}}}{M^{\frac{1}{p^{2}}}}\right)^{-r}. \end{equation}

  (39)
• (ii) If \(p< 0\), \(q< 0\), \(r> 0\), then
  \begin{equation} \label{4.14} \left(\sum^{n}_{k=1}x_{k}\right)^{\frac{1}{q}} \left(\sum^{n}_{k=1}y_{k}\right)^{\frac{1}{p}} \left(\sum^{n}_{k=1}z_{k}\right)^{\frac{1}{r}} \leq c\left(\frac{m^{\frac{1}{q^{2}}}}{M^{\frac{1}{p^{2}}}}\right)^{-r}. \end{equation}

  (40)

Proof. Putting \(\alpha=1\), \(\mathbb{T}=\mathbb{Z}\), \(a=1\), \(x=b=n+1\) and \(w\equiv1\) in Theorem 9, we obtain the inequalities (39) and (40).

5. Conclusion

Young's inequalities on fractional calculus by means of generalized fractional integrals can be found in [15]. Such inequalities on fractional calculus by Hadamard fractional integral operator can be found in [16]. Motivated by the work, we have obtained dynamic Young's inequalities on fractional calculus of time scales, which has become a significant way in pure and applied mathematics. We have also developed several versions of dynamic Qi's inequalities on fractional calculus of time scales.

Conflict of Interests

The author declares no conflict of interest.

1. Introduction and Preliminaries

The concept of dislocated metric space was introduced by Hitzler [1] in an effort to generalize the well known Banach contraction principle. Later his work was generalized by Zeyada [2] and many papers covering fixed point results for a single and a pair of mappings satisfying various types of contraction conditions are also published, see [2,3,4]. Similarly, Bhaskar and Lakshmikantham [5] introduced the concept of coupled fixed point for non-linear contractions in partially ordered metric spaces. After wards, Lakshmikantham and Ciric [6] proved coupled coincidence and coupled common fixed point theorems for nonlinear contractive mappings in a complete partially ordered metric space. This area of research has attracted the interest of many researchers and a number of works has been published in different spaces, see [7,8,9,10]. Most recently, Mohammad et al., [11] has obtained coupled fixed point finding in the context of dislocated quasi metric space. In this paper, we have established and proved existence and uniqueness of coupled coincidence and coupled common fixed points for a pair of maps in the context of dislocated quasi metric spaces.

2. Preliminaries

Now, we present relevant definitions and results that will be retrieved in the sequel and throughout this paper \(\Re^{+}\) will denote the set of non negative real numbers.

Definition 1. [1] Let \(X\) be a non-empty set and let \(d : X\times X\rightarrow \Re^{+}\cup \{0\}\) be a function satisfying the conditions

• (i) \(d(x,y)=d(y,x)=0 \Rightarrow x=y.\)
• (ii) \(d(x,y)\leq d(x,z)+d(z,y)\) for all \(x,y,z \in X.\)

Then \(d\) is known as dislocated quasi-metric on \(X\) and the pair \((X,d)\) is called a dislocated quasi-metric space.

Definition 2. [2] A sequence \(\{x_n\}\) in a dislocated quasi metric space \((X, d)\) is said to converge to a point \(x \in X\) if and only if \(\lim\limits_{n\to\infty} d(x_{n}, x)=\lim\limits_{n\to\infty} d(x,x_{n}) = 0.\)

Definition 3. [2] A sequence \(\{x_n\}\) in a dislocated quasi metric space \((X, d)\) is called a Cauchy sequence if for every \(\epsilon>0\), there exists a positive integer \(n_{0}\) such that for \(m,n>n_{0}\), we have \(d(x_{n},x_{m})< \epsilon\). That is, \(\lim\limits_{n,m\to\infty} d(x_{n}, x_{m})=0.\)

Definition 4. [2] A dislocated quasi metric space is called complete if every Cauchy sequence converges to an element in the same metric space.

Definition 5. [12] Let \((X,d)\) be a metric space and \(T : X \rightarrow X\) be a self-map, then \(T\) is said to be a contraction mapping if there exists a constant \(k \in [0, 1)\) called a contraction factor, such that \(d(Tx,Ty) \leq kd(x,y)\)for all \(x,y \in X.\)

Definition 6. [12] Let X be a nonempty set and \(T : X \rightarrow X\) a self-map. We say that x is a fixed point of T if Tx = x.

Theorem 7. [12] Suppose \((X,d)\) be a complete metric space and \(T : X\rightarrow X\) be a contraction, then \(T\) has a unique fixed point.

Definition 8.[5] An element \((x,y) \in X\times X\) , where \(X\) is any non-empty set, is called a coupled fixed point of the mapping \(F:X\times X \rightarrow X\) if \(F(x,y)=x\) and \(F(y,x)=y.\)

Definition 9. [6] An element \((x, y) \in X \times X\) is called a coupled coincidence point of the mappings \(F:X \times X \rightarrow X\) and \(g : X \rightarrow X\) if F(x, y) = g(x) and F(y, x) = g(y), and \((gx,gy)\) is called coupled point of coincidence.

Definition 10. [6] An element \((x,y) \in X\times X\), where \(X\) is any non-empty set, is called a coupled common fixed point of the mappings \(F:X\times X \rightarrow X\) and and \(g : X \rightarrow X\) if \(F(x,y)=g(x)=x\) and \(F(y,x)=g(y)=y\).

Definition 11. [6] The mappings \(F : X \times X \rightarrow X\) and \(g : X \rightarrow X\) are called commutative if \(g(F(x, y)) = F(gx, gy)\) for all \(x,y \in X\).

Definition 12. [6] The mappings \(F : X \times X \rightarrow X\) and \(g : X \rightarrow X\) are called w-Compatible if \(g(F(x, y)) = F(gx, gy)\) and \(g(F(y, x)) = F(gy, gx)\) whenever \(gx = F(x, y)\) and \(gy = F(y, x)\).

Theorem 13. [11] Let \((X,d)\) be a complete dislocated quasi-metric space and \(T : X \rightarrow X\) be a continuous mapping satisfying the following rational type contractive condition \begin{align*} &d[T(x,y),T(u,v)] \leq a_{1}\left[d(x,u)+d(y,v)\right]+a_{2}\left[d\left(x,T(x,y)\right)+d(u,T(u,v))\right]+a_{3}\left[d\left(x,T(u,v)\right)+d\left(u,T(x,y)\right)\right]\\ &\ +a_{4}\left[\frac{d\left(x,T(x,y)\right)d\left(u,T(u,v)\right)}{d(x,u)+d(y,v)}\right] +a_{5}\left[\frac{\left(d(x,u)+d(y,v)\right)\times\left(d\left(x,T(x,y)\right)+d\left(u,T(u,v)\right)\right)}{1+d(x,u)+d(y,v)}\right]\\ &\ +a_{6}\left[\frac{d\left(x,T(x,y)\right)+d\left(x,T(u,v)\right)}{1+d\left(u,T(u,v)\right)d\left(u,T(x,y)\right)}\right] \end{align*} for all \(x,y,u,v \in X\) and \(a_{1}, a_{2}, a_{3},a_{4}, a_{5}\), and \(a_{6}\) are non-negative constants with \(2(a_{1}+a_{2}+a_{5})+4(a_{3}+a_{6})+a_{4}< 1\), then \(T\) has a unique coupled fixed point in \(X\times X\).

3. Main results

At this stage, we state our theorem and come up with the main findings.

Theorem 14. Let \((X,d)\) be a dislocated quasi-metric space and \(T : X\times X \rightarrow X\) and \(g : X \rightarrow X\) be a continuous and commutative mappings satisfying the following rational type contractive condition

where \(x,y,u,v \in X\) and \(a_{1}, a_{2}, a_{3},a_{4}, a_{5}, a_{6}, a_{7} \in \Re^{+}\) with \(2(a_{1}+a_{2}+a_{5})+4(a_{3}+a_{6})+a_{4}+a_{7}< 1\), \(T(X\times X) \subseteq g(X)\), and \(g(X)\) is complete, then \(T\) and \(g\) have a unique coupled coincidence point. Moreover, if \(T\) and \(g\) are weakly compatible, then \(T\) and \(g\) have unique coupled common fixed point of the form \((u, u).\)

Proof. Let \(x_{0}\) and \(y_{0} \in X\) and set \(gx_{1}=T(x_{0},y_{0}) \mbox{ and }gy_{1}=T(y_{0},x_{0}).\) This is possible since \(T(X\times X) \subseteq g(X)\). Proceeding this way, we can construct two sequences \(\{x_{n}\}\) and \(\{y_{n}\}\) in \(X\) such that \(gx_{n+1}=T(x_{n},y_{n}) \mbox{ and } gy_{n+1}=T(y_{n},x_{n}).\) Consider \(d(gx_{n},gx_{n+1})=d\bigl[T(x_{n-1},y_{n-1}),T(x_{n},y_{n})\bigr].\) This is in order to show that \(\{gx_{n}\}\) and \(\{gy_{n}\}\) are Cauchy sequences in \(g(X)\). Now applying (1), we get \begin{eqnarray*} && d(gx_{n},gx_{n+1}) \leq a_{1} \left[d(gx_{n-1},gx_{n})+d(gy_{n-1},gy_{n})\right] +a_{2}\left[d(gx_{n-1},T(x_{n-1},y_{n-1}))+d(gx_{n},T(x_{n},y_{n}))\right]\nonumber\\&&{} +a_{3}\left[d(gx_{n-1},T(x_{n},y_{n}))+d(gx_{n},T(x_{n-1},y_{n-1}))\right]+a_{4}\left[\frac{d(gx_{n-1},T(x_{n-1},y_{n-1}))d(gx_{n},T(x_{n},y_{n}))}{d(gx_{n-1},gx_{n}) +d(gy_{n-1},gy_{n})}\right]\nonumber\\&&{} +a_{5}\left[\frac{\left(d(gx_{n-1},gx_{n})+d(gy_{n-1},gy_{n})\right)\times \left(d\bigl(gx_{n-1},T(x_{n-1},y_{n-1})) +d\left(gx_{n},T(x_{n},y_{n})\right)\right)}{1+d(gx_{n-1},gx_{n})+d(gy_{n-1},gy_{n})}\right]\nonumber\\&&{} +a_{6}\left[\frac{d(gx_{n-1},T(x_{n-1},y_{n-1}))+d(gx_{n-1},T(x_{n},y_{n}))}{1+d(gx_{n},T(x_{n},y_{n})) d(gx_{n},T(x_{n-1},y_{n-1}))}\right] +a_{7}\left[\frac{d(gx_{n-1},T(x_{n-1},y_{n-1}))d(gx_{n},T(x_{n},y_{n}))}{1+d(gx_{n-1},gx_{n}) +d(gx_{n},T(x_{n},y_{n}))}\right]. \end{eqnarray*} At this point, we are going to make use of the definitions of the sequences \(\{gx_{n}\}\) and \(\{gy_{n}\}\) to get \begin{eqnarray*} && d(gx_{n},gx_{n+1}) \leq a_{1} \left[d(gx_{n-1},gx_{n})+d(gy_{n-1},gy_{n})\right]+a_{2}\left[d(gx_{n-1},gx_{n})+d(gx_{n},gx_{n+1})\right]\nonumber\\&&{} +a_{3}\left[d(gx_{n-1},gx_{n+1})+d(gx_{n},gx_{n})\right] +a_{4}\left[\frac{d(gx_{n-1},gx_{n})d(gx_{n},gx_{n+1})}{d(gx_{n-1},gx_{n})+d(gy_{n-1},gy_{n})}\right]\nonumber\\&&{} +a_{5}\left[\frac{(d(gx_{n-1},gx_{n})+d(gy_{n-1},gy_{n}))\times(d(gx_{n-1},gx_{n})+d(gx_{n},gx_{n+1}))} {1+d(gx_{n-1},gx_{n})+(gy_{n-1},gy_{n})}\right]\nonumber\\&&{} +a_{6}\left[\frac{d(gx_{n-1},gx_{n})+d(gx_{n-1},gx_{n+1})}{1+d(gx_{n},gx_{n+1})d(gx_{n},gx_{n})}\right] +a_{7}\left[\frac{d(gx_{n-1},gx_{n})d(gx_{n},gx_{n+1})}{1+d(gx_{n-1},gx_{n})+d(gx_{n},gx_{n+1})}\right]. \end{eqnarray*} Applying the triangle inequality and the fact that \(d(x,y) \geq 0\), we obtain \begin{eqnarray*} && d(gx_{n},gx_{n+1}) \leq a_{1} \bigl[d(gx_{n-1},gx_{n})+d(gy_{n-1},gy_{n})\bigr]+a_{2}\bigl[d(gx_{n-1},gx_{n})+d(gx_{n},gx_{n+1})\bigr]\nonumber\\ &&{}+a_{3}\bigl[d(gx_{n-1},gx_{n})+d(gx_{n},gx_{n+1})+d(gx_{n-1},gx_{n})+d(gx_{n},gx_{n+1})\bigr]+a_{4}d(gx_{n},gx_{n+1})\nonumber\\ &&{}+a_{5}\bigl[d(gx_{n-1},gx_{n})+d(gx_{n},gx_{n+1})\bigr]+a_{6}\bigl[d(gx_{n-1},gx_{n})+d(gx_{n},gx_{n+1})+d(gx_{n-1},gx_{n})\nonumber\\ &&{}+d(gx_{n},gx_{n+1})\bigr]+a_{7}d(gx_{n-1},gx_{n}). \end{eqnarray*} Simplification yields \[ \alpha d(gx_{n},gx_{n+1}) \leq \beta d(gx_{n-1},gx_{n})+a_{1} d(gy_{n-1},gy_{n}) \] where \(\alpha = 1-(a_{2}+2a_{3}+a_{4}+a_{5}+2a_{6}),\) and \(\beta = a_{1}+a_{2}+2a_{3}+a_{5}+2a_{6}+a_{7}.\) It follows that

