1. Introduction

Let \(G=(V;E)\) be a simple connected graph. In chemical graph theory, the sets of vertices and edges of \(G\) are denoted by \(V=V(G)\) and \(E=E(G)\), respectively. A molecular graph is a simple finite graph such that its vertices correspond to the atoms and the edges to the bonds. A general reference for the notation in graph theory is [1, 2, 3].

In chemical graph theory, we have many different topological index of arbitrary molecular graph \(G\). A topological index of a graph is a number related to a graph which is invariant under graph automorphisms. Obviously, every topological index defines a counting polynomial and vice versa.

A graph can be recognized by a numeric number, a polynomial, a sequence of numbers or a matrix which represents the whole graph, and these representations are aimed to be uniquely defined for that graph. A topological index is a numeric quantity associated with a graph which characterizes the topology of the graph and is invariant under graph automorphism. There are some major classes of topological indices such as distance based topological indices, degree based topological indices and counting related polynomials and indices of graphs. Among these classes degree based topological indices are of great importance and play a vital role in chemical graph theory and particularly in chemistry.

Among topological descriptors, connectivity indices are very important and they have a prominent role in chemistry. One of the best known and widely used is the connectivity index \(R(G)=\sum_{uv \in E(G)} \frac{1}{\sqrt{d_ud_v}}\) introduced in 1975 by Milan Randić (see [4]), who has shown this index to reflect molecular branching.

The Sanskruti index \(S(G)\) of a graph \(G\) is defined as follows (see [5]):

where \(S_u\) is the summation of degrees of all neighbors of vertex \(u\) in \(G\). In this paper, we compute the Sanskruti index for an infinite class of Titania nanotubes. For more details about Sanskruti index we refer to the articles [6, 7, 8, 9, 10, 11].

2. Main results and discussion

As a well-known semiconductor with numerous technological applications, Titania is comprehensively studied in materials science. Titania nanotubes were systematically synthesized during the last 10-15 years using different methods and carefully studied as prospective technological materials. Since the growth mechanism for \(TiO_2\) nanotubes is still not well defined, their comprehensive theoretical studies attract enhanced attention. \(TiO_2\) sheets with a thickness of a few atomic layers were found to be remarkably stable [12]. Recently Malik and Imran [13] and some others researchers computed multiple Zagreb index of \(TiO_2\) nanotubes, readers can see the paper series [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27].

The graph of the Titania nanotube \(TiO_2[m, n]\) is presented in Fig. 1 where \(m\) denotes the number of octagons in a row and \(n\) denotes the number of octagons in a column of the Titania nanotube.

Let us define the partitions for the vertex set and the edge set of the Titania nanotube \(TiO_2\), for \(\delta(G)\leq k\leq \Delta(G)\), \(2\delta(G)\leq i\leq 2\Delta(G)\), and \(\delta(G)^2\leq j\leq \Delta(G)^2\), then we have [28]

$$V_k=\{v\in V(G)|deg(v)=k\},$$ $$E_i=\{e=uv\in E(G)|d(u)+d(v)=i\}.$$ In the molecular graph of \(TiO_2\)-nanotube, we can see that \(2\leq d(G)\leq5\). So, we have the vertex partitions as follows.
\(V_2=\{u\in V(G)|d(u)=2\}\), \(V_3=\{u\in V(G)|d(u)=3\}\).
\(V_4=\{u\in V(G)|d(u)=4\}\), \(V_5=\{u \in V(G)|d(u)=5\}\).
It is easy to see that \(|V_2|=2mn+4n\), \(|V_3|=2mn\), \(|V_4|=2n\), \(|V_5|=2mn\), So we have \(|V(TiO_2)|=6n(m+1)\). Similarly, the edge partitions of the graph \(TiO_2\) are as follows.
\(E_6=\{e=uv\in E(G)|d(u)=2\) and \(d(v)=4\}.\)
\(E_7=\{e=uv\in E(G)|d(u)=2\) and \(d(v)=5\}\)
\(\cup\{e=uv\in E(G)|d(u)=3\) and \(d(v)=4\}\).
\(E_8=\{e=uv\in E(G)|d(u)=3\) and \(d(v)=5\}\).
The vertex partition \(V_k\) and the edge partitions \(E_i\) are collectively exhaustive, that is
$$\bigcup \limits_ {k=\delta(G)}^{\Delta(G)} V_k=V(G), \bigcup \limits_ {i=2\delta(G)}^{2\Delta(G)-2} E_i=E(G).$$

Table 1. The edge partitions based on degrees of end vertices

Edge partition	\(E_6\)	\(E_7\)	\(E_8\)
Cardinality	\(6n\)	\(4mn+4n\)	\(6mn-2n\)

Table 2. The edge partitions based on degree sum of neighbors of end vertices

\((S_u,S_v)\) where \(uv\in E(TiO_2[m,n])\)	(10,5)	(7,5)	(7,9)	(8,9)	(10,9)	(11,9)
Numbers of edges	\(2\)	\(2\)	\(2n\)	\(4n\)	\(2n\)	\(6m\)

Table 3. The edge partitions based on degree sum of neighbors of end vertices

\((S_u,S_v)\) \(\forall uv\in E(TiO_2[m,n])\)	(13,9)	(7,13)	(10,13)	(11,13)	(13,13)
Numbers of edges	\(3m\)	\(2n\)	\(4mn+2n\)	\(2mn-2n\)	\(6mn-4n\)

Theorem 2.1. The Sanskruti index \(S(G)\) of Titania nanotube \(G=TiO_2[m,n]\) is given by $$S(G)=\frac{1753571}{8788}-\frac{5732099098294657}{19655694576000}n+\frac{12790839}{8000}m+\frac{8788000}{9261}mn+\frac{6092281}{2304}mn.$$

Proof. To compute the Sanskruti index \(S(G)\) of Titania nanotube \(TiO_2[m, n]\), we need an edge partition of the Titania nanotube \(TiO_2[m, n]\), based on degree sum of neighbors of end vertices of each edge. We presented these partitions with their cardinalities in Tables 2 and 3. Now using Eq. (1) and Tables 2 and 3 we get the required result as follows. \begin{eqnarray*} S(TiO_2[m, n])&=& \sum_{uv \in E(G)} (\frac{S_uS_v}{S_u+S_v-2})^3\\ &=&2(\frac{10\times5}{10+5-2})^3+2(\frac{7\times5}{7+5-2})^3+2n(\frac{7\times9}{7+9-2})^3\\ &&+4n(\frac{8\times9}{8+9-2})^3+2n(\frac{10\times9}{10+9-2})^3+6m(\frac{11\times9}{11+9-2})^3\\ &&+3m(\frac{13\times9}{13+9-2})^3+2n(\frac{7\times13}{7+13-2})^3+(4mn+2n)(\frac{10\times13}{10+13-2})^3\\ &&+(2mn-2n)(\frac{10\times5}{10+5-2})^3+2(\frac{10\times5}{10+5-2})^3\\ &=&\frac{1753571}{8788}-\frac{5732099098294657}{19655694576000}n+\frac{12790839}{8000}m\\ &&+\frac{8788000}{9261}mn+\frac{6092281}{2304}mn. \end{eqnarray*} The proof of theorem is completed.

Competing Interests

The author(s) do not have any competing interests in the manuscript.

1. Introduction

In this paper, we shall take \(H\) as a real Hilbert space, \(\langle, \rangle\) as inner product, \(\|.\|\) as the induced norm, and \(C\) as a nonempty closed subset of \(H\).

Definition 1.1. Let \(T : H \to H\) be a mapping. \(T\) is called non-expansive if $$\|T(x) - T(y)\|\leq\|x-y\|, \quad \forall x,y \in H$$

Definition 1.2. A mapping \(f:H\rightarrow H\) is called a contraction if for all \(x,y\in H\) and \(\theta\in[0,1)\) $$\|f(x)-f(y)\|\leq \theta\|x-y\|.$$

Definition 1.3. \(P_c: H\rightarrow C\) is called a metric projection if for every \(x\in H\) there exists a unique nearest point in \(C\), denoted by \(P_cx\), such that $$\|x-P_cx\|\leq \|x-y\|, \quad \forall\,\, y\in C.$$

In order to verify the weak convergence of an algorithm to a fixed point of a non-expansive mapping we need the demiclosedness principle:

Theorem 1.4. [1] (The demiclosedness principle) Let \(C\) be a nonempty closed convex subset of the real Hilbert space \(H\) and \(T:C\rightarrow C\) such that $$x_n\rightharpoonup x^\ast \in C\,\, \mbox{and}\,\, (I-T)x_n \rightarrow 0$$ Then \(x^\ast=Tx^\ast\). Here, \(\rightarrow\) and \(\rightharpoonup\) denotes strong and weak convergence respectively.

Moreover, the following result gives the conditions for the convergence of a nonnegative real sequence.