where \( \eta = \frac{\beta}{\alpha}\) and \(\theta =\frac{a_{1}}{\alpha}\).

Similarly, we can show that

Adding (2) and (3), we get where \( \lambda=\eta +\theta\). Similarly, we have \begin{eqnarray} \bigl[d(gx_{n-1},gx_{n})+d(gy_{n-1},gy_{n})\bigr] &\leq& \lambda\bigl[d(gx_{n-2},gx_{n-1})+d(gy_{n-2},gy_{n-1})\bigr]. \nonumber \end{eqnarray} Also \begin{eqnarray} \bigl[d(gx_{n-1},gx_{n})+d(gy_{n-1},gy_{n})\bigr] &\leq& \lambda^2\bigl[d(gx_{n-3},gx_{n-2})+d(gy_{n-3},gy_{n-2})\bigr]. \nonumber \end{eqnarray} Continuing this procedure, we obtain Since \(0< \lambda < 1\), we have \(\lambda^{n}\rightarrow 0\) as \(n\rightarrow \infty\) and \(\bigl[d(gx_{n},gx_{n+1})+d(gy_{n},gy_{n+1})\bigr]\rightarrow 0.\) So \(d(gx_{n},gx_{n+1})\rightarrow 0\) and \(d(gy_{n},gy_{n+1})\rightarrow 0\). Applying triangle inequality, using (5), and letting \(m>n\geq 1\), it follows that \begin{eqnarray*} &&[d(gx_{n},gx_{m})+d(gy_{n},gy_{m})] \leq [d(gx_{n},gx_{n+1})+d(gx_{n+1},gx_{n+2})+d(gx_{n+2},gx_{n+3})+...+d(gx_{m-1},gx_{m}) \\&&\;\;+[d(gy_{n},gy_{n+1})+d(gy_{n+1},gy_{n+2})+d(gy_{n+2},gy_{n+3})+...+d(gy_{m-1},gy_{m})\\ &&\leq \lambda^{n}\left[d(gx_{0},gx_{1})+d(gy_{0},gy_{1})\right]+\lambda^{n+1}[d(gx_{0},gx_{1})+d(gy_{0},gy_{1}) +\lambda^{n+2}[d(gx_{0},gx_{1})+d(gy_{0},gy_{1})+...\\&& \;\;+\lambda^{m-1}[d(gx_{0},gx_{1})+d(gy_{0},gy_{1})\leq \frac{\lambda^{n}}{1-\lambda}[d(gx_{0},gx_{1})+d(gy_{0},gy_{1})]. \end{eqnarray*} It follows that \(\bigl[d(gx_{n},gx_{m})+d(gy_{n},gy_{m})\bigr]\rightarrow 0\) as \(n,m\rightarrow \infty\) Hence \(d(gx_{n},gx_{m})\rightarrow 0\) and \(d(gy_{n},gy_{m})\rightarrow 0\) as \(n,m\rightarrow \infty.\) Thus, \(\{gx_{n}\}\) and \(\{gy_{n}\}\) are Cauchy sequences in \(g( X)\). By completeness of \( g(X)\;\exists\; x,y \in g( X) \) such that \(\{gx_{n}\}\) and \(\{gy_{n}\}\) converge to \(x\) and \(y\) respectively.Now, we prove that \(T(x,y)=gx\) and \(T(y,x)=gy\). Since \(T\) and \(g\) are commuting, it follows that Using (6) and continuity of \(T\) and \(g\), we have \(\lim\limits_{n\to\infty}ggx_{n}=\lim\limits_{n\to\infty}T(gx_{n},gy_{n}),\) implies \(g\left(\lim\limits_{n\to\infty}gx_{n}\right)=T\left(\lim\limits_{n\to\infty}gx_{n},\lim\limits_{n\to\infty} gy_{n}\right).\) Hence \(g(x)=T(x,y).\)

Similarly, we can show that \(g(y)=T(y,x)\). Hence, \((gx,gy)\) is coupled point of coincidence of \(T\) and \(g\). Now, we claim that \((gx,gy)\) is the unique coupled point of coincidence of \(T\) and \(g\). Suppose, we have another coupled point of coincidence say \((gx_{1},gy_{1})\) where \((x_{1},y_{1}) \in X^{2}\) with \(gx_{1}=T(x_{1},y_{1})\) and \(gy_{1}=T(y_{1},x_{1})\).

Using (1), we have \begin{align*} &d(gx,gx) = d\left[T(x,y),T(x,y)\right] \leq a_{1}\left[d(gx,gx)+d(gy,gy)\right]+a_{2}\left[d(gx,gx)+d(gx,gx)\right] +a_{3}\left[d(gx,gx)+d(gx,gx)\right]\\&{} +a_{4}\left[\frac{d(gx,gx)d(gx,gx)}{d(gx,gx)+d(gy,gy)}\right]+a_{5}\left[\frac{[d(gx,gx)+d(gy,gy)][d(gx,gx)+d(gx,gx)]}{1+d(gx,gx)+d(gy,gy)}\right]+a_{6}\left[\frac{d(gx,gx)+d(gx,gx)} {1+d(gx,gx)d(gx,gx)}\right]\\&{} +a_{7}\left[\frac{d(gx,gx)d(gx,gx)}{1+d(gx,gx)+d(gx,gx)}\right]. \end{align*} Now, we have \begin{align*} & d(gx,gx)\leq a_{1}\bigl[d(gx,gx)+d(gy,gy)\bigr] +a_{2}\bigl[d(gx,gx)+d(gx,gx)\bigr]+a_{3}\bigl[d(gx,gx)+d(gx,gx)\bigr]+a_{4}d(gx,gx)\\&{}+a_{5}\bigl[d(gx,gx)+d(gx,gx)\bigr]+a_{6}\bigl[d(gx,gx)+d(gx,gx)\bigr]+a_{7}d(gx,gx). \end{align*} It follows that where \(\phi=a_{1}+2a_{2}+2a_{3}+a_{4}+2a_{5}+2a_{6}+a_{7}\). Similarly Adding (7) and (8), we get \begin{equation*} \bigl[d(gx,gx)+d(gy,gy)\bigr] \leq \psi \bigl[d(gx,gx)+d(gy,gy)\bigr]. \end{equation*} where \(\psi=\phi+a_{1}\). This is possible only when \(d(gx,gx)+d(gy,gy)=0\) since \(\psi< 1\) which implies that \(d(gx,gx)=0\) and \(d(gy,gy)=0\). Similarly \(d(gx_{1},gx_{1})=0\) and \(d(gy_{1},gy_{1})=0\). Now, we shall show the uniqueness of the coupled point of coincidence of \(T\) and \(g\). For this task, we consider \(d(gx,gx_{1})\). Using (1), we have \begin{align*} &d(gx,gx_{1}) = d\left[T(x,y),T(x_{1},y_{1})\right]\leq a_{1}\left[d(gx,gx_{1})+d(gy,gy_{1})\right] +a_{2}\left[d(gx,T(x,y))+d(gx_{1},T(x_{1},y_{1}))\right]\nonumber\\&{} +a_{3}\left[d(gx,T(x_{1},y_{1}))+d(gx_{1},T(x,y))\right] +a_{4}\left[\frac{d(gx,T(x,y))d(gx_{1},T(x_{1},y_{1}))}{d(gx,gx_{1})+d(gy,gy_{1})}\right]\nonumber\\&{} +a_{5}\left[\frac{[d(gx,gx_{1})+d(gy,gy_{1})]\times[d(gx,T(x,y))+d(gx_{1},T(x_{1},y_{1}))]} {1+d(gx,gx_{1})+d(gy,gy_{1})}\right]\nonumber\\&{} +a_{6}\left[\frac{d(gx,T(x,y))+d(gx,T(x_{1},y_{1}))}{1+d(gx_{1},T(x_{1},y_{1}))d(gx_{1},T(x_{1},y_{1}))}\right] +a_{7}\left[\frac{d(gx,T(x,y))d(gx_{1},T(x_{1},y_{1}))}{1+d(gx,gx_{1})+d(gx_{1},T(x_{1},y_{1}))}\right]. \end{align*} Using the fact that \(gx=T(x,y)\) and \(gx_{1}=T(x_{1},y_{1})\), we have \begin{align*} & d(gx,gx_{1}) \leq a_{1} \left[d(gx,gx_{1})+d(gy,gy_{1})\right]+a_{2}\left[d(gx,gx)+d(gx_{1},gx_{1})\right] +a_{3}\left[d(gx,gx_{1})+d(gx_{1},gx)\right]\nonumber\\&{} +a_{4}\left[\frac{d(gx,gx)d(gx_{1},gx_{1})}{d(gx,gx_{1})+d(gy,gy_{1})}\right] +a_{5}\left[\frac{[d(gx,gx_{1})+d(gy,gy_{1})][d(gx,gx)+d(gx_{1},gx_{1})]}{1+d(gx,gx_{1})+d(gy,gy_{1})}\right]\nonumber\\&{} +a_{6}\left[\frac{d(gx,gx)+d(gx,gx_{1})}{1+d(gx_{1},gx_{1})d(gx_{1},gx_{1})}\right]+ a_{7}\left[\frac{d(gx,gx)d(gx_{1},gx_{1})}{1+d(gx,gx_{1})+d(gx_{1},gx_{1})}\right]. \end{align*} Thus, we have \[ d(gx,gx_{1}) \leq (a_{1} + a_{3} + a_{6})d(gx,gx_{1}) + a_{1} d(gy,gy_{1}) +( a_{3}+ a_{6} ) d(gx_{1},gx),\] implies Similarly Adding (9) and (10) and then simplifying, we get where \(\omega = \left[\frac{a_{3}+ a_{6} }{1-(2a_{1} + a_{3} +a_{6})}\right]\). Similarly Adding (11) and (12), we get So, \(\bigl[d(gx_{1},gx)+d(gy_{1},gy)+d(gx,gx_{1})+d(gy,gy_{1})\bigr]=0\), since \(\omega< 1\). It follows that \(d(gx_{1},gx)=d(gy_{1},gy)=d(gx,gx_{1})=d(gy,gy_{1})=0.\) Now, applying Definition (1), we get \(gx_{1}=gx \mbox{ and } gy_{1}=gy.\) Thus, \((gx,gy)\) is the unique coupled point of coincidence of \(T\) and \(g\).

Next, we show that \(gx=gy\).

\begin{align*} & d(gx,gy)= d\bigl[T(x,y),T(y,x)\bigr] \leq a_{1}\bigl[d(gx,gy)+d(gy,gx)\bigr] +a_{2}\Bigl[d\bigl(gx,T(x,y)\bigr)+d\bigl(gy,T(y,x)\bigr)\Bigr]\nonumber\\&{}+a_{3}\Bigl[d\bigl(gx,T(y,x)\bigr)+d\bigl(gy,T(x,y)\bigr)\Bigr] +a_{4}\left[\frac{d\bigl(gx,T(x,y)\bigr)d\bigl(gy,T(y,x)\bigr)}{d(gx,gy)+d(gy,gx)}\right]\nonumber\\ &{}+a_{5}\Biggl[\frac{\Bigl[d(gx,gy)+d(gy,gx)\Bigr]\Bigl[d(gx,T(x,y))+d(gy,T(y,x))\Bigr]}{1+d(gx,gy)+d(gy,gx)}\Biggr]\nonumber\\ &{}+a_{6}\left[\frac{d\bigl(gx,T(x,y)\bigr)+d\bigl(gx,T(y,x)\bigr)}{1+d\bigl(gy,T(y,x)\bigr)d\bigl(gy,T(y,x)\bigr)}\right]+a_{7}\left[\frac{d\bigl(gx,T(x,y)\bigr)d\bigl(gy,T(y,x)\bigr)}{1+d(gx,gy)+d\bigl(gy,T(y,x)\bigr)}\right].\nonumber \end{align*} Using (1) and the fact that \(gx=T(x,y)\) and \(gy=T(y,x)\), we have \begin{align*} &d(gx,gy)\leq a_{1}\bigl[d(gx,gy)+d(gy,gx)\bigr] +a_{2}\bigl[d(gx,gx)+d(gy,gy)\bigr]+a_{3}\bigl[d(gx,gy)+d(gy,gx)\bigr]\\ &{}+a_{4}\left[\frac{d(gx,gx)d(gy,gy)}{d(gx,gy)+d(gy,gx)}\right] +a_{5}\left[\frac{\Bigl[d(gx,gy)+d(gy,gx)\Bigr]\Bigl[d(gx,gx)+d(gy,gy)\Bigr]}{1+d(gx,gy)+d(gy,gx)}\right]\\ &{}+a_{6}\left[\frac{d(gx,gx)+d(gx,gy)}{1+d(gy,gy)d(gy,gy)}\right] +a_{7}\left[\frac{d(gx,gx)d(gy,gy)}{1+d(gx,gy)+d(gy,gy)}\right].\nonumber \end{align*} Thus, we have where \(\sigma=\left[\frac{ a_{1}+a_{3} }{1-(a_{1} + a_{3} +a_{6})}\right]\). Similarly, we can show that Adding (14) and (15), we have \(\bigl[d(gx,gy)+d(gy,gx)\bigr] \leq \sigma \bigl[d(gx,gy)+d(gy,gx)\bigr].\) Since \(\sigma< 1\), the above inequality is only possible if \(d(gx,gy)=d(gy,gx)=0.\) That is, \(gx=gy.\) Now, we show that \(T\) and \(g\) have coupled common fixed point. To do so, first let \(u=gx=T(x,y)\). Due to the fact that \(T\) and \(g\) are weakly compatible, we have \(gu=g(gx)=gT(x,y)=T(gx,gy)=T(u,u).\) Hence \((gu,gu)\) is a coupled point of coincidence and \((u,u)\) is a coupled coincidence point of \(T\) and \(g\). Applying the uniqueness property of coupled point of coincidence of \(T\) and \(g\), we get \(gu=u=gx=gy.\) Therefore \(T(u,u)=gu=u.\) That is \((u,u)\) is a coupled common fixed point of \(T\) and \(g\). Now it remains to show the uniqueness of a coupled common fixed point of \(T\) and \(g\). Assume, we have another coupled common fixed point of \(T\) and \(g\) say \((u_{1},u_{1}) \in X^{2}\). It follows that \(u_{1}=gu_{1}=T(u_{1},u_{1})\). Hence \((gu,gu)\) and \((gu_{1},gu_{1})\) are two coupled points of coincidence of \(T\) and \(g\). But due to the uniqueness of coupled point of coincidence, we get \(gu=gu_{1}\) and so \(u_{1} = T(u_{1},u_{1})=T(u,u)=u.\) Therefore \((u,u)\) is the unique coupled common fixed point of \(T\) and \(g\).

Remark 1. If we take \(g=I\) (the identity map) and \(a_{7}=0\) in Theorem 14, we get Theorem 13 of [11].

The following example supports our main theorem.

Example 1. Let \(X=[0,1)\) and \(d : X\times X\rightarrow \Re^{+}\) be defined by \(d(x,y)=|x-y|+|y|\) for all \(x,y \in X\). Then \((X,d)\) is \(dq\)-metric space. We define the functions \(T : X\times X\rightarrow X\) and \(g : X\rightarrow X\) by \[gx=\begin{cases} \frac{1}{3}x &\quad \text{ if \(0\leq x< \frac{9}{10}\),} \\ \frac{3}{10} &\quad \text{ if \(\frac{9}{10}\leq x < 1,\)} \end{cases} \] and \[T(x,y)=\begin{cases} \frac{x+y}{27} &\quad \text{ if \(0\leq x,y< \frac{9}{10},\)} \\ \frac{1}{30}y &\quad \text{ if \(\frac{9}{10}\leq x< 1\) and \(0 \leq y < \frac{9}{10},\)}\\ \frac{1}{30}x &\quad \text{ if \(\frac{9}{10}< y< 1\) and \(0\leq x< \frac{9}{10},\)}\\ \frac{1}{15} &\quad \text{ if \(\frac{9}{10}\leq x< 1\) and \(\frac{9}{10}\leq y < 1.\)} \end{cases} \] Clearly \(T\) and \(g\) are continuous, \(T(X\times X) \subseteq g(X)\), and \(g(X)\) is complete. Following four cases will arise for \(x\), \(u\), \(v\), and \(y\);

• Case (1): \(0\leq x, u, y, v< \frac{9}{10}\).
• Case (2): \(\frac{9}{10}\leq x,u< 1\) and \(0 \leq y,v < \frac{9}{10}\).
• Case (3): \(\frac{9}{10}< y, v< 1\) and \(0\leq x,u< \frac{9}{10}\).
• Case (4): \(\frac{9}{10}\leq x, u< 1\) and \(\frac{9}{10}\leq y, v < 1\).

Case 1: For \(0\leq x, u, y, v< \frac{9}{10}\), we have \begin{eqnarray*} d[T(x,y),T(u,v)]&=& d\left(\frac{x+y}{27},\frac{u+v}{27}\right) \nonumber\\&=&\left|\frac{x+y}{27}-\frac{u+v}{27}\right|+\left|\frac{u+v}{27}\right|\nonumber\\ &=& \left|\frac{x}{27}+\frac{y}{27}-\frac{u}{27}-\frac{v}{27}\right|+\left|\frac{u}{27}+\frac{v}{27}\right|\nonumber\\ &\leq & \left|\frac{x}{27}-\frac{u}{27}\right|+\left|\frac{y}{27}-\frac{v}{27}\right|+\left|\frac{u}{27}\right|+\left|\frac{v}{27}\right|\nonumber\\ &=&\frac{1}{9}\left[\left(\left|\frac{x}{3}-\frac{u}{3}\right|+\left|\frac{u}{3}\right|\right)+\left(\left|\frac{y}{3}-\frac{v}{3}\right|+\left|\frac{v}{3}\right|\right)\right]\nonumber\\ &\leq & \frac{1}{9}\bigl[d(gx,gu)+d(gy,gv)\bigr]\nonumber\\ &\leq & \frac{2}{9}\bigl[d(gx,gu)+d(gy,gv)\bigr]. \end{eqnarray*} Similarly, for Cases (2) to (4), we obtain \begin{equation*} d\bigl[T(x,y),T(u,v)\bigr]\leq \frac{2}{9}\bigl[d(gx,gu)+d(gy,gv)\bigr]. \end{equation*} Hence all the conditions of Theorem 13 are satisfied with \(a_{1}=\frac{2}{9}, a_{2}=\frac{1}{120}, a_{3}=\frac{1}{64}, a_{4}=\frac{1}{80}, a_{5}=\frac{1}{100}, a_{6}=\frac{1}{128}\), and \(a_{7}=\frac{1}{32}\). Therefore, \(T\) and \(g\) have unique coupled point of coincidence and unique coupled common fixed point which are \((g0,g0)\) and \((0,0)\) respectively. This is due to the fact that \(g T(0,0)=T(g0,g0)=T(0,0)=0.\)

4. Conclusion

In 2018, Mohammed established the existence of coupled fixed point for mapping satisfying certain rational type contraction condition in a complete dislocated quasi metric space. In this paper, we explored the properties of dislocated quasi-metric spaces and also discuss the difference between metric space and dislocated metric space. We established and proved existence of coupled coincidence point and existence and uniqueness of coupled common fixed point theorem for a pair of maps \(T\) and \(g\) in the setting of dislocated quasi metric spaces. Also, we provided an example in support of our main result. Our work extended coupled fixed point result to common coupled fixed point result. The presented theorem extends and generalizes several well-known comparable results in literature.