Theorem 1.5. [2] Assume that \(\{a_n\}\) is a sequence of nonnegative real numbers such that \(a_{n+1}\leq(1-\gamma_n)a_n+\delta_n, \forall n\geq0\), where \(\{\gamma_n\}\) is a sequence in \((0,1)\) and \(\{\delta_n\}\) is a sequence with

1. \(\sum_{n=0}^\infty\gamma_n=\infty\),
2. \( \lim_{n\rightarrow\infty}\sup\frac{\delta_n}{\gamma_n}\leq0\) or \(\sum_{n=0}^{\infty}|\delta_n|< \infty\).

Then \(a_n\rightarrow0\).

The following strong convergence theorem, which is also called the viscosity approximation method, for non-expansive mappings in real Hilbert spaces is given by Moudafi, [3], in (2000).

Theorem 1.6. Let \(C\) be a non-empty closed convex subset of the real Hilbert space \(H\). Let \(T\) be a non-expansive mapping of \(C\) into itself such that \(F(T) = \{x \in C : T(x) = x \}\) is nonempty. Let \(f\) be a contraction of \(C\) into itself. Consider the sequence $$x_{n+1} = \frac{\epsilon_n}{1+\epsilon_n}f(x_n)+\frac{1}{1+\epsilon_n}T(x_n), \quad n\geq 0,$$ where the sequence \(\{\epsilon_n\}\in (0,1)\) satisfies

1. \(\lim_{n\rightarrow\infty} \epsilon_n= 0\),
2. \(\sum_{n=0}^{\infty}\epsilon_n=\infty\), and
3. \(\lim_{n\rightarrow\infty}|\frac{1}{\epsilon_{n+1}}- \frac{1}{\epsilon_n}|=0\).

Then \({x_n}\) converges strongly to a fixed point \(x^\ast\) of the non-expansive mapping \(T\), which is also the unique solution of the variational inequality \begin{equation*} \langle(I-f)x, y-x\rangle \geq 0, \quad \forall\in F(T). \end{equation*}

In (2015), Xu et al. [2] applied viscosity method on the midpoint rule for non-expansive mappings and give the generalized viscosity implicit rule (GVIR): $$x_{n+1}=\alpha_nf(x_n)+(1-\alpha_n)T\left(\frac{x_n+x_{n+1}}{2}\right), \quad \forall n \geq 0$$ This, using contraction, regularizes the implicit midpoint rule for nonexpansive mappings. They also proved that the sequence generated by GMIR converges strongly to a fixed point of \(T\). Ke et al. [4], motivated and inspired by the idea of Xu et al. [2], proposed two generalized viscosity implicit rules: \begin{equation*} x_{n+1}=\alpha_nf(x_n)+(1-\alpha_n)T\left(s_nx_n+(1-s_n)x_{n+1}\right), \end{equation*} \begin{equation*} x_{n+1}=\alpha_nx_n+\beta f(x_n)+\gamma_nT(s_nx_n+(1-s_n)x_{n+1}). \end{equation*} Our contribution in this direction is the following viscosity rule for common fixed points of two nonexpansive mappings in Hilbert spaces: \begin{equation*} x_{n+1} = \alpha_n f(x_n) + \beta_n S\left(\frac{x_{n+1} + x_n}{2}\right) + \gamma_n T\left(\frac{x_{n+1}+x_n}{2}\right) \end{equation*}

2. Main Result

Theorem 2.1. Let \(C\) be a nonempty closed convex subset of the real Hilbert space \(H\). Let \(S:C\rightarrow C\) and \(T:C\rightarrow C\) be two nonexpansive mappings with \(U:=F(T)\cap F(S)\neq \phi\) and \(f:C\rightarrow C\) be a contraction with coefficient \(\theta \in [0,1)\). Let \(\{x_n\}\) be a sequence in \(C\) generated by \begin{equation}\label{eq3.1} x_{n+1}=\alpha_nf(x_n)+\beta_nS\left(\frac{x_{n+1}+x_n}{2}\right)+\gamma_nT\left(\frac{x_{n+1}+x_n}{2}\right) , \end{equation} where \(\{\alpha_n\},\{\beta_n\},\{\gamma_n\}\subset(0,1)\), satisfying the following conditions:

1. \(\alpha_n + \beta_n + \gamma_n = 1\) and \(\lim\limits_{n\rightarrow \infty} \gamma_n = 1\),
2. \(\sum\limits_{n=0}^\infty |\alpha_{n+1}-\alpha_n|< \infty\) and \(\sum\limits_{n=0}^\infty |\beta_{n+1}-\beta_n|< \infty\),
3. \(\sum\limits_{n=0}^\infty\alpha_n=\infty\),
4. \(\lim\limits_{n\rightarrow\infty}||{T\left(\frac{x_{n+1}+x_n}{2}\right)-S\left(\frac{x_{n+1}+x_n}{2}\right)}=0\) and \(\lim\limits_{n\rightarrow \infty}\alpha_n=\lim\limits_{n\rightarrow \infty}\beta_n=0\).

Then \(\{x_n\}\) converges strongly to a common fixed point \(x^\ast\) of the nonexpansive mappings \(T\) and \(S\) which is also the unique solution of the variational inequality $$\langle(I-f)x,y-x\rangle, \quad \forall \quad y\in U$$ In other words, \(x^\ast\) is the unique fixed point of the contraction \(P_Uf\).