Acknowledgments

The authors would like to thank the College of Natural Sciences, Jimma University for funding this research work.

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflict of Interests

The authors declare no conflict of interest.

1. Introduction

Group theory has applications in physics, chemistry, and computer science, and even puzzles like Rubik's Cube can be represented using group theory. Fuzzy sets, proposed by Zadeh [1], are sets whose elements have degrees of membership. Rosenfeld [2] introduced fuzzy sets in the realm of group theory and formulated the concepts of fuzzy subgroups of a group. Many authors have worked on fuzzy group theory [2,3,4], especially, some authors considered the fuzzy subgroups with respect to norms [5,6,7]. Alsarahead and Ahmad [8] defined the complex fuzzy subgroup and investigate some of its characteristics. The author by using norms, investigated some properties of fuzzy algebraic structures [9,10,11].

In this paper, by using \(s\)-norms, we define and investigate some properties of anti complex fuzzy subgroups of group \( G \) under \(s\)-norm \(S\) as \(ACFS(G).\) Also we define the composition and intersection of two \( \mu_{1},\mu_{2} \in ACFS(G)\) and obtain some of their characteristics. Later, we introduce and investigate the normality of \( \mu \in ACFS(G)\) denoted by \(NACFS(G).\) Finally, we introduce the normality between two \( \mu_{1},\mu_{2} \in ACFS(G)\) as \( \mu_{1} \bowtie \mu_{2} \) and investigate some important properties of them. By using a group homomorphism \( f: G \to H,\) we prove that if \( \mu \in ACFS(G)\) and \( \nu \in ACFS(H),\) then \( f(\mu) \in ACFS(H)\) and \( f^{-1}(\nu) \in ACFS(G).\) Also if \( \mu \in NACFS(G)\) and \( \nu \in NACFS(H),\) then we prove that \( f(\mu) \in NACFS(H)\) and \( f^{-1}(\nu) \in NACFS(G).\) Also we show that if \( \mu_{1},\mu_{2} \in ACFS(G)\) such that \(\mu_{1} \bowtie \mu_{2},\) then we show that \(f(\mu_{1}) \bowtie f(\mu_{2})\) and if \( \mu_{1},\mu_{2} \in ACFS(H)\) such that \(\mu_{1} \bowtie \mu_{2},\) then we obtain \(f^{-1}(\mu_{1}) \bowtie f^{-1}(\mu_{2}).\)

2. Preliminaries

The following definitions and preliminaries are required in the sequel of our work and hence presented in brief. For details we refer readers to [6,12,13,14,15].

Definition 1. A group is a non-empty set \( G\), on which there is a binary operation \((a,b) \to ab\) such that

• if \( a \) and \( b\) belong to \( G \) then \( ab\) is also in \( G\) (closure),
• \(a(bc) = (ab)c\) for all \(a,b,c \in G\) (associativity),
• there is an element \( e_{G} \in G\) such that \(ae_{G} = ee_{G}a = a\) for all \(a \in G\) (identity),
• if \(a \in G,\) then there is an element \(a^{-1} \in G\) such that \(aa^{-1} = a^{-1}a =e_{G}\) (inverse).

One can easily check that this implies the unicity of the identity and of the inverse. A group \( G \) is called abelian if the binary operation is commutative, i.e., \(ab = ba\) for all \(a,b \in G.\)

Remark 1. There are two standard notations for the binary group operation: either the additive notation, that is \((a,b) \to a + b\) in which case the identity is denoted by \( 0\), or the multiplicative notation, that is \((a,b) \to ab\) for which the identity is denoted by \( e. \)

Definition 2. Let \(G\) be an arbitrary group with a multiplicative binary operation and identity \(e\). By a fuzzy subset of \(G\), we mean a function from \(G\) into \( [0,1]\). The set of all fuzzy subsets of \(G\) is called the \( [0,1]\)-power set of \(G\) and is denoted \( [0,1]^G.\)

Definition 3. Let \( X\) be a nonempty set. A complex fuzzy set \( A\) on \( X\) is an object having the form \(A =\lbrace (x, \mu_{A}(x)) | x \in X \rbrace,\) where \( \mu_{A}\) denotes the degree of membership function that assigns each element \( x \in X \), a complex number \( \mu_{A}(x)) \) lies within the unit circle in the complex plane. We shall assume that \( \mu_{A}(x) \) will be represented by \(r_{A}(x)e^{iw_{A}(x)}\) , where \(i = \sqrt{-1},\) and \( r:X \to [0,1] \) and \( w:X \to [0,2\pi] \). Note that by setting \( w(x)=0 \) in the definition above, we return back to the traditional fuzzy subset. Let \( \mu_{1}=r_{1}e^{w_{1}} \) and \( \mu_{2}=r_{2}e^{w_{2}} \) be two complex numbers lie within the unit circle in the complex plane. By \( \mu_{1} \leq \mu_{2} \), we mean \(r_{1} \leq r_{2} \) and \( w_{1} \leq w_{2}.\)

Definition 4. An \(s\)-norm \(S\) is a function \(S : [0,1]\times [0,1] \to [0,1]\) having the following four properties:

• \(S(x,0)=x\),
• \(S(x,y)\leq S(x,z)\) if \(y\leq z\),
• \(S(x,y)= S(y,x)\),
• \(S(x,S(y,z))=S(S(x,y),z)\)

for all \(x,y,z \in[0,1].\)

We say that \(S\) is idempotent if for all \(x \in [0,1]\),\( S(x, x) =x.\)

Example 1. The basic \(s\)-norms are \(S_m(x,y) = \max \{ x,y \}\),\(S_b(x,y) = \min\{1, x+y \}\) and \(S_p(x, y) = x+y-xy \) for all \(x,y \in [0,1].\) \(S_m\) is standard union, \(S_b\) is bounded sum, \(S_p\) is algebraic sum.

Lemma 1. Let \(S\) be an \(s\)-norm. Then \[S(S(x,y),S(w,z))= S(S(x,w),S(y,z)), \] for all \(x,y,w,z\in [0,1].\)

3. Anti complex fuzzy subgroups under \(s\)-norms

Definition 5. Let \( G\) be a group and \( \mu: G \to [0,1] \) be a complex fuzzy set on \(G.\) Then \( \mu=re^{iw}\) is said to be an anti complex fuzzy subgroup of \( G \) under \(s\)-norm \(S\) if the following conditions hold:

• \(r(xy) \leq S(r(x),r(y)),\)
• \(r(x^{-1}) \leq r(x),\)
• \(w(xy) \leq \max \lbrace w(x),w(y) \rbrace,\)
• \(w(x^{-1}) \leq w(x),\)

for all \( x,y \in G. \) The set of all anti complex fuzzy subgroups of \( G \) under \(s\)-norm \(S\) is denoted by \(ACFS(G)\).

Example 2. Let \(G = \{ 0,a,b,c\} \) be the Klein's group. Every element is its own inverse, and the product of any two distinct non-identity elements is the remaining non-identity element. Thus the Klein 4-group admits the elegant presentation \(a^2=b^2=c^2=abc=0.\) Define \( r: G \to [0,1] \) by \begin{equation*} r(x) = \left\{ \begin{array}{rl} 0.35 &\text{if }x=a, \\ 0.45 &\text{if }x=b, \\ 0.65 &\text{if }x=c, \\ 0.85 &\text{if }x=0,\\ \end{array} \right. \end{equation*} and \( w: G \to [0,2\pi] \) by \begin{equation*} w(x) = \left\{ \begin{array}{rl} 0.4 \pi &\text{if }x=a, \\ 0.4 \pi &\text{if }x=b, \\ 0.5 \pi &\text{if }x=c, \\ 0.6 \pi &\text{if }x=0. \\ \end{array} \right. \end{equation*} Let \(S(a, b) = S_p(a, b) =a+b-ab \) for all \(a,b\in [0,1]\), then \(\mu(x) =r(x)e^{iw(x)}\in ACFS(G)\) for all \( x \in G. \)

Proposition 1. Let \(\mu=re^{iw}\in ACFS(G)\) such that \(C\) be idempotent \(s\)-norm. Then

• \(\mu(e) \leq \mu(x)\) for all \(x \in G \),
• \(\mu(x^n) \leq \mu(x)\) for all \(x \in G \) and \(n\geq 1,\)
• \(\mu(x)=\mu(x^{-1})\) for all \(x \in G.\)

Proof. Let \(\mu=re^{iw}\in ACFS(G)\) and \(x \in G \) and \(n\geq 1\). Then \begin{eqnarray*} (1) r(e)&=&r(xx^{-1}) \leq S(r(x),r(x^{-1})) \leq S(r(x),r(x))=r(x),\\ w(e)&=&w(xx^{-1}) \leq \max \lbrace w (x),w(x^{-1}) \rbrace \leq \max \lbrace w (x),w(x) \rbrace=w(x),\\ \mu(e)&=&r(e)e^{iw(x)} \leq r(x)e^{iw(x)} = \mu(x),\\ r(x^n)&=&r(\underbrace{xx...x}_{n}) \leq S(\underbrace{r(x),r(x),...,r(x)}_{n})= r(x),\\ w(x^n)&=&w(\underbrace{xx...x}_{n}) \leq \max \lbrace \underbrace{r(x),r(x),...,r(x)}_{n} \rbrace= w(x),\\ \mu(x^n)&=&r(x^n)e^{iw(x^n)} \leq r(x)e^{iw(x)}=\mu(x),\\ r(x)&=&r((x^{-1}))^{-1} \leq r(x^{-1}) \leq r(x),\\ r(x)&=& r(x^{-1}),\\ w(x)&=&w((x^{-1}))^{-1} \leq w(x^{-1}) \leq w(x),\\ w(x)&=&w(x^{-1}),\\ \mu(x)&=&r(x)e^{iw(x)}=r(x^{-1})e^{iw(x^{-1})}=\mu(x^{-1}). \end{eqnarray*}

Proposition 2. Let \(\mu=re^{iw}\in ACFS(G)\) and \(x\in G\) such that \(S\) be idempotent \(s\)-norm. Then \(\mu (xy)=\mu(y)\) \(\forall y \in G\) if and only if \(\mu (x)=\mu(e). \)

Proof. Let \(\mu(xy)= \mu(y)\) \(\forall y \in G.\) As \(y=e\), so \(\mu (x)=\mu(e). \) Conversely, let \(\mu (x)=\mu(e)\), then \(r(x)=r(e) \) and \(w(x)=w(e)\). From Proposition 1, we get \(r(x) \leq r(y) \) and \( r(x) \leq r(xy) \). Also \(w(x) \leq w(y) \) and \( w(x) \leq w(xy).\) Now \(r(xy) \leq S(r(x),r(y)) \leq S(r(y),r(y))=r(y)=r(x^{-1}xy) \leq S(r(x),r(xy)) \leq S(r(xy),r(xy))=r(xy).\)

Also \(w(xy) \leq \max \lbrace w(x),w(y) \rbrace \leq \max \lbrace w(y),w(y) \rbrace=w(y)=w(x^{-1}xy) \leq \max \lbrace w(x),w(xy) \rbrace \leq \max \lbrace w(xy),w(xy) \rbrace=w(xy).\) Therefore \(\mu(xy)=r(xy)e^{iw(xy)}=r(y)e^{iw(y)}=\mu(y).\)