Proof. We will prove this theorem into the following five steps:
Step 1. Firstly, we want to show that the sequence \(\{x_n\}\) is bounded. Indeed, take \(p\in U\) arbitrarily, we have \begin{eqnarray*} \|x_{n+1}-p\|&=&\left\|- p + \alpha_nf(x_n)+\beta_nS\left(\frac{x_{n+1}+x_n}{2}\right)+\gamma_nT\left(\frac{x_{n+1}+x_n}{2}\right)\right\|\\ &=&\left\| -(\alpha_n+\beta_n+\gamma_n)p + \alpha_nf(x_n)+\beta_nS\left(\frac{x_{n+1}+x_n}{2}\right) \right. \\ && \left. + \gamma_nT\left(\frac{x_{n+1}+x_n}{2}\right) \right\| \\ &\leq&\alpha_n\|f(x_n)-p\|+\beta_n\left\|S\left(\frac{x_{n+1}+x_n}{2}\right)-p\right\| \\ && + \gamma_n \left\| T\left(\frac{x_{n+1}+x_n}{2}\right) - p \right\| \\ &\leq&\alpha_n\|f(x_n)-f(p)\|+\alpha_n\|f(p)-p\|+\beta_n\left\|\frac{x_{n+1}+x_n}{2}-p\right\| \\ && + \gamma_n\left\|\frac{x_{n+1}+x_n}{2}-p\right\|\\ &\leq&\theta\alpha_n\|x_n-p\|+\alpha_n\|f(p)-p\|+(\beta_n+\gamma_n)\left\|\frac{x_{n+1}+x_n}{2}-p\right\|\\ &=&\theta\alpha_n\|x_n-p\|+\alpha_n\|f(p)-p\|+(1-\alpha_n)\left\|\frac{x_{n+1}+x_n}{2}-p\right\|\\ &\leq&\theta\alpha_n\|x_n-p\|+\alpha_n\|f(p)-p\|+\frac{1-\alpha_n}{2}\|x_{n+1}-p\| \\ && + \frac{1-\alpha_n}{2}\|x_n-p\|. \end{eqnarray*} This is equivalent to \begin{equation*} \left(1-\frac{1-\alpha_n}{2}\right)\|x_{n+1}-p\| \leq \left(\frac{1-\alpha_n}{2}+\alpha_n\theta\right)\|x_n-p\|+\alpha_n\|f(p)-p\| \end{equation*} \(\Rightarrow\) \begin{equation*} (1+\alpha_n)\|x_{n+1}-p\| \leq (1-\alpha_n+2\alpha_n\theta)\|x_n-p\|+2\alpha_n\|f(p)-p\| \end{equation*} \(\Rightarrow\) \begin{eqnarray*} \|x_{n+1}-p\| &\leq& \frac{1+\alpha_n-2\alpha_n+2\alpha_n\theta}{1+\alpha_n}\|x_n-p\| + \frac{2\alpha_n}{1+\alpha_n}\|f(p)-p\| \\ &=&\left(1-\frac{2\alpha_n(1-\theta)}{1+\alpha_n}\right)\|x_n-p\| \\ && + \frac{2\alpha_n(1-\theta)}{1+\alpha_n} \left(\frac{1}{1-\theta} \|f(p) - p\| \right). \end{eqnarray*} Thus, $$\|x_{n+1}-p\|\leq\max\left\{\|x_n-p\|,\frac{1}{1-\theta}\|f(p)-p\|\right\}.$$ Similarly, $$\|x_n-p\|\leq\max\left\{\|x_{n-1}-p\|,\left(\frac{1}{1-\theta}\|f(p)-p\|\right)\right\}.$$ From this, we obtain, \begin{eqnarray*} \|x_{n+1}-p\|&\leq&\max\left\{\|x_n-p\|,\frac{1}{1-\theta}\|f(p)-p\| \right\} \\ &\leq& \max\left\{\|x_{n-1}-p\|, \frac{1}{1-\theta}\|f(p)-p\| \right\} \\ &.&\\ &.&\\ &.&\\ &.&\\ &\leq&\max\left\{\|x_0-p\|, \frac{1}{1-\theta}\|f(p)-p\| \right\}. \end{eqnarray*} Hence, we concluded that \(\{x_n\}\) is a bounded sequence. Consequently, \(\{f(x_n)\}\), \(\big\{S\big(\frac{x_{n+1}+x_n}{2}\big)\big\}\) and \(\big\{T\big(\frac{x_{n+1}+x_n}{2}\big)\big\}\) are bounded.
Step 2. Now, we prove that \(\lim\limits_{n\rightarrow\infty}\|x_{n+1}-x_n\| = 0\). \begin{eqnarray*} &&\|x_{n+1}-x_n\|\\ &=&\left\|\alpha_nf(x_n)+\beta_nS\left(\frac{x_{n+1}+x_n}{2}\right)+\gamma_nT\left(\frac{x_{n+1}+x_n}{2}\right) \right.\\ && - \left. \alpha_{n-1}f(x_{n-1})+\beta_{n-1}S\left(\frac{x_n+x_{n-1}}{2}\right)+\gamma_{n-1}T\left(\frac{x_n+x_{n-1}}{2}\right) \right\|\\ &=& \left\|\alpha_n(f(x_n)-f(x_{n-1}))+(\alpha_n-\alpha_{n-1})f(x_{n-1}) + \beta_n\left(S\left(\frac{x_{n+1}+x_n}{2}\right) \right. \right.\\ && - \left. S\left(\frac{x_n+x_{n-1}}{2}\right)\right) + (\beta_n-\beta_{n-1})S\left(\frac{x_n+x_{n-1}}{2}\right) \\ && + \gamma_n\left(T\left(\frac{x_{n+1}+x_n}{2}\right)-T\left(\frac{x_n+x_{n-1}}{2}\right)\right) \\ && + \left. (\gamma_n-\gamma_{n-1})T\left(\frac{x_n+x_{n-1}}{2}\right)\right\| \\ &=& \left\|\alpha_n(f(x_n)-f(x_{n-1}))+(\alpha_n-\alpha_{n-1})f(x_{n-1}) + \beta_n\left(S\left(\frac{x_{n+1}+x_n}{2}\right) \right. \right. \\ && - \left. S\left(\frac{x_n+x_{n-1}}{2}\right) \right) + (\beta_n-\beta_{n-1})S\left(\frac{x_n+x_{n-1}}{2}\right) \\ && + \gamma_n\left(T\left(\frac{x_{n+1}+x_n}{2}\right)-T\left(\frac{x_n+x_{n-1}}{2}\right)\right)\\ && + \left.(\alpha_n-\alpha_{n-1}+\beta_n-\beta_{n-1})T\left(\frac{x_n+x_{n-1}}{2}\right)\right\| \\ &=& \left\|\alpha_n(f(x_n)-f(x_{n-1}))+(\alpha_n-\alpha_{n-1})\left(f(x_{n-1})-T\left(\frac{x_n+x_{n-1}}{2}\right)\right) \right. \\ && + \beta_n\left(S\left(\frac{x_{n+1}+x_n}{2}\right)-S\left(\frac{x_n+x_{n-1}}{2}\right)\right) \\ && + (\beta_n-\beta_{n-1})\left(S\left(\frac{x_n+x_{n-1}}{2}\right)-T\left(\frac{x_n+x_{n-1}}{2}\right)\right)\\ && + \left. \gamma_n\left(T\left(\frac{x_{n+1}+x_n}{2}\right)-T\left(\frac{x_n+x_{n-1}}{2}\right)\right)\right\| \\ &\leq& \alpha_n \big\|f(x_n)-f(x_{n-1})\big\|+|\alpha_n-\alpha_{n-1}|\left\|f(x_{n-1})-T\left(\frac{x_n+x_{n-1}}{2}\right)\right\| \\ && + \beta_n\left\|S\left(\frac{x_{n+1}+x_n}{2}\right)-S\left(\frac{x_n+x_{n-1}}{2}\right)\right\| \\ && + |\beta_n-\beta_{n-1}| \left\|S\left(\frac{x_n+x_{n-1}}{2}\right)-T\left(\frac{x_n+x_{n-1}}{2}\right)\right\| \\ && + \gamma_n \left\| T\left(\frac{x_{n+1}+x_n}{2}\right)-T\left(\frac{x_n+x_{n-1}}{2}\right)\right\|. \end{eqnarray*} Let \(M_2\) be a number such that \(M_2 \geq \max \left\{\sup\limits_{n\geq0} \left\|S\left(\frac{x_{n+1}+x_n}{2}\right)-T\left(\frac{x_{n+1}+x_n}{2}\right)\right\|, \right.\) \(\left.\sup\limits_{n\geq0} \left\|f(x_n) - T\left(\frac{x_{n+1}+x_n}{2}\right)\right\| \right\}\). Thus, the above is equivalent to \begin{eqnarray*} && \|x_{n+1}-x_n\| \\ &\leq& \alpha_n \theta \left\|x_n - x_{n-1}\right\| + \beta_n \left\|\frac{x_{n+1} + x_n}{2} - \frac{x_n + x_{n-1}}{2}\right\| \\ && + \gamma_n \left\|\frac{x_{n+1} + x_n}{2} - \frac{x_n + x_{n-1}}{2}\right\| + \{|\alpha_n - \alpha_{n-1}| + |\beta_n - \beta_{n-1}|\} M_2 \\ &\leq& \alpha_n \theta \left\|x_n - x_{n-1}\right\| \\ &&+ \frac{\beta_n}{2} \left\|x_{n+1} - x_n \right\|+ \frac{\beta_n}{2} \left\|x_n - x_{n-1}\right\| + \frac{\gamma_n}{2} \left\|x_{n+1} - x_n\right\| \\ && + \frac{\gamma_n}{2} \left\|x_n - x_{n-1}\right\| + \{|\alpha_n - \alpha_{n-1}| + |\beta_n - \beta_{n-1}|\}M_2 \\ &=&\left(\alpha_n\theta+\frac{\beta_n}{2}+\frac{\gamma_n}{2}\right)\|x_n-x_{n-1}\|+\left(\frac{\beta_n}{2}+\frac{\gamma_n}{2}\right)\|x_{n+1}-x_n\| \\ && + (|\alpha_n-\alpha_{n-1}|+|\beta_n-\beta_{n-1}|)M_2\\ &=&\left(\alpha_n\theta+\frac{1-\alpha_n}{2}\right)\|x_n-x_{n-1}\|+\frac{1-\alpha_n}{2}\|x_{n+1}-x_n\| \\ && + (|\alpha_n-\alpha_{n-1}|+|\beta_n-\beta_{n-1}|)M_2. \end{eqnarray*} Combining the common terms from left and right hand sides, we get, \begin{eqnarray*} \left(1-\frac{1-\alpha_n}{2}\right)\|x_{n+1}-x_n\| &\leq& \left(\alpha_n\theta+\frac{1-\alpha_n}{2}\right)\|x_n-x_{n-1}\| \\ && + (|\alpha_n-\alpha_{n-1}|+|\beta_n-\beta_{n-1}|)M_2. \end{eqnarray*} This implies that \begin{eqnarray*} &&\|x_{n+1}-x_n\| \\ &\leq& \frac{1+\alpha_n-2\alpha_n+2\alpha_n\theta}{1+\alpha_n}\|x_n-x_{n-1}\|\\&& + \{\frac{2(|\alpha_n-\alpha_{n-1}|+|\beta_n-\beta_{n-1}|)}{1+\alpha_n}\} M_2 \\ &=&\left(1-\frac{2\alpha_n(1-\theta)}{1+\alpha_n}\right)\|x_n-x_{n-1}\|\\&& + \{\frac{2(|\alpha_n-\alpha_{n-1}|+|\beta_n-\beta_{n-1}|)}{1+\alpha_n}\} M_2. \end{eqnarray*} Note that \(\sum\limits_{n=0}^\infty \alpha_n = \infty\), \(\sum\limits_{n=0}^\infty |\alpha_{n+1}-\alpha_n| < \infty\) and \(\sum\limits_{n=0}^\infty|\beta_{n+1}-\beta_n| < \infty \). Using Theorem 1.5, we have \(\|x_{n+1}-x_n\|\rightarrow0\) as \(n\rightarrow\infty\).
Step 3. Now, we will show that \(\lim\limits_{n\rightarrow\infty} \|x_n-Sx_n\|=0\) and \(\lim\limits_{n\rightarrow\infty} \|x_n-Tx_n\|=0\). Consider \begin{eqnarray*} && \|x_n-S(x_n)\| \\ &=& \left\|x_n-x_{n+1}+x_{n+1}-T\left(\frac{x_{n+1}+x_n}{2}\right)+T\left(\frac{x_{n+1}+x_n}{2}\right) \right. \\ && - \left. S \left(\frac{x_{n+1}+x_n}{2}\right)+S\left(\frac{x_{n+1}+x_n}{2}\right)-S(x_n) \right\| \\ &\leq& \|x_n-x_{n+1}\|+ \left\|x_{n+1} - T\left(\frac{x_{n+1}+x_n}{2}\right)\right\| + \left\| T\left(\frac{x_{n+1} + x_n}{2}\right) \right. \\ && \left. - S\left(\frac{x_{n+1}+x_n}{2}\right) \right\| + \left\|S\left(\frac{x_{n+1}+x_n}{2}\right)-S(x_n)\right\| \\ &\leq& \|x_n-x_{n+1}\| + \left\| \alpha_n f(x_n) + \beta_n S\left(\frac{x_{n+1} + x_n}{2} \right) + \gamma_n T\left(\frac{x_{n+1} + x_n}{2} \right) \right. \\ && \left. - T\left(\frac{x_{n+1} + x_n}{2}\right) \right\| + \left\|\frac{x_{n+1}+x_n}{2} - x_n\right\| \\ && + \left\|T\left(\frac{x_{n+1}+x_n}{2}\right)-S\left(\frac{x_{n+1}+x_n}{2}\right)\right\| \\ &\leq& \|x_n-x_{n+1}\| + \alpha_n \left\| f(x_n) - T\left(\frac{x_{n+1}+x_n}{2}\right) \right\| \\ && + \beta_n \left\| S\left(\frac{x_{n+1}+x_n}{2}\right) - T\left(\frac{x_{n+1}+x_n}{2}\right)\right\| + \frac{1}{2}\|x_{n+1}-x_n\|\\ && + \left\|S\left(\frac{x_{n+1}+x_n}{2}\right)-T\left(\frac{x_{n+1}+x_n}{2}\right)\right\| \\ &\leq& \frac{3}{2}\|x_n-x_{n+1}\| + \alpha_n \left\|f(x_n) - T\left(\frac{x_{n+1} + x_n}{2}\right)\right\| \\ && + (1+\beta_n)\left\|S\left(\frac{x_{n+1}+x_n}{2}\right) - T\left(\frac{x_{n+1}+x_n}{2}\right)\right\|. \end{eqnarray*} Since, \(\lim\limits_{n\rightarrow\infty} \alpha_n = \lim\limits_{n\rightarrow\infty} \left\|T\left(\frac{x_{n+1} + x_n}{2}\right) - S\left(\frac{x_{n+1} + x_n}{2}\right)\right\| = 0\) and \(\lim\limits_{n\rightarrow\infty} \|x_{n+1} - x_n\| \rightarrow 0\), we get \(\|x_n-S(x_n)\|\rightarrow0\) as \(n\rightarrow\infty\) Moreover, we have \begin{eqnarray*} \left\|S \left(\frac{x_{n+1} + x_n}{2}\right) - x_n\right\| &=& \left\|S\left(\frac{x_{n+1} + x_n}{2} \right) - S(X_n) + S(x_n) - x_n\right\| \\ &\leq& \left\|S \left(\frac{x_{n+1} + x_n}{2}\right) - S(x_n)\right\| + \left\|S(x_n) - x_n\right\| \\ &\leq& \left\|\frac{x_{n+1} + x_n}{2} - x_n\right\| + \|S(x_n)-x_n\| \\ &=& \frac{1}{2}\|x_{n+1} - x_n\| + \|S(x_n) - x_n\| \\ &\to& 0, \quad \quad \text{ as } (n \to \infty). \end{eqnarray*} Now, consider \begin{eqnarray*} &&\|x_n-T(x_n)\|\\ &=& \left\|x_n-x_{n+1}+x_{n+1}-S\left(\frac{x_{n+1}+x_n}{2}\right)+S\left(\frac{x_{n+1}+x_n}{2}\right) \right.\\ && - \left. T\left(\frac{x_{n+1}+x_n}{2}\right)+T\left(\frac{x_{n+1}+x_n}{2}\right)-T(x_n) \right\| \\ &\leq& \|x_n-x_{n+1}\| + \left\|x_{n+1} - S\left(\frac{x_{n+1}+x_n}{2}\right)\right\| + \left\|T\left(\frac{x_{n+1}+x_n}{2}\right)-T(x_n)\right\| \\ && + \left\|S\left(\frac{x_{n+1}+x_n}{2}\right) - T\left(\frac{x_{n+1}+x_n}{2}\right)\right\| \\ &\leq& \|x_n-x_{n+1}\| + \left\|T\left(\frac{x_{n+1}+x_n}{2}\right) - S\left(\frac{x_{n+1}+x_n}{2}\right)\right\| + \\ && \left\|\alpha_nf(x_n) + \beta_nS\left(\frac{x_{n+1}+x_n}{2}\right) + \gamma_nT\left(\frac{x_{n+1}+x_n}{2}\right) - S\left(\frac{x_{n+1}+x_n}{2}\right) \right\| \\ && + \left\|\frac{x_{n+1}+x_n}{2}-x_n\right\| \\ &\leq& \left\|x_n-x_{n+1}\right\| + \alpha_n\left\|f(x_n)-T\left(\frac{x_{n+1}+x_n}{2}\right)\right\| +\frac{1}{2}\|x_{n+1}-x_n\| \\ && + \gamma_n\left\|S\left(\frac{x_{n+1}+x_n}{2}\right)-T\left(\frac{x_{n+1}+x_n}{2}\right)\right\| \\ && + \left\|S\left(\frac{x_{n+1}+x_n}{2}\right)-T\left(\frac{x_{n+1}+x_n}{2}\right)\right\| \\ &\leq& \frac{3}{2}\|x_n-x_{n+1}\| + \alpha_n \left\|f(x_n)-T\left(\frac{x_{n+1}+x_n}{2}\right)\right\| \\ && + (1+\gamma_n)\left\|S\left(\frac{x_{n+1}+x_n}{2}\right)-T\left(\frac{x_{n+1}+x_n}{2}\right)\right\| . \end{eqnarray*} Since, \(\lim\limits_{n \to \infty} \alpha_n = \lim\limits_{n \to \infty} \left\|T\left(\frac{x_{n+1} + x_n}{2}\right) - S\left(\frac{x_{n+1} + x_n}{2}\right)\right\| = 0\) and \(\lim\limits_{n \to \infty} \|x_{n+1}-x_n\| \to 0\), we get \(\|x_n-Tx_n\| \to 0\) as \(n \to \infty\). Also, \begin{eqnarray*} \left\|T\left(\frac{x_{n+1}+x_n}{2}\right)-x_n\right\| &=& \left\|T\left(\frac{x_{n+1}+x_n}{2}\right)-T(X_n)+T(x_n)-x_n\right\| \\ &\leq& \left\|T \left(\frac{x_{n+1} + x_n}{2}\right) - T(x_n)\right\| + \left\|T(x_n) - x_n\right\| \\ &\leq& \left\|\frac{x_{n+1}+x_n}{2}-x_n\right\| + \left\|T(x_n) - x_n\right\| \\ &=& \frac{1}{2}\left\|x_{n+1} - x_n\right\| + \left\|T(x_n) - x_n\right\| \\ &\to& 0 \quad \quad \quad (\text{ as } n \to \infty) \end{eqnarray*} Step 4. In this step, we will show that \(\limsup\limits_{n \to \infty} \langle x^\ast - f(x^\ast), x^\ast - x_n \rangle \leq 0\), where, \(x^\ast = P_Uf(x^\ast)\).