Definition 6. Let \(G\) be a set and \(\mu_{1}=r_{1}e^{iw_{1}}\), \(\mu_{2}=r_{2}e^{iw_{2}}\) be two complex fuzzy sets on \(G.\) Denote the composition of \( \mu_{1} \) and \( \mu_{2} \) as \( \mu_{1} \circ \mu_{2}=(r_{1} \circ r_{2})e^{i(w_{1} \circ w_{2})}\) such that \( r_{1} \circ r_{2} : G \to [0,1] \) and \( w_{1} \circ w_{2}: G \to [0,2\pi] \) and define by \((\mu_{1} \circ \mu_{2})(x)=(r_{1} \circ r_{2})(x)e^{i(w_{1} \circ w_{2})(x)}(x)\) such that \begin{equation*} (r_{1} \circ r_{2})(x) = \left\{ \begin{array}{rl} \inf\limits_{x=ab}S(r_{1}(a),r_{2}(b)) &\text{if }x=ab, \\ 0 &\text{if }x \neq ab, \end{array} \right. \end{equation*} and \begin{equation*} (w_{1} \circ w_{2})(x) = \left\{ \begin{array}{rl} \max\limits_{x=ab} \lbrace w_{1}(a),w_{2}(b) \rbrace &\text{if }x=ab, \\ 0 &\text{if }x \neq ab, \end{array} \right. \end{equation*} thus \((\mu_{1} \circ \mu_{2})(x)=\inf_{x=ab}S(r_{1}(a),r_{2}(b))e^.{i \max_{x=ab}} \{ w_{1}(a),w_{2}(b) \} .\)

Proposition 3. Let \(\mu^{-1}\) be the inverse of \(\mu\) such that \(\mu^{-1}(x)=\mu (x^{-1}).\) Then \(\mu\in ACFS(G)\) if and only if \(\mu\) satisfies the following conditions:

• (1) \( \mu \leq \mu \circ \mu ;\)
• (2) \(\mu^{-1}=\mu.\)

Proof. Let \(x,y,z \in G\) with \(x=yz\) and \(\mu\in ACFS(G).\) Then \begin{eqnarray*} r(x)&=&r (yz) \leq S(r(y),r(z))=(r \circ r)(x),\\ w(x)&=&w (yz) \leq \max \lbrace r(y),r(z) \rbrace=(w \circ w)(x),\\ \mu(x)&=&r(x)e^{iw(x)} \leq (r \circ r)(x)e^{i(w \circ w)(x)}=(\mu \circ \mu)(x), \end{eqnarray*} so \(\mu\leq \mu \circ \mu.\)

Also from Proposition 1, for all \( x \in G \), we have that \( \mu^{-1}(x)=\mu(x^{-1})=\mu(x) \) and so \(\mu^{-1}=\mu.\) Conversely let \( \mu \leq \mu \circ \mu \) and \(\mu^{-1}=\mu. \) We prove that \(\mu\in ACFS(G).\) As \( \mu \leq \mu \circ \mu \) so \(r(x)\leq (r \circ r)(x)\) and \(w(x)\leq (w \circ w)(x)\). Thus

\[r(yz)=r(x)\leq (r \circ r)(x)=\inf_{x=yz}S(r(y),r(z))\leq S(r(y),r(z)) \] and \[w(yz)=w(x)\geq (w \circ w)(x)=\max_{x=yz} \lbrace w(y),w(z) \rbrace \geq \lbrace w(y),w(z) \rbrace. \] Since \(\mu^{-1}=\mu\) so \( r^{-1}(x) =r(x)\) and \( w^{-1}(x) =w(x).\) Therefore \( r(x^{-1})=r^{-1}(x) =r(x) \) and \( w(x^{-1})=w^{-1}(x) =w(x).\) Then \(\mu\in ACFS(G).\)

Corollary 1. Let \(\mu_{1},\mu_{2} \in ACFS(G)\) and \(G\) be commutative group. Then \(\mu_{1} \circ \mu_{2} \in ACFS(G)\) if and only if \(\mu_{1}\circ \mu_{2} = \mu_{2}\circ \mu_{1}.\)

Proof. As \(\mu_{1},\mu_{2} \in CFST(G)\) and \(\mu_{1} \circ \mu_{2} \in ACFS(G)\), so from Proposition 3, we get \(\mu^{-1}_{1} =\mu_{1} \) and \(\mu^{-1}_{2} =\mu_{2} \) and \( (\mu_{2}\circ \mu_{1})^{-1}=\mu_{2}\circ \mu_{1}. \) Then \( \mu_{1}\circ \mu_{2}=\mu^{-1}_{1}\circ \mu^{-1}_{2}=(\mu_{2}\circ \mu_{1})^{-1}=\mu_{2}\circ \mu_{1}.\) Conversely, let \(\mu_{1}\circ \mu_{2} = \mu_{2}\circ \mu_{1}\), then \((\mu_{1} \circ \mu_{2})\circ(\mu_{1} \circ \mu_{2})=\mu_{1} \circ (\mu_{2} \circ \mu_{1}) \circ \mu_{2}= \mu_{1} \circ (\mu_{1} \circ \mu_{2}) \circ \mu_{2}= (\mu_{1} \circ \mu_{1}) \circ (\mu_{2} \circ \mu_{2}) \geq \mu_{1} \circ \mu_{2}.\) Also \((\mu_{1} \circ \mu_{2})^{-1} =( \mu_{2} \circ \mu_{1})^{-1}=\mu_{1}^{-1} \circ \mu_{2}^{-1}=\mu_{1} \circ \mu_{2}.\) Then the Proposition 3 gives us that \(\mu_{1} \circ \mu_{2} \in ACFS(G).\)

Definition 7. Let \(\mu_{1}=r_{1}e^{iw_{1}} \in ACFS(G)\) and \(\mu_{2}=r_{2}e^{iw_{2}} \in ACFS(G).\) Define the intersection \(\mu_{1} \cap \mu_{2}\) as \( \mu_{1} \cap \mu_{2} = r_{1}e^{iw_{1}} \cap r_{2}e^{iw_{2}}=(r_{1} \cap r_{2})e^{i(w_{1} \cap w_{2})}\) such that \(r_{1} \cap r_{2}: G \to [0,1] \) and \( w_{1} \cap w_{2}: G \to [0,2\pi] \) and for all \( x \in G \), define \((r_{1} \cap r_{2})(x)=S(r_{1}(x),r_{2}(x))\) and \((w_{1} \cap r_{2})(x)=\max \lbrace w_{1}(x),w_{2}(x) \rbrace.\)

Proposition 4. Let \(\mu_{1}=r_{1}e^{iw_{1}} \in ACFS(G)\) and \(\mu_{2}=r_{2}e^{iw_{2}} \in ACFS(G).\) Then \(\mu_{1} \cap \mu_{2} \in ACFS(G).\)

Proof.

• (1) Let \( g_{1},g_{2} \in G.\) Then \begin{eqnarray*} (r_{1} \cap r_{2})(g_{1}g_{2})&=&S(r_{1}(g_{1}g_{2}) ,r_{2}(g_{1}g_{2})) \leq S(S(r_{1}(g_{1}) ,r_{1}(g_{2})),S(r_{2}(g_{1}) ,r_{2}(g_{2})))\\ &=&S(S(r_{1}(g_{1}) ,r_{2}(g_{1})),T(r_{1}(g_{2}) ,r_{2}(g_{2}))) \\ &=&S((r_{1} \cap r_{2})(g_{1}),(r_{1} \cap r_{2})(g_{2})), \end{eqnarray*} and thus \((r_{1} \cap r_{2})(g_{1}g_{2}) \leq S((r_{1} \cap r_{2})(g_{1}),(r_{1} \cap r_{2})(g_{2})).\)
• (2) If \( g \in G,\) then \begin{eqnarray*} (r_{1} \cap r_{2})(g^{-1})=S(r_{1}(g^{-1}),r_{2}(g^{-1})) &\leq& S(r_{1}(g),r_{2}(g))=(r_{1} \cap r_{2})(g)\\ (r_{1} \cap r_{2})(g^{-1}) &\geq& (r_{1} \cap r_{2})(g). \end{eqnarray*}
• (3) Let \( g_{1},g_{2} \in G.\) Then \begin{eqnarray*} (w_{1} \cap w_{2})(g_{1}g_{2})&=&\max \lbrace w_{1}(g_{1}g_{2}) ,w_{2}(g_{1}g_{2}) \rbrace\\ &\leq& \max \lbrace \max \lbrace w_{1}(g_{1}) ,w_{1}(g_{2}) \rbrace,\max \lbrace w_{2}(g_{1}) ,w_{2}(g_{2}) \rbrace \rbrace\\ &=&\max \lbrace \max \lbrace w_{1}(g_{1}) ,w_{2}(g_{1}) \rbrace,\max \lbrace w_{1}(g_{2}) ,w_{2}(g_{2})) \rbrace \\ &=&\max \lbrace (w_{1} \cap w_{2})(g_{1}),(w_{1} \cap w_{2})(g_{2}))\\ (w_{1} \cap w_{2})(g_{1}g_{2}) &\leq& \max \lbrace (w_{1} \cap w_{2})(g_{1}),(w_{1} \cap w_{2})(g_{2})). \end{eqnarray*}
• (4) Let \( g \in G \) so \begin{eqnarray*} (w_{1} \cap w_{2})(g^{-1})&=&\max \lbrace w_{1}(g^{-1}), w_{21}(g^{-1}) \rbrace\\ &\leq& \max \lbrace w_{1}(g), w_{2}(g) \rbrace\\ &=& (w_{1} \cap w_{2})(g)\\ (w_{1} \cap w_{2})(g^{-1}) &\leq& (w_{1} \cap w_{2})(g). \end{eqnarray*}

Thus from (1)-(4) we give that \(\mu_{1} \cap \mu_{2} \in ACFS(G).\)

Corollary 2. Let \(I_{n}=\{1,2,...,n\}.\) If \(\{\mu_{i}\hspace{0.1cm} | \hspace{0.1cm}i\in I_{n}\} \subseteq ACFS(G)\) then \(\mu=\cap_{i\in I_{n}}\mu_{i}\in ACFS(G).\)

Definition 8. \(\mu\in ACFS(G)\) is called normal if for all \(x,y\in G\) we have that \(\mu(xyx^{-1}) = \mu (y).\) The set of all normal anti complex fuzzy subgroups of \( G \) under \(s\)-norm \(S\) ia denoted by \(NACFS(G)\).

Proposition 5. Let \(\mu_{1}=r_{1}e^{iw_{1}} \in NACFS(G)\) and \(\mu_{2}=r_{2}e^{iw_{2}} \in NACFS(G).\) Then \(\mu_{1} \cap \mu_{2} \in NACFS(G).\)

Proof. From Proposition 4 we will have that \(\mu_{1} \cap \mu_{2} \in ACFS(G).\) Let \( g_{1} , g_{2} \in G \) then \begin{eqnarray*} (r_{1} \cap r_{2})(g_{1}g_{2}g^{-1}_{1})&=&S(r_{1} (g_{1}g_{2}g^{-1}_{1}),r_{2} (g_{1}g_{2}g^{-1}_{1}))=S(r_{1} (g_{2}),r_{2} (g_{2}))=(r_{1} \cap r_{2})(g_{2})\\ (w_{1} \cap w_{2})(g_{1}g_{2}g^{-1}_{1})&=&\max \lbrace w_{1} (g_{1}g_{2}g^{-1}_{1}),w_{2} (g_{1}g_{2}g^{-1}_{1}) \rbrace=\max \lbrace w_{1} (g_{2}),w_{2} (g_{2}) \rbrace =(w_{1} \cap w_{2})(g_{2})\\ (\mu_{1} \cap \mu_{2})(g_{1}g_{2}g^{-1}_{1})&=&(r_{1} \cap r_{2})(g_{1}g_{2}g^{-1}_{1})e^{i(w_{1} \cap w_{2})(g_{1}g_{2}g^{-1}_{1})}=(r_{1} \cap r_{2})(g_{2})e^{i(w_{1} \cap w_{2})(g_{2})}=(\mu_{1} \cap \mu_{2})(g_{2}) \end{eqnarray*} and therefore \(\mu_{1} \cap \mu_{2} \in NACFS(G).\)

Corollary 3. Let \(I_{n}=\{1,2,...,n\}.\) If \(\{\mu_{i}\hspace{0.1cm} | \hspace{0.1cm}i\in I_{n}\} \subseteq NACFS(G)\), then \(\mu=\cap_{i\in I_{n}}\mu_{i}\in NACFS(G).\)

Definition 9. Let \(\mu_{1}=r_{1}e^{iw_{1}} \in ACFS(G)\) and \(\mu_{2}=r_{2}e^{iw_{2}} \in ACFS(G)\) such that \( \mu_{1} \subseteq \mu_{2}.\) We say that \(\mu_{1}\) is normal of the \(\mu_{2}\), written \(\mu_{1} \bowtie \mu_{2}\), if \(r_{1}(g_{1}g_{2}g_{1}^{-1}) \leq S(r_{1}(g_{2}), r_{2}(g_{1})) \hspace{0.1cm} and \hspace{0.1cm} w_{1}(g_{1}g_{2}g_{1}^{-1}) \leq \max \lbrace w_{1}(g_{2}), w_{2}(g_{1}) \rbrace \) for all \( g_{1} , g_{2} \in G. \)

Proposition 6. If \(S\) be idempotent \(s\)-norm, then every \(\mu=re^{iw} \in ACFS(G)\) will be normal of itself.

Proof. Let \( g_{1},g_{2} \in G \) and \(\mu=re^{iw} \in ACFS(G),\) then \begin{eqnarray*} r(g_{1}g_{2}g^{-1}_{1}) &\leq& S(r(g_{1}) ,r(g_{2}g^{-1}_{1}))\\ &\leq& S(r(g_{1}),S(r(g_{2}),r(g^{-1}_{1})))\\ &\leq& S(r(g_{1}),S(r(g_{2}),r(g_{1})))\\ &=&S(r(g_{2}),S(r(g_{1}),r(g_{1})))\\ &=&S(r(g_{2}),r(g_{1}))\\ (g_{1}g_{2}g^{-1}_{1}) &\leq& S(r(g_{2}),r(g_{1})).\end{eqnarray*} Also \begin{eqnarray*} w(g_{1}g_{2}g^{-1}_{1}) &\leq& \max \lbrace w(g_{1}) ,w(g_{2}g^{-1}_{1})) \rbrace\\ &\leq& \max \lbrace w(g_{1}),\max \lbrace w(g_{2}),w(g^{-1}_{1}) \rbrace \rbrace \\ &\leq& \max \lbrace w(g_{1}),\max \lbrace w(g_{2}),w(g_{1})))\\ &=&\max \lbrace w(g_{2}),\max \lbrace w(g_{1}),w(g_{1}) \rbrace \rbrace\\ & =&\max \lbrace w(g_{2}),w(g_{1}) \rbrace \\ w(g_{1}g_{2}g^{-1}_{1}) &\leq& \max \lbrace w(g_{2}),w(g_{1}) \rbrace.\end{eqnarray*} Therefore \( \mu=re^{iw} \bowtie \mu=re^{iw}.\)

Proposition 7. Let \(\mu_{1} =r_{1}e^{iw_{1}} \in NACFS(G)\) and \(\mu_{2} =r_{2}e^{iw_{2}} \in ACFS(G)\) such that \(S\) be idempotent \(s\)-norm, then \(\mu_{1} \cap\mu_{2} \bowtie \mu_{2}.\)

Proof. As Proposition 4 \((\mu_{1} \cap\mu_{2}) \leq \mu_{2}\) and \((\mu_{1} \cap\mu_{2})\in ACFS(G).\) Let \( g_{1},g_{2} \in G \) and \( \mu_{1} \cap\mu_{2}=(r_{1} \cap r_{2})e^{i(w_{1} \cap w_{2})},\) then \begin{eqnarray*} (r_{1} \cap r_{2})(g_{1}g_{2}g^{-1}_{1})&=&S(r_{1}(g_{1}g_{2}g^{-1}_{1}), r_{2}(g_{1}g_{2}g^{-1}_{1}))\\ &=&S(r_{1}(g_{2}), r_{2}(g_{1}g_{2}g^{-1}_{1}))\\ &\leq& S(r_{1}(g_{2}),S( r_{2}(g_{1}g_{2}), r_{2}(g^{-1}_{1})))\\ &\leq& S(r_{1}(g_{2}),S( r_{2}(g_{1}g_{2}), r_{2}(g_{1})))\\ &\leq& S(r_{1}(g_{2}),S(S(r_{2}(g_{1}),r_{2}(g_{2})), r_{2}(g_{1})))\\ &=&S(r_{1}(g_{2}),S(S(r_{2}(g_{1}),r_{2}(g_{1})), r_{2}(g_{2})))\\ &=&S(r_{1}(g_{2}),S(r_{2}(g_{1}), r_{2}(g_{2})))\\ &=&S(S(r_{1}(g_{2}),r_{2}(g_{2})),r_{2}(g_{1}))\\ &=&S((r_{1} \cap r_{2})(g_{2}),r_{2}(g_{1})),\end{eqnarray*} and thus \( (r_{1} \cap r_{2})(g_{1}g_{2}g^{-1}_{1}) \leq S((r_{1} \cap r_{2})(g_{2}),r_{2}(g_{1})).\)

Also

\begin{eqnarray*} (w_{1} \cap w_{2})(g_{1}g_{2}g^{-1}_{1})&=&\max \lbrace w_{1}(g_{1}g_{2}g^{-1}_{1}), w_{2}(g_{1}g_{2}g^{-1}_{1}) \rbrace\\ &=&\max \lbrace w_{1}(g_{2}), w_{2}(g_{1}g_{2}g^{-1}_{1}) \rbrace \\ &\leq& \max \lbrace w_{1}(g_{2}),\max \lbrace w_{2}(g_{1}g_{2}), w_{2}(g^{-1}_{1}) \rbrace \rbrace\\ &\leq& \max \lbrace w_{1}(g_{2}),\max \lbrace w_{2}(g_{1}g_{2}), w_{2}(g_{1}) \rbrace \rbrace \\ &\leq& \max \lbrace w_{1}(g_{2}),\max \lbrace \max \lbrace w_{2}(g_{1}),w_{2}(g_{2}) \rbrace, w_{2}(g_{1}) \rbrace \rbrace\\ &=&\max \lbrace w_{1}(g_{2}),\max \lbrace \max \lbrace w_{2}(g_{1}),w_{2}(g_{1})) \rbrace, w_{2}(g_{2}) \rbrace \rbrace \\ &=&\max \lbrace w_{1}(g_{2}),\max \lbrace w_{2}(g_{1}), w_{2}(g_{2}) \rbrace \rbrace \\ &=&\max \lbrace \max \lbrace w_{1}(g_{2}),w_{2}(g_{2}) \rbrace,w_{2}(g_{1}) \rbrace \\ &=&\max \lbrace (w_{1} \cap w_{2})(g_{2}),w_{2}(g_{1}) \rbrace,\end{eqnarray*} and then \( (w_{1} \cap w_{2})(g_{1}g_{2}g^{-1}_{1}) \leq \max \lbrace (w_{1} \cap w_{2})(g_{2}),w_{2}(g_{1}) \rbrace.\) Therefore \( \mu_{1} \cap\mu_{2}=(r_{1} \cap r_{2})e^{i(w_{1} \cap w_{2})} \bowtie \mu_{2}. \)

Proposition 8. Let \(\mu_{1} =r_{1}e^{iw_{1}} \in ACFS(G)\) and \(\mu_{2} =r_{2}e^{iw_{2}} \in ACFS(G)\) and \(\mu_{3} =r_{3}e^{iw_{3}} \in ACFS(G)\) and \(S\) be idempotent \(s\)-norm. If \( \mu_{1} \bowtie \mu_{3}\) and \( \mu_{2} \bowtie \mu_{3},\) then \( \mu_{1} \cap \mu_{2} \bowtie \mu_{3}.\)

Proof. By Proposition 4, we have \( \mu_{1} \cap \mu_{2} \in ACFS(G)\) and \( \mu_{1} \cap \mu_{2} \leq \mu_{3}. \) Let \( g_{1},g_{2} \in G.\) As \( \mu_{1}\bowtie \mu_{3}\), so \( r_{1}(g_{1}g_{2}g^{-1}_{1}) \leq S(r_{1}(g_{2}),r_{3}(g_{1}))\) and \( w_{1}(g_{1}g_{2}g^{-1}_{1}) \leq \max \lbrace r_{1}(g_{2}),r_{3}(g_{1}) \rbrace\) and as \( \mu_{2} \bowtie \mu_{3}\) so \( r_{2}(g_{1}g_{2}g^{-1}_{1}) \leq S(r_{2}(g_{2}),r_{3}(g_{1}))\) and \( w_{2}(g_{1}g_{2}g^{-1}_{1}) \leq \max \lbrace w_{2}(g_{2}),w_{3}(g_{1}) \rbrace.\) Now \begin{eqnarray*} (r_{1} \cap r_{2})(g_{1}g_{2}g^{-1}_{1})&=&S(r_{1} (g_{1}g_{2}g^{-1}_{1}),r_{2} (g_{1}g_{2}g^{-1}_{1}))\\ &\leq& S(S(r_{1}(g_{2}),r_{3}(g_{1})),S(r_{2}(g_{2}),r_{3}(g_{1})))\\ &=&S(S(r_{1}(g_{2}),r_{2}(g_{2})),S(r_{3}(g_{1}),r_{3}(g_{1})))\\ &=&S(S(r_{1}(g_{2}),r_{2}(g_{2})),r_{3}(g_{1}))\\ &=&S((r_{1} \cap r_{2})(g_{2}),r_{3}(g_{1})),\end{eqnarray*} and then \( (r_{1} \cap r_{2})(g_{1}g_{2}g^{-1}_{1}) \leq S((r_{1} \cap r_{2})(g_{2}),r_{3}(g_{1})).\) Also\begin{eqnarray*} (w_{1} \cap w_{2})(g_{1}g_{2}g^{-1}_{1})&=&\max \lbrace w_{1} (g_{1}g_{2}g^{-1}_{1}),w_{2} (g_{1}g_{2}g^{-1}_{1}) \rbrace\\ &\geq& \max \lbrace \max \lbrace w_{1}(g_{2}),w_{3}(g_{1}) \rbrace ,\max \lbrace w_{2}(g_{2}),w_{3}(g_{1}) \rbrace \rbrace\\ &=&\max \lbrace \max \lbrace w_{1}(g_{2}),w_{2}(g_{2}) \rbrace ,\max \lbrace w_{3}(g_{1}),w_{3}(g_{1})\rbrace \rbrace \\ &=&\max \lbrace \max \lbrace w_{1}(g_{2}),w_{2}(g_{2}) \rbrace,w_{3}(g_{1}) \rbrace\\ &=&\max \lbrace (w_{1} \cap w_{2})(g_{2}),w_{3}(g_{1}) \rbrace,\end{eqnarray*} and so \( (w_{1} \cap w_{2})(g_{1}g_{2}g^{-1}_{1}) \leq \max \lbrace (w_{1} \cap w_{2})(g_{2}),w_{3}(g_{1}) \rbrace.\) Thus \( \mu_{1} \cap\mu_{2}=(r_{1} \cap r_{2})e^{i(w_{1} \cap w_{2})} \bowtie \mu_{3}. \)