Indeed, we take a subsequence, \(\{x_{n_i}\}\) of \(\{x_n\}\), which converges weakly to a fixed point \(p \in U = F(T) \cap F(S)\). Without loss of generality, we may assume that \(\{x_{n_i}\} \rightharpoonup p\). From \(\lim\limits_{n \to \infty} ||{x_n - S(x_n)} = 0\), \(\lim\limits_{n \to \infty} ||{x_n - T(x_n)} = 0\) and Theorem 1.4, we have \(p = S(p)\) and \(p = T(p)\). This together with the property of the metric projection implies that \begin{equation*} \limsup\limits_{n \to \infty} \langle x^\ast - f(x^\ast), x^\ast - x_n \rangle = \langle x^\ast - f(x^\ast), x^\ast - p \rangle \leq 0. \end{equation*} Step 5. Finally, we show that \(x_n \rightarrow x^\ast\) as \(n\rightarrow\infty\). Again, take \)x^\ast \in U\) to be the unique fixed point of the contraction \(P_Uf\). Consider \begin{eqnarray*} &&\|x_{n+1}-x_n\|^2 \nonumber \\ &=& \left\|\alpha_nf(x_n)+\beta_nS\left(\frac{x_{n+1}+x_n}{2}\right)+\gamma_nT\left(\frac{x_{n+1}+x_n}{2}\right)-x^\ast\right\| ^2\nonumber\\ &=& \left\|\alpha_nf(x_n)+\beta_nS\left(\frac{x_{n+1}+x_n}{2}\right)+\gamma_nT\left(\frac{x_{n+1}+x_n}{2}\right)-(\alpha_n+\beta_n+\gamma_n)x^\ast\right\|^2\nonumber\\ &=& \left\|\alpha_n(f(x_n)-x^\ast)+\beta_n \{S\left(\frac{x_{n+1}+x_n}{2}\right)-x^\ast\} + \gamma_n \{T\left(\frac{x_{n+1}+x_n}{2}\right)-x^\ast\}\right\|^2 \nonumber\\ &=& \alpha^2_n \left\|f(x_n) - x^\ast\right\|^2 + \beta^2_n \left\|S\left(\frac{x_{n+1}+x_n}{2}\right) - x^\ast\right\|^2 \\ &&+ \gamma^2_n \left\|T\left(\frac{x_{n+1} + x_n}{2}\right) - x^\ast\right\|^2\nonumber\\ && + 2\alpha_n \beta_n \left\langle f(x_n) - x^\ast, S\left(\frac{x_{n+1}+x_n}{2}\right)-x^\ast \right\rangle \nonumber \\ && + 2\alpha_n \gamma_n \left\langle f(x_n)-x^\ast,T\left(\frac{x_{n+1}+x_n}{2}\right)-x^\ast \right\rangle \nonumber \\ && + 2\beta_n \gamma_n \left\langle S\left(\frac{x_{n+1} + x_n}{2}\right) - x^\ast, T\left(\frac{x_{n+1} + x_n}{2}\right) - x^\ast \right\rangle \nonumber \\ &\leq& \alpha^2_n \|f(x_n) - x^\ast\|^2 + \beta^2_n \left\|\frac{x_{n+1} + x_n}{2} - x^\ast\right\|^2 + \gamma^2_n \left\|\frac{x_{n+1} + x_n}{2} - x^\ast\right\|^2 \nonumber \\ && + 2\alpha_n \beta_n \left\langle f(x_n) - f(x^\ast), S\left(\frac{x_{n+1} + x_n}{2}\right) - x^\ast \right\rangle \nonumber \\ && + 2\alpha_n \beta_n \left\langle f(x^\ast) - x^\ast, S\left(\frac{x_{n+1} + x_n}{2}\right) - x^\ast \right\rangle \nonumber \\ && + 2 \alpha_n \gamma_n \left\langle f(x_n) - f(x^\ast), T\left(\frac{x_{n+1} + x_n}{2}\right) - x^\ast \right\rangle \nonumber \\ && + 2 \alpha_n \gamma_n \left\langle f(x^\ast) - x^\ast, T\left(\frac{x_{n+1} + x_n}{2}\right) - x^\ast \right\rangle \nonumber \\ && + 2\beta_n \gamma_n \left\langle S\left(\frac{x_{n+1} + x_n}{2}\right) - x^\ast, T\left(\frac{x_{n+1} + x_n}{2}\right) - x^\ast \right\rangle \nonumber \\ &\leq& \{\beta^2_n + \gamma^2_n\} \left\|\frac{x_{n+1} + x_n}{2} - x^\ast\right\|^2 + 2\alpha_n\beta_n\| f(x_n) - f(x^\ast)\|.\nonumber\\ &&\left\|S\left(\frac{x_{n+1}+x_n}{2}\right)-x^\ast\right\|\nonumber\\ && + 2\alpha_n \gamma_n \|f(x_n) - f(x^\ast)\|.\left\|T\left(\frac{x_{n+1} + x_n}{2}\right) - x^\ast\right\| \nonumber \\ && + 2\beta_n \gamma_n \left\|S\left(\frac{x_{n+1} + x_n}{2}\right) - x^\ast\right\|.\left\|T\left(\frac{x_{n+1} + x_n}{2}\right) - x^\ast\right\| + K_n, \nonumber \end{eqnarray*} where, \begin{eqnarray*} K_n &=& \alpha^2_n \|f(x_n)-x^\ast\|^2 + 2\alpha_n\beta_n \left\langle f(x^\ast) - x^\ast, S\left(\frac{x_{n+1} + x_n}{2}\right) - x^\ast \right\rangle \\ && + 2\alpha_n\gamma_n \left\langle f(x^\ast)-x^\ast,T\left(\frac{x_{n+1}+x_n}{2}\right)-x^\ast \right\rangle \end{eqnarray*} This implies that \begin{eqnarray*} &&\|x_{n+1}-x_n\|^2 \nonumber \\ &\leq& (\beta^2_n + \gamma^2_n) \left\|\frac{x_{n+1} + x_n}{2} - x^\ast\right\|^2 + 2\alpha_n\beta_n \theta \left\|x_n - x^\ast\right\|.\left\|\frac{x_{n+1} + x_n}{2} - x^\ast\right\| \nonumber \\ && + 2\alpha_n\gamma_n\theta\left\|x_n - x^\ast\right\|.\left\|\frac{x_{n+1}+x_n}{2}-x^\ast\right\| \nonumber \\ && + 2\beta_n\gamma_n\left\|\frac{x_{n+1}+x_n}{2}-x^\ast\right\|.\left\|\frac{x_{n+1}+x_n}{2}-x^\ast\right\| + K_n \nonumber \\ &=&(\beta^2_n+\gamma^2_n+2\beta_n\gamma_n)\left\|\frac{x_{n+1}+x_n}{2}-x^\ast\right\|^2 \\ && + 2 \alpha_n\theta(\beta_n+\gamma_n)\left\|x_n-x^\ast\right\|.\left\|\frac{x_{n+1}+x_n}{2}-x^\ast\right\| + K_n \nonumber\\ &=&(\beta_n+\gamma_n)^2 \left\|\frac{x_{n+1}+x_n}{2}-x^\ast\right\|^2 \\ && + 2\alpha_n\theta(\beta_n+\gamma_n)\left\|x_n-x^\ast\right\|.\left\|\frac{x_{n+1}+x_n}{2}-x^\ast\right\| + K_n\nonumber\\ &=& (1-\alpha_n)^2\left\|\frac{x_{n+1}+x_n}{2}-x^\ast\right\|^2 \\ && + 2\alpha_n\theta(1-\alpha_n)\left\|x_n-x^\ast\right\|.\left\|\frac{x_{n+1}+x_n}{2}-x^\ast\right\|+K_n. \end{eqnarray*} The above calculation shows that \begin{eqnarray*} 0 &\leq& 2\alpha_n\theta(1-\alpha_n)\left\|x_n-x^\ast\right\|.\left\|\frac{x_{n+1}+x_n}{2}-x^\ast\right\| \\ && + (1-\alpha_n)^2\left\|\frac{x_{n+1}+x_n}{2}-x^\ast\right\|^2 - \left\|x_{n+1}-x^\ast\right\|^2 + K_n, \end{eqnarray*} which is a quadratic inequality in \(||{\frac{x_{n+1}+x_n}{2}-x^\ast}\). Solving the above inequality for \(||{\frac{x_{n+1}+x_n}{2}-x^\ast}\), we have, \begin{eqnarray*} ||{\frac{x_{n+1}+x_n}{2}-x^\ast} &\geq& \frac{-2\theta\alpha_n(1-\alpha_n)\|x_n-x^\ast\|}{2(1-\alpha_n)^2} \end{eqnarray*} \begin{eqnarray*} && + \frac{\sqrt{4\theta^2\alpha^2_n(1-\alpha_n)^2\|x_n-x^\ast\|^2-4(1-\alpha_n)^2(K_n-\|x_{n+1}-x^\ast\|^2 )}}{2(1-\alpha_n)^2} \\ &=& \frac{-\theta\alpha_n\|x_n-x^\ast\|+\sqrt{\theta^2\alpha^2_n\|x_n-x^\ast\|^2-K_n+\|x_{n+1}-x^\ast\|^2 }}{1-\alpha_n}. \end{eqnarray*} This will give \begin{eqnarray*} && \frac{1}{2}\left(\|x_{n+1}-x^\ast\|+\|x_n-x^\ast\|\right) \\ &\geq& \frac{-\theta\alpha_n\|x_n-x^\ast\|+\sqrt{\theta^2\alpha^2_n\|x_n-x^\ast\|^2-K_n+\|x_{n+1}-x^\ast\|^2 }}{1-\alpha_n} \end{eqnarray*} \(\Rightarrow\) \begin{eqnarray*} && \frac{1}{2}\left((1-\alpha_n)||{x_{n+1}-x^\ast} + (1 + (2\theta - 