Corollary 4. Let \(I_{n}=\{1,2,...,n\}\) and \(\{\mu_{i}\hspace{0.1cm} | \hspace{0.1cm}i\in I_{n}\} \subseteq ACFS(G)\) such that \(\{\mu_{i}\hspace{0.1cm} | \hspace{0.1cm}i\in I_{n}\} \bowtie \xi.\) Then \(\mu=\cap_{i\in I_{n}}\mu_{i} \bowtie \xi.\)

4. Group homomorphisms and anti complex fuzzy subgroups under \(s\)-norms

Definition 10. Let \(f:G \to H\) be a mapping such that \( \mu_{G}=r_{G}e^{iw_{G}}\) and \( \mu_{H}=r_{H}e^{iw_{H}}\) be two complex fuzzy sets on \(G\) and \(H,\) respectively. Define \( f(\mu_{G}): H \to [0,1]\) as \(f(\mu_{G})=f(r_{G}e^{iw_{G}})=f(r_{G})e^{if(w_{G})}\) such that for all \( h \in H \) we define \(f(r_{G})(h)=\inf \lbrace r_{G}(g) \hspace{0.1cm}|\hspace{0.1cm} g \in G, f(g)=h \rbrace\) and \(f(w_{G})(h)=\inf \lbrace w_{G}(g) \hspace{0.1cm}|\hspace{0.1cm} g \in G, f(g)=h \rbrace.\) Also define \( f^{-1}(\mu_{H}): G \to [0,1]\) as \( f^{-1}(r_{H}e^{iw_{H}})=f^{-1}(r_{H}) e^{if^{-1}(w_{H})}\) such that for all \(g \in G\), we define \(f^{-1}(r_{H}e^{iw_{H}})(g)=r_{H}(f(g))e^{iw_{H}(f(g))}.\)

Proposition 9. Let \(\mu_{G}=r_{G}e^{iw_{G}} \in ACFS(G)\) and \(f: G \to H\) be a group homomorphism, then \(f(\mu_{G})\in ACFS(H).\)

Proof.

• (1) Let \(h_{1} , h_{2}\in H\) and \(g_{1},g_{2} \in G\) such that \( h_{1}=f(g_{1}) \) and \( h_{2}=f(g_{2}).\) Then \begin{eqnarray*} f(r_{G})(h_{1}h_{2})&=&\inf \lbrace r_{G}(g_{1}g_{2}) \hspace{0.1cm}|\hspace{0.1cm} g_{1},g_{2} \in G, f(g_{1})=h_{1}, f(g_{2})=h_{2} \rbrace\\ &\leq& \inf \lbrace S( r_{G}(g_{1}),r_{G}(g_{2})) \hspace{0.1cm}|\hspace{0.1cm} g_{1},g_{2} \in G, f(g_{1})=h_{1}, f(g_{2})=h_{2} \rbrace\\ &=&S(\inf \lbrace r_{G}(g_{1}) \hspace{0.1cm}|\hspace{0.1cm} g_{1} \in G, f(g_{1})=h_{1} \rbrace,\inf \lbrace r_{G}(g_{2}) \hspace{0.1cm}|\hspace{0.1cm} g_{2} \in G, f(g_{2})=h_{2} \rbrace)\\ &=&S(f(r_{G})(h_{1}),f(r_{G})(h_{2})),\end{eqnarray*} and so \( f(r_{G})(h_{1}h_{2}) \leq S(f(r_{G})(h_{1}),f(r_{G})(h_{2})). \)
• (2) Let \(h\in H\) and \(g \in G\) such that \( h=f(g).\) Then \begin{eqnarray*} f(r_{G})(h^{-1})&=&\inf \lbrace r_{G}(g^{-1}) \hspace{0.1cm}|\hspace{0.1cm} g^{-1} \in G, f(g^{-1})=h^{-1} \rbrace \\ &\leq& \inf \lbrace r_{G}(g) \hspace{0.1cm}|\hspace{0.1cm} g \in G, f^{-1}(g)=h^{-1} \rbrace \\ &=&\inf \lbrace r_{G}(g) \hspace{0.1cm}|\hspace{0.1cm} g \in G, f(g)=h \rbrace =f(r_{G})(h),\end{eqnarray*} and so \( f(r_{G})(h^{-1}) \leq f(r_{G})(h). \)
• (3) Let \(h_{1} , h_{2}\in H\) and \(g_{1},g_{2} \in G\) such that \( h_{1}=f(g_{1}) \) and \( h_{2}=f(g_{2}).\) Then \begin{eqnarray*} f(w_{G})(h_{1}h_{2})&=&\inf \lbrace w_{G}(g_{1}g_{2}) \hspace{0.1cm}|\hspace{0.1cm} g_{1},g_{2} \in G, f(g_{1})=h_{1}, f(g_{2})=h_{2} \rbrace \\ &\leq& \inf \lbrace \max \lbrace w_{G}(g_{1}),w_{G}(g_{2}) \rbrace \hspace{0.1cm}|\hspace{0.1cm} g_{1},g_{2} \in G, f(g_{1})=h_{1}, f(g_{2})=h_{2} \rbrace \\ &=&\max \lbrace \inf \lbrace w_{G}(g_{1}) \hspace{0.1cm}|\hspace{0.1cm} g_{1} \in G, f(g_{1})=h_{1} \rbrace,\inf \lbrace w_{G}(g_{2}) \hspace{0.1cm}|\hspace{0.1cm} g_{2} \in G, f(g_{2})=h_{2} \rbrace) \\ &=&\max \lbrace f(w_{G})(h_{1}),f(w_{G})(h_{2}) \rbrace,\end{eqnarray*} and thus \( f(w_{G})(h_{1}h_{2}) \geq \max \lbrace f(w_{G})(h_{1}),f(w_{G})(h_{2}) \rbrace. \)
• (4) Let \(h\in H\) and \(g \in G\) such that \( h=f(g).\) Now \begin{eqnarray*} f(w_{G})(h^{-1})&=&\inf \lbrace w_{G}(g^{-1}) \hspace{0.1cm}|\hspace{0.1cm} g^{-1} \in G, f(g^{-1})=h^{-1} \rbrace \\ &\geq& \inf \lbrace w_{G}(g) \hspace{0.1cm}|\hspace{0.1cm} g^{-1} \in G, f^{-1}(g)=h^{-1} \rbrace \\ &=&\inf \lbrace w_{G}(g) \hspace{0.1cm}|\hspace{0.1cm} g \in G, f(g)=h \rbrace =f(w_{G})(h),\end{eqnarray*} and therefore \( f(w_{G})(h^{-1}) \geq f(w_{G})(h).\)

Thus (1) - (4) mean that \(f(\mu_{G})=f(r_{G}e^{iw_{G}})=f(r_{G})e^{if(w_{G})} \in ACFS(H).\)

Proposition 10. Let \(\mu_{H}=r_{H}e^{iw_{H}} \in ACFS(H)\) and \(f: G \to H\) be a group homomorphism, then \(f^{-1}(\mu_{H})\in ACFS(G).\)

Proof.

• (1) Let \( g_{1},g_{2} \in G \), then \( f^{-1}(r_{H})(g_{1}g_{2})=r_{H}(f(g_{1}g_{2}))=r_{H}(f(g_{1})f(g_{2})) \leq S(r_{H}(f(g_{1})) ,r_{H}(f(g_{2})))=S(f^{-1}(r_{H})(g_{1}),f^{-1}(r_{H})(g_{2})),\) and then \( f^{-1}(r_{H})(g_{1}g_{2}) \leq S(f^{-1}(r_{H})(g_{1}),f^{-1}(r_{H})(g_{2})). \)
• (2) Let \( g \in G \), then \(f^{-1}(r_{H})(g^{-1})=r_{H}(f(g^{-1}))=r_{H}(f^{-1}(g)) \leq r_{H}(f(g)) =f^{-1}(r_{H})(g),\) and thus \( f^{-1}(r_{H})(g^{-1}) \leq f^{-1}(r_{H})(g). \)
• (3) Let \( g_{1},g_{2} \in G \), so \( f^{-1}(w_{H})(g_{1}g_{2})=w_{H}(f(g_{1}g_{2}))=w_{H}(f(g_{1})f(g_{2})) \leq \max \lbrace w_{H}(f(g_{1})) ,w_{H}(f(g_{2})) \rbrace = \max \lbrace f^{-1}(w_{H})(g_{1}),f^{-1}(w_{H})(g_{2}) \rbrace,\) and then \( f^{-1}(w_{H})(g_{1}g_{2}) \leq \max \lbrace f^{-1}(w_{H})(g_{1}),f^{-1}(w_{H})(g_{2})) \rbrace. \)
• (4) Let \( g \in G \), then \( f^{-1}(w_{H})(g^{-1})=w_{H}(f^{-1}(g))\leq w_{H}(f(g)) =f^{-1}(w_{H})(g)\) and then \( f^{-1}(w_{H})(g^{-1}) \leq f^{-1}(w_{H})(g). \)

Therefore (1)-(4) give us \(f^{-1}(r_{H}e^{iw_{H}})(g)=r_{H}(f(g))e^{iw_{H}(f(g))} \in ACFS(G).\)

Proposition 11. Let \(\mu_{G}=r_{G}e^{iw_{G}} \in NACFS(G)\) and \(f: G \to H\) be a group homomorphism. Then \(f(\mu_{G})\in NACFS(H).\)