1) \alpha_n) ||{x_n-x^\ast} \right) \\ &\geq&\sqrt{\theta^2\alpha^2_n\|x_n-x^\ast\|^2-K_n+\|x_{n+1}-x^\ast\|^2} \end{eqnarray*} \(\Rightarrow\) \begin{eqnarray*} && \frac{1}{4}\left((1-\alpha_n)\|x_{n+1}-x^\ast\|+(1+(2\theta-1)\alpha_n)\|x_n-x^\ast\|\right)^2 \\ &\geq&\theta^2\alpha^2_n\|x_n-x^\ast\|^2-K_n+\|x_{n+1}-x^\ast\|^2, \end{eqnarray*} which is reduced to \begin{eqnarray*} &&\frac{1}{4}(1-\alpha_n)^2\|x_{n+1}-x^\ast\|^2+\frac{1}{4}(1+(2\theta-1)\alpha_n)^2\|x_n-x^\ast\|^2\\ &+&\frac{1}{2}(1-\alpha_n)(1+(2\theta-1)\alpha_n)\|x_{n+1}-x^\ast\|\|x_n-x^\ast\|\\ &\geq&\theta^2\alpha^2_n\|x_n-x^\ast\|^2-K_n+\|x_{n+1}-x^\ast\|^2. \end{eqnarray*} This inequality is further reduced by using the elementary inequality \begin{eqnarray*} 2\|x_{n+1}-x^\ast\|\|x_n-x^\ast\|&\leq&\|x_{n+1}-x^\ast\|^2+\|x_n-x^\ast\|^2, \end{eqnarray*} to the following inequality \begin{eqnarray*} &&\frac{1}{4}(1-\alpha_n)^2\|x_{n+1}-x^\ast\|^2+\frac{1}{4}(1+(2\theta-1)\alpha_n)^2\|x_n-x^\ast\|^2\\ &+&\frac{1}{4}(1-\alpha_n)(1+(2\theta-1)\alpha_n)(\|x_{n+1}-x^\ast\|^2+\|x_n-x^\ast\|^2)\\ &\geq&\theta^2\alpha^2_n\|x_n-x^\ast\|^2-K_n+\|x_{n+1}-x^\ast\|^2. \end{eqnarray*} This implies that \begin{eqnarray*} &&\left(1-\frac{1}{4}(1-\alpha_n)^2-\frac{1}{4}(1-\alpha_n)(1+(2\theta-1)\alpha_n)\right)\|x_{n+1}-x^\ast\|^2\\ &\leq&\left(\frac{1}{4}(1+(2\theta-1)\alpha_n)^2+\frac{1}{4}(1-\alpha_n)(1+(2\theta-1)\alpha_n)-\theta^2\alpha^2_n\right)\|x_n-x^\ast\|^2 \\ && + K_n, \end{eqnarray*} or \begin{eqnarray} \|x_{n+1}-x^\ast\|^2 &\leq& \frac{\frac{1}{4}(1-\alpha_n)(1+(2\theta-1)\alpha_n)-\theta^2\alpha^2_n}{1-\frac{1}{4}(1-\alpha_n)^2-\frac{1}{4}(1-\alpha_n)(1+(2\theta-1)\alpha_n)}\|x_n-x^\ast\|^2 \nonumber\\ && + \frac{\frac{1}{4}(1+(2\theta-1)\alpha_n)^2}{1-\frac{1}{4}(1-\alpha_n)^2-\frac{1}{4}(1-\alpha_n)(1+(2\theta-1)\alpha_n)} + K'_n,\label{a} \end{eqnarray} where, \begin{eqnarray*} K'_n &=& \frac{K_n}{1-\frac{1}{4}(1-\alpha_n)^2-\frac{1}{4}(1-\alpha_n)(1+(2\theta-1)\alpha_n)}. \end{eqnarray*} Note that, \begin{eqnarray*} && 1-\frac{1}{4}(1-\alpha_n)^2-\frac{1}{4}(1-\alpha_n)(1+(2\theta-1)\alpha_n) \\ &=&1-\frac{1}{4}(1-\alpha_n)(1-\alpha_n+1+(2\theta-1)\alpha_n)\\ &=&1-\frac{1}{4}(1-\alpha_n)(1-\alpha_n+1+2\theta\alpha_n-\alpha_n)\\ &=&1-\frac{1}{4}(1-\alpha_n)(2-2\alpha_n+2\theta\alpha_n)\\ &=&1-\frac{1}{2}(1-\alpha_n)(1-\alpha_n+\theta\alpha_n)\\ &=&1-\frac{1}{2}(1-\alpha_n)(1-\alpha_n(1-\theta)), \end{eqnarray*} and \begin{eqnarray*} &&\frac{1}{4}(1+(2\theta-1)\alpha_n)^2+\frac{1}{4}(1-\alpha_n)(1+(2\theta-1)\alpha_n)-\theta^2\alpha^2_n\\ &=&\frac{1}{4}(1+(2\theta-1)\alpha_n)(1+(2\theta-1)\alpha_n+1-\alpha_n)-\theta^2\alpha^2_n\\ &=&\frac{1}{4}(1+(2\theta-1)\alpha_n)(2+2\theta\alpha_n-2\alpha_n)-\theta^2\alpha^2_n\\ &=&\frac{1}{2}(1+(2\theta-1)\alpha_n)(1+\theta\alpha_n-\alpha_n)-\theta^2\alpha^2_n\\ &=&\frac{1}{2}(1+(2\theta-1)\alpha_n)(1-(1-\theta)\alpha_n)-\theta^2\alpha^2_n. \end{eqnarray*} Now from (2), \begin{eqnarray} && \|x_{n+1}-x^\ast\|^2 \nonumber\\ &\leq& \frac{\frac{1}{2}(1 + (2\theta - 1)\alpha_n)(1 - (1 - \theta)\alpha_n) - \theta^2 \alpha^2_n}{1 - \frac{1}{2}(1 - \alpha_n)(1 - \alpha_n(1 - \theta))}\|x_n - x^\ast\|^2 + K'_n.\label{b} \end{eqnarray} Consider the following function, for \(t > 0\). $$g(t):=\frac{1}{t}\left\{1-\frac{\frac{1}{2}(1+(2\theta-1)t)(1-(1-\theta)t)-\theta^2t^2}{1-\frac{1}{2}(1-t)(1-t(1-\theta))}\right\}$$ \begin{eqnarray*} g(t)&=&\frac{1}{t}\left\{\frac{1-\frac{1}{2}(1-t)(1-t(1-\theta))-\frac{1}{2}(1+(2\theta-1)t)(1-(1-\theta)t)+\theta^2t^2}{1-\frac{1}{2}(1-t)(1-t(1-\theta))}\right\}\\ &=&\frac{1}{t}\left\{\frac{1-\frac{1}{2}(1-t(1-\theta))(1-t+1+2\theta t-t)+\theta^2t^2}{1-\frac{1}{2}(1-t)(1-t(1-\theta))}\right\}\\ &=&\frac{1}{t}\left\{\frac{1-\frac{1}{2}(1-t(1-\theta))(2-2t+2\theta t)+\theta^2t^2}{1-\frac{1}{2}(1-t)(1-t(1-\theta))}\right\}\\ &=&\frac{1}{t}\left\{\frac{1-(1-t+\theta t))(1-t+\theta t ))+\theta^2t^2}{1-\frac{1}{2}(1-t)(1-t(1-\theta))}\right\}\\ &=&\frac{1}{t}\left\{\frac{1-(1+t^2+\theta^2 t^2-2t-2\theta t^2+2\theta t)+\theta^2t^2}{1-\frac{1}{2}(1-t)(1-t(1-\theta))}\right\}\\ &=&\frac{1}{t}\left\{\frac{1-1-t^2-\theta^2 t^2+2t+2\theta t^2-2\theta t+\theta^2t^2}{1-\frac{1}{2}(1-t)(1-t(1-\theta))}\right\}\\ &=&\frac{-t+2+2\theta t-2\theta }{1-\frac{1}{2}(1-t)(1-t(1-\theta))}. \end{eqnarray*} By applying limit \(t\rightarrow0\), we have \begin{eqnarray*} \lim_{t\rightarrow0}g(t)=4(1-\theta)>0. \end{eqnarray*} Let \(\delta>0\) be such that for all \(0< t < \delta \), \(g(t) > \epsilon := 4(1-\theta) > 0\). This is equivalent to \begin{eqnarray*} \frac{1}{t}\left\{1-\frac{\frac{1}{2}(1+(2\theta-1)t)(1-(1-\theta)t)-\theta^2t^2}{1-\frac{1}{2}(1-t)(1-t(1-\theta))}\right\} &>& \epsilon \end{eqnarray*} This implies, \begin{eqnarray*} 1-t\epsilon>\frac{\frac{1}{2}(1+(2\theta-1)t)(1-(1-\theta)t)-\theta^2t^2}{1-\frac{1}{2}(1-t)(1-t(1-\theta))}. \end{eqnarray*} Since \(\alpha_n\rightarrow0\) as \(n\rightarrow\infty\), there exist some integer \(N\), such that \(\alpha_n< \delta\), \(\forall\) \(n\geq N\). From (3), we have \begin{equation*} \|x_{n+1}-x^\ast\|^2\leq(1-\alpha_n\epsilon)\|x_n-x^\ast\|^2+K'_n \end{equation*} On the other hand, we have \begin{eqnarray*} \limsup\limits_{n \to \infty} \frac{K_n}{\alpha_n} &=& \limsup\limits_{n \to \infty} \left\{ 2\beta_n \left\langle f(x^\ast)-x^\ast,S\left(\frac{x_{n+1}+x_n}{2}\right)-x^\ast \right\rangle \right. \\ && \hspace{11.5mm} + 2\gamma_n \left\langle f(x^\ast)-x^\ast,T\left(\frac{x_{n+1}+x_n}{2}\right)-x^\ast \right\rangle \\ && \left. \hspace{11.5mm} + \alpha_n\|f(x_n)-x^\ast\|^2\right\}\leq 0. \end{eqnarray*} The above inequality implies that \begin{eqnarray*} \limsup\limits_{n\rightarrow\infty}\frac{K'_n}{\alpha_n} &\leq& 0. \end{eqnarray*} From the above two inequalities and Theorem 1.4 we have $$\lim_{n\rightarrow\infty}\|x_{n+1}-x^\ast\|^2=0,$$ which implies that \(x_n\rightarrow x^\ast\) as \(n\rightarrow\infty\). This completes the proof.