Proof. From Proposition 9, we have \(f(\mu_{G})\in ACFS(H).\) Let \(g_{1},g_{2} \in G\) and \(h_{1},h_{2} \in H\) such that \( f(g_{1})=h_{1} \) and \( f(g_{2})=h_{2}.\) Now \begin{eqnarray*} f(r_{G})(h_{1}h_{2}h^{-1}_{1})&=&\inf \lbrace r_{G}(g_{1}g_{2}g^{-1}_{1}) \hspace{0.1cm}|\hspace{0.1cm} f(g_{1}g_{2}g^{-1}_{1})=h_{1}h_{2}h^{-1}_{1} \rbrace\\ &=&\inf \lbrace r_{G}(g_{2}) \hspace{0.1cm}|\hspace{0.1cm} f(g_{1})f(g_{2})f(g^{-1}_{1})=h_{1}h_{2}h^{-1}_{1} \rbrace \\ &=&\inf \lbrace r_{G}(g_{2}) \hspace{0.1cm}|\hspace{0.1cm} f(g_{1})f(g_{2})f^{-1}(g_{1})=h_{1}h_{2}h^{-1}_{1} \rbrace \\ &=&\inf \lbrace r_{G}(g_{2}) \hspace{0.1cm}|\hspace{0.1cm} f(g_{2})=h_{2} \rbrace =f(r_{G})(h_{2}).\end{eqnarray*} Also\begin{eqnarray*} f(w_{G})(h_{1}h_{2}h^{-1}_{1})&=&\inf \lbrace w_{G}(g_{1}g_{2}g^{-1}_{1}) \hspace{0.1cm}|\hspace{0.1cm} f(g_{1}g_{2}g^{-1}_{1})=h_{1}h_{2}h^{-1}_{1} \rbrace \\ &=&\inf \lbrace w_{G}(g_{2}) \hspace{0.1cm}|\hspace{0.1cm} f(g_{1})f(g_{2})f(g^{-1}_{1})=h_{1}h_{2}h^{-1}_{1} \rbrace \\ &=&\inf \lbrace w_{G}(g_{2}) \hspace{0.1cm}|\hspace{0.1cm} f(g_{1})f(g_{2})f^{-1}(g_{1})=h_{1}h_{2}h^{-1}_{1} \rbrace \\ &=&\inf \lbrace w_{G}(g_{2}) \hspace{0.1cm}|\hspace{0.1cm} f(g_{2})=h_{2} \rbrace =f(w_{G})(h_{2}).\end{eqnarray*} Then \(f(\mu_{G})(h_{1}h_{2}h^{-1}_{1})=f(r_{G})(h_{1}h_{2}h^{-1}_{1})e^{if(w_{G})(h_{1}h_{2}h^{-1}_{1})}=f(r_{G})(h_{2})e^{if(w_{G})(h_{2})}=f(\mu_{G})(h_{2}).\) and so \(f(\mu_{G})\in NACFS(H).\)

Proposition 12. Let \(\mu_{H}=r_{H}e^{iw_{H}} \in NACFS(H)\) and \(f: G \to H\) be a group homomorphism, then \(f^{-1}(\mu_{H})\in NACFS(G).\)

Proof. Using Proposition 10, we get \(f^{-1}(\mu_{H})\in ACFS(G).\) Let \( g_{1} , g_{2} \in G \), then \( f^{-1}(r_{H})(g_{1}g_{2}g^{-1}_{1})=r_{H}(f(g_{1}g_{2}g^{-1}_{1})) =r_{H}(f(g_{1})f(g_{2})f(g^{-1}_{1}))=r_{H}(f(g_{1})f(g_{2})f^{-1}(g_{1}))=r_{H}(f(g_{2}))=f^{-1}(r_{H})(g_{2}).\) Also \(f^{-1}(w_{H})(g_{1}g_{2}g^{-1}_{1})=w_{H}(f(g_{1}g_{2}g^{-1}_{1}))=w_{H}(f(g_{1})f(g_{2})f(g^{-1}_{1}))=w_{H}(f(g_{1})f(g_{2})f^{-1}(g_{1})) =w_{H}(f(g_{2}))=f^{-1}(w_{H})(g_{2}).\) Thus \(f^{-1}(\mu_{H})(g_{1}g_{2}g^{-1}_{1})=f^{-1}(r_{H})(g_{1}g_{2}g^{-1}_{1})e^{if^{-1}(w_{H})(g_{1}g_{2}g^{-1}_{1})} =f^{-1}(r_{H})(g_{2})e^{if^{-1}(w_{H})(g_{2})} =f^{-1}(\mu_{H})(g_{2}) \) and thus \(f^{-1}(\mu_{H})\in NACFS(G).\)

Proposition 13. Let \(\mu_{1} =r_{1}e^{iw_{1}} \in ACFS(G)\) and \(\mu_{2} =r_{2}e^{iw_{2}} \in ACFS(G)\) and \(f: G \to H\) be a group homomorphism. If \( \mu_{1} \bowtie \mu_{2},\) then \( f(\mu_{1}) \bowtie f(\mu_{2}).\)

Proof. We know that \( f(\mu_{1})=f(r_{1})e^{if(w_{1})}\) and \( f(\mu_{2})=f(r_{2})e^{if(w_{2})}.\) By Proposition 9, we have \( f(\mu_{1}) \in ACFS(H) \) and \( f(\mu_{2}) \in ACFS(H).\) Let \( g_{1},g_{2} \in G\) and \( h_{1},h_{2} \in H\) such that \( f(g_{1})=h_{1} \) and \( f(g_{2})=h_{2}.\) Since \( \mu_{1} \bowtie \mu_{2}\) so \( r_{1}(g_{1}g_{2}g^{-1}_{1}) \leq S(r_{1}(g_{2}),r_{2}(g_{1})) \) and \( w_{1}(g_{1}g_{2}g^{-1}_{1}) \leq \max \lbrace w_{1}(g_{2}),w_{2}(g_{1}) \rbrace. \) Now \begin{eqnarray*} f(r_{1})(h_{1}h_{2}h^{-1}_{1})&=&\inf \lbrace r_{1}(g_{1}g_{2}g^{-1}_{1}) \hspace{0.1cm}|\hspace{0.1cm} f(g_{1}g_{2}g^{-1}_{1})=h_{1}h_{2}h^{-1}_{1} \rbrace \\ &\leq& \inf \lbrace S(r_{1}(g_{2}) ,r_{2}(g_{1}) ) \hspace{0.1cm}|\hspace{0.1cm} f(g_{1})f(g_{2})f(g^{-1}_{1})=h_{1}h_{2}h^{-1}_{1} \rbrace \\ &=&\inf \lbrace T(r_{1}(g_{2}) ,r_{2}(g_{1}) ) \hspace{0.1cm}|\hspace{0.1cm} f(g_{1})f(g_{2})f^{-1}(g_{1})=h_{1}h_{2}h^{-1}_{1} \rbrace \\ &=&S(\inf \lbrace r_{1}(g_{2}) \hspace{0.1cm}|\hspace{0.1cm} f(g_{2})=h_{2} \rbrace,\inf \lbrace r_{2}(g_{1}) \hspace{0.1cm}|\hspace{0.1cm} f(g_{1})=h_{1} \rbrace)\\ &=&S(f(r_{1})(h_{2}),f(r_{2})(h_{1})),\end{eqnarray*} and then \( f(r_{1})(h_{1}h_{2}h^{-1}_{1}) \leq S(f(r_{1})(h_{2}),f(r_{2})(h_{1})).\) Also \begin{eqnarray*} f(w_{1})(h_{1}h_{2}h^{-1}_{1})&=&\inf \lbrace w_{1}(g_{1}g_{2}g^{-1}_{1}) \hspace{0.1cm}|\hspace{0.1cm} f(g_{1}g_{2}g^{-1}_{1})=h_{1}h_{2}h^{-1}_{1} \rbrace \\ &\leq& \inf \lbrace \max \lbrace w_{1}(g_{2}) ,w_{2}(g_{1}) \rbrace \hspace{0.1cm}|\hspace{0.1cm} f(g_{1})f(g_{2})f(g^{-1}_{1})=h_{1}h_{2}h^{-1}_{1} \rbrace \\ &=&\inf \lbrace \max \lbrace w_{1}(g_{2}) ,w_{2}(g_{1}) \rbrace \hspace{0.1cm}|\hspace{0.1cm} f(g_{1})f(g_{2})f^{-1}(g_{1})=h_{1}h_{2}h^{-1}_{1} \rbrace \\ &=&\max \lbrace \inf \lbrace w_{1}(g_{2}) \hspace{0.1cm}|\hspace{0.1cm} f(g_{2})=h_{2} \rbrace,\inf \lbrace w_{2}(g_{1}) \hspace{0.1cm}|\hspace{0.1cm} f(g_{1})=h_{1} \rbrace \rbrace \\ &=&\max \lbrace f(w_{1})(h_{2}),f(w_{2})(h_{1}) \rbrace,\end{eqnarray*} and so \(f(w_{1})(h_{1}h_{2}h^{-1}_{1}) \leq \max \lbrace f(w_{1})(h_{2}),f(w_{2})(h_{1}) \rbrace .\) Hence \( f(\mu_{1}) \bowtie f(\mu_{2}).\)

Proposition 14. Let \(\mu_{1} =r_{1}e^{iw_{1}} \in ACFS(H)\), \(\mu_{2} =r_{2}e^{iw_{2}} \in ACFS(H)\) and \(f: G \to H\) be a group homomorphism. If \( \mu_{1} \bowtie \mu_{2},\) then \( f^{-1}(\mu_{1}) \bowtie f^{-1}(\mu_{2}).\)

Proof. Let \( f^{-1}(\mu_{1})=f^{-1}(r_{1})e^{if^{-1}(w_{1})}\) and \( f^{-1}(\mu_{2})=f^{-1}(r_{2})e^{if^{-1}(w_{2})}\). From Proposition 10, we obtain \( f^{-1}(\mu_{1}) \in ACFS(G) \) and \( f^{-1}(\mu_{2}) \in ACFS(G).\) Let \( g_{1},g_{2} \in G \), then \begin{eqnarray*} f^{-1}(r_{1})(g_{1}g_{2}g^{-1}_{1})&=&r_{1}(f(g_{1}g_{2}g^{-1}_{1}))=r_{1}(f(g_{1})f(g_{2})f(g^{-1}_{1}))=r_{1}(f(g_{1})f(g_{2})f^{-1}(g_{1}))\\ &\leq& S(r_{1}(f(g_{2})),r_{2}(f(g_{1})))=S(f^{-1}(r_{1})(g_{2}),f^{-1}(r_{2})(g_{1}).\end{eqnarray*} Also \begin{eqnarray*}f^{-1}(w_{1})(g_{1}g_{2}g^{-1}_{1})&=&w_{1}(f(g_{1}g_{2}g^{-1}_{1}))=w_{1}(f(g_{1})f(g_{2})f(g^{-1}_{1}))=w_{1}(f(g_{1})f(g_{2})f^{-1}(g_{1}))\\ &\leq& \max \lbrace w_{1}(f(g_{2})),w_{2}(f(g_{1})) \rbrace=\max \lbrace f^{-1}(w_{1})(g_{2}),f^{-1}(w_{2})(g_{1} \rbrace.\end{eqnarray*} Therefore \( f^{-1}(\mu_{1}) \bowtie f^{-1}(\mu_{2}). \)

Acknowledgments

I would like to thank the referees for carefully reading the manuscript and making several helpful comments to increase the quality of the paper.

Conflict of Interests

The author declares no conflict of interest.