Competing Interests

The author(s) do not have any competing interests in the manuscript.

1. Introduction

In 1938 Ostrowski [1] proved an inequality stated in the following result (see also [2, p.468]).

Theorem 1.1. Let \( f:I \rightarrow \mathbb{R} \) where \(I\) is interval in \( \mathbb{R} \), be a mapping differentiable in \( I^{\circ} \) the interior of \(I\) and \(a, b \in I^{\circ}\), \(a < b \). If \( \big|f^{'}(t) \big| \leq M \), for all \(t \in [a , b] \), then we have \begin{equation*} \bigg| f(x)-\dfrac{1}{b-a}\int_{a} ^{b}f(t)dt \bigg|\leq \left[\frac{1}{4}+\frac{(x-\frac{a+b}{2})^{2}}{(b-a)^{2}}\right](b-a)M, x\in[a,b]. \end{equation*}

Ostrowski inequality gives bounds of integral average of a function \(f\) over an interval \([a,b]\) to its value \(f(x)\) at point \(x\in [a,b].\) Ostrowski and Ostrowski type inequalities have great importance in numerical analysis as they provide the error bound of many quaderature rules [3]. Therefore in recent years, so many such type of inequalities have been obtained and generalized (see [4, 5]).

As fractional calculus is a generalization of classical calculus concerned with operations of integration and differentiation of fractional order so in this research article we will use Katugampola fractional integrals to generalize the Ostrowski type inequalities given in [4].

In [6] Laurent give definition of Riemann-Liouville fractional integrals.

Definition 1.2. [6] Let \(f\in L_{1}[a,b]. \) The Riemann-Liouville fractional integrals \(J_{a+}^{\alpha}f\) and \(J_{b-}^{\alpha}f\) of order \(\alpha>0 \) with \(a\geq0\) are defined by \begin{equation*} J_{a+}^{\alpha}f(x)=\dfrac{1}{\Gamma(\alpha)}\int_{a}^{x}(x-t)^{\alpha-1}f(t)dt, x>a \end{equation*} and \begin{equation*} J_{b-}^{\alpha}f(x)=\dfrac{1}{\Gamma(\alpha)}\int_{x}^{b}(t-x)^{\alpha-1}f(t)dt, x< b, \end{equation*} respectively, where \( \Gamma(\alpha)=\int_{0}^{\infty}e^{-u}u^{\alpha-1}du.\) Here \(\Gamma(\alpha+1)=\alpha\Gamma(\alpha)\),
\( J_{a+}^{0}f(x)=J_{b-}^{0}f(x)=f(x).\) In case of \( \alpha=1 \), the fractional integral reduces to the classical integral.

Definition 1.3. J. Hadamard introduced the Hadamard fractional integral in [7], and is given by \begin{equation*} I_{a^{+}}^{\alpha}f(x)=\dfrac{1}{\Gamma(\alpha)}\int_{a}^{x}\left(log\dfrac{x}{\tau}\right)^{\alpha-1}f(\tau)\dfrac{d\tau}{\tau}, \end{equation*} for \(Re(\alpha)>0,\,x>a\geq0 \).

Recently Katugampola generalized Riemann-Liouville and Hadamard fractional integrals into a single form called Katugampola fractional integrals.

Definition 1.4. [8] Let \([a,b]\) be a finite interval in \( \mathbb{R}\). Then Katugampola fractional integrals of order \(\alpha>0\) for a real valued function \(f\) are defined by \begin{equation*} ^{\rho}I_{a+}^{\alpha}f\left(x\right)=\frac{\rho^{1-\alpha}}{\Gamma\left(\alpha\right)}\int_{a}^{x}t^{\rho-1} \left(x^{\rho}-t^{\rho}\right)^{\alpha-1}f\left(t\right)dt \end{equation*} and \begin{equation*} ^{\rho}I_{b-}^{\alpha}f\left(x\right)=\frac{\rho^{1-\alpha}}{\Gamma\left(\alpha\right)} \int_{x}^{b}t^{\rho-1}\left(t^{\rho}-x^{\rho}\right)^{\alpha-1}f\left(t\right)dt \end{equation*} with \(a< x< b\) and \(\rho >0 \).
Where \(\Gamma\left(\alpha\right)\) is the Euler gamma function. For \(\rho=1\), Katugampola fractional integrals give Riemann-Liouville fractional integrals, while \(\rho \rightarrow 0^+\) produces the Hadamard fractional integral. For its proof one can check [8].

The \(\rho \)-Gamma function [9] for any two positive numbers \(x, y\) denoted by \( ^{\rho}\Gamma(x,y)\), is defined by \begin{equation*} ^{\rho}\Gamma(\alpha)=\int_{0}^{\infty}e^{-t^{\rho}}(t^{\rho})^{\alpha-\frac{1}{\rho}}dt. \end{equation*} We can have the following relation

Definition 1.5. [10] A non-negative function \(f:I\rightarrow \mathbb{R}\) is said to be \(p\)-function, if for any two points \(x,y \in I\) and \(t \in [0,1]\) \begin{align*} f\left(tx+(1-t)y\right)\leq f(x)+f(y). \end{align*}

Definition 1.6. [11] A function \(f:I\rightarrow \mathbb{R}\) is said to be Godunova-Levin function, if for any two points \(x,y \in I\) and \(t \in (0,1)\) \begin{align*} f\left(tx+(1-t)y\right)\leq \dfrac{f(x)}{t}+\dfrac{f(y)}{1-t}. \end{align*}

Definition 1.7. [12] A function \(f:I\rightarrow \mathbb{R}\) is said to be \(s\)-Godunova-Levin function of first kind, if \( s\in(0,1] \), for all \(x,y \in I\) and \(t \in (0,1)\) then we have \begin{align*} f\left(tx+(1-t)y\right)\leq \dfrac{f(x)}{t^{s}}+\dfrac{f(y)}{1-t^{s}}. \end{align*}

Definition 1.8. [13] A function \(f:I\rightarrow \mathbb{R}\) is said to be \(s\)-Godunova-Levin function of second kind, if \(s\in[0,1]\), for all \(x,y \in I\) and \(t \in (0,1)\) then we have \begin{align*} f\left(tx+(1-t)y\right)\leq \dfrac{f(x)}{t^{s}}+\dfrac{f(y)}{(1-t)^{s}}. \end{align*}

We organize the paper in such a way that in the following section we prove some Ostrowski type fractional integral inequalities for \(s\)-Godunova-Levin functions of second kind via Katugampola fractional integrals. Also we will obtain some corollaries for \(p\)-functions and Godunova-Levin functions and deduce some known results of [14].

2. Ostrowski type fractional integral inequalities for mappings whose derivatives are s-Godunova-Levin of second kind via Katugampola fractional integrals

The following lemma (given and also proved in [9]) is very useful to obtain our results.

Lemma 2.1. Let \(f:[a^{\rho},b^{\rho}] \rightarrow \mathbb{R} \) be a differentiable mapping on \((a^{\rho},b^{\rho})\) with \( a< b \) such that \(f^{'} \in L_{1}[a,b]\), where \(\rho>0 \). Then we have the following equality

Theorem 2.2. Let \(f:[a^{\rho},\,b^{\rho}]\rightarrow \mathbb{R} \), \( a,b\geq 0 \), \( a< b \) be a differentiable function on \((a^{\rho},b^{\rho})\) and \(f^{'} \in L_{1}[a,b]\). If \( \big|f^{'}\big| \) is \(s\)-Godunova-Levin function of second kind and \(\big|f^{'}(x^{\rho})\big|\leq M\), \(x \in [a,b],\) then the following inequality holds

Proof. Using Lemma 2.1 and the fact that \(\big|f^{'}\big| \) is \(s\)-Godunova-Levin function of second kind, we have \begin{align*} &\bigg|\left(\dfrac{(x^{\rho}-a^{\rho})^{\alpha}+(b^{\rho}-x^{\rho})^{\alpha}}{b-a}\right)f(x^{\rho})-\dfrac{(\alpha \rho+\rho-1)\Gamma(\alpha)}{\rho^{1-\alpha}(b-a)}\times\nonumber\\&\left[^{\rho}I_{x^{-}}^{\alpha}f(a^{\rho})+\,\,^{\rho}I_{x^{+}}^{\alpha}f(b^{\rho})\right] \nonumber \bigg|\nonumber \\&\leq \frac{\rho(x^{\rho}-a^{\rho})^{\alpha+1}}{b-a}\int_{0}^{1}t^{\alpha \rho+\rho-1}\big|f^{'}(t^{\rho}x^{\rho}+(1-t^{\rho})a^{\rho})\big|dt \\&+\frac{\rho(b^{\rho}-x^{\rho})^{\alpha+1}}{b-a}\int_{0} ^{1}t^{\alpha \rho+\rho-1}\big|f^{'}(t^{\rho}x^{\rho}+(1-t^{\rho})b^{\rho})\big|dt \\& \leq \frac{\rho(x^{\rho}-a^{\rho})^{\alpha+1}}{b-a}\int_{0}^{1}\left[\frac{t^{\alpha \rho+\rho-1}}{(t^{\rho})^{s}}\big|f^{'}(x^{\rho})\big|+\frac{t^{\alpha \rho+\rho-1}}{(1-t^{\rho})^{s}}\big|f^{'}(a^{\rho})\big|\right]dt \\&+\frac{\rho(b^{\rho}-x^{\rho})^{\alpha+1}}{b-a}\int_{0}^{1}\left[\frac{t^{\alpha \rho+\rho-1}}{(t^{\rho})^{s}}\big|f^{'}(x^{\rho})\big|+\frac{t^{\alpha \rho+\rho-1}}{(1-t^{\rho})^{s}}\big|f^{'}(b^{\rho})\big|\right]dt \\& \leq \frac{M\rho(x^{\rho}-a^{\rho})^{\alpha+1}}{b-a}\int_{0}^{1}\left[\frac{t^{\alpha \rho+\rho-1}}{(t^{\rho})^{s}}+\frac{t^{\alpha \rho+\rho-1}}{(1-t^{\rho})^{s}}\right]dt \\&+\frac{M\rho(b^{\rho}-x^{\rho})^{\alpha+1}}{b-a}\int_{0}^{1}\left[\frac{t^{\alpha \rho+\rho-1}}{(t^{\rho})^{s}}+\frac{t^{\alpha \rho+\rho-1}}{(1-t^{\rho})^{s}}\right]dt \\&=M\rho\left[\frac{(x^{\rho}-a^{\rho})^{\alpha+1}+(b^{\rho}-x^{\rho})^{\alpha+1}}{2(b-a)}\right]\times\nonumber\\&\int_{0}^{1}\left[t^{\alpha\rho-\rho s+\rho-1}+t^{\alpha\rho+\rho-1}(1-t^{\rho})^{-s}\right]dt. \\&=M\rho\left[\frac{(x^{\rho}-a^{\rho})^{\alpha+1}+(b^{\rho}-x^{\rho})^{\alpha+1}}{2(b-a)}\right]\times\nonumber\\&\left[\dfrac{1}{\rho(\alpha+1-s)}+\dfrac{^{\rho}\Gamma(\alpha+1)\,\,^{\rho}\Gamma(1-s)}{\rho\,\,^{\rho}\Gamma(\alpha+2-s)}\right] \\&= M\bigg[\frac{(x^{\rho}-a^{\rho})^{\alpha+1}+(b^{\rho}-x^{\rho})^{\alpha+1}}{b-a}\bigg]\times\nonumber\\&\left[\dfrac{1}{\alpha+1-s}+\dfrac{^{\rho}\Gamma(\alpha+1)\,\,^{\rho}\Gamma(1-s)}{\,\,^{\rho}\Gamma(\alpha+2-s)}\right]. \end{align*} Here we use (1). The proof is completed.

Lemma 2.3. (i) If we put \( \rho=1 \) in (3), then we get [4, Theorem 3.1].
(ii) If we put \( \rho=1\) and \( \alpha=1 \) in (3), then we get [4, Corollary 3.1].

Corollary 2.4. In Theorem 2.2 , if we take \(s=0\), which means that \( \big|f^{'}\big| \) is \(p\)-function, then (3) becomes the following inequality \begin{align*} &\bigg|\left(\dfrac{(x^{\rho}-a^{\rho})^{\alpha}+(b^{\rho}-x^{\rho})^{\alpha}}{b-a}\right)f(x^{\rho})-\dfrac{(\alpha \rho+\rho-1)\Gamma(\alpha)}{\rho^{1-\alpha}(b-a)}\times\nonumber\\&\left[^{\rho}I_{x^{-}}^{\alpha}f(a^{\rho})+\,\,^{\rho}I_{x^{+}}^{\alpha}f(b^{\rho})\right] \nonumber \bigg|\nonumber\\ &\leq \dfrac{2M}{\alpha+1}\bigg[\frac{(x^{\rho}-a^{\rho})^{\alpha+1}+(b^{\rho}-x^{\rho})^{\alpha+1}}{b-a}\bigg];\,x\in[a,b]. \end{align*}

Corollary 2.5. In Theorem 2.2, if we take \(s=1\), which means that \(\big|f^{'}\big|\) is Godunova-Levin function, then (3) becomes the following inequality \begin{align*} &\bigg|\left(\dfrac{(x^{\rho}-a^{\rho})^{\alpha}+(b^{\rho}-x^{\rho})^{\alpha}}{b-a}\right)f(x^{\rho})-\dfrac{(\alpha \rho+\rho-1)\Gamma(\alpha)}{\rho^{1-\alpha}(b-a)}\times\nonumber\\&\left[^{\rho}I_{x^{-}}^{\alpha}f(a^{\rho})+\,\,^{\rho}I_{x^{+}}^{\alpha}f(b^{\rho})\right] \nonumber \bigg|\nonumber\\ &\leq \dfrac{M(\alpha+1)}{\alpha}\bigg[\frac{(x^{\rho}-a^{\rho})^{\alpha+1}+(b^{\rho}-x^{\rho})^{\alpha+1}}{b-a}\bigg];\,x\in[a,b]. \end{align*}

Theorem 2.6. Let \(f:[a^{\rho},b^{\rho}]\rightarrow \mathbb{R}\), \(a,b\geq 0\), \(a< b\) be a differentiable function on \((a^{\rho},b^{\rho})\) and \(f^{'} \in L_{1}[a,b]\). If \(\big|f^{'}\big|^{q},\) is \(s\)-Godunova-Levin function of second kind and \(\big|f^{'}(x^{\rho})\big|\leq M\), \(x \in [a,b]\) then the following inequality for Katugampola fractional integrals holds

with \(\frac{1}{p}+\frac{1}{q}=1\) where \(q>1\).

Proof. Using Lemma 2.1 and then Holder's inequality, we have

Since \(\big|f^{'}\big|^{q}\) is \(s\)-Godunova-Levin function of second kind and \(\big|f^{'}(x^{\rho})\big|\leq M\), we get similarly We also have Using (6), (7) and (8) in (5) we can get (4).

Remark 2.7. (i) If we put \(\rho=1\) in (4), then we get [4, Theorem 3.2].
(ii) If we put \(\rho=1\) and \(\alpha=1\) in (4), then we get [4, Corollary 3.2].

Corollary 2.8. In Theorem 2.6, if we take \(s=0\), which means that \(\big|f^{'}\big|\) is \(p\)-function, then (4) becomes the following inequality \begin{align*} &\bigg|\left(\dfrac{(x^{\rho}-a^{\rho})^{\alpha}+(b^{\rho}-x^{\rho})^{\alpha}}{b-a}\right)f(x^{\rho})-\dfrac{(\alpha \rho+\rho-1)\Gamma(\alpha)}{\rho^{1-\alpha}(b-a)}\times\nonumber\\&\left[^{\rho}I_{x^{-}}^{\alpha}f(a^{\rho})+\,\,^{\rho}I_{x^{+}}^{\alpha}f(b^{\rho})\right] \nonumber \bigg|\nonumber\\ &\leq \dfrac{M}{\left(1+p(\alpha\rho+\rho-1)\right)^{\frac{1}{p}}}\bigg[\frac{(x^{\rho}-a^{\rho})^{\alpha+1}+(b^{\rho}-x^{\rho})^{\alpha+1}}{b-a}\bigg];\,x\in[a,b]. \end{align*}

Corollary 2.9. In Theorem 2.6, if we take \(s=1\), which means that \(\big|f^{'}\big|\) is Godunova-Levin function, then (4) becomes the following inequality \begin{align*} &\bigg|\left(\dfrac{(x^{\rho}-a^{\rho})^{\alpha}+(b^{\rho}-x^{\rho})^{\alpha}}{b-a}\right)f(x^{\rho})-\dfrac{(\alpha \rho+\rho-1)\Gamma(\alpha)}{\rho^{1-\alpha}(b-a)}\times\nonumber\\&\left[^{\rho}I_{x^{-}}^{\alpha}f(a^{\rho})+\,\,^{\rho}I_{x^{+}}^{\alpha}f(b^{\rho})\right] \nonumber \bigg|\leq \dfrac{M\rho}{\left(1+p(\alpha\rho+\rho-1)\right)^{\frac{1}{p}}}\times\nonumber\\&\bigg[\frac{(x^{\rho}-a^{\rho})^{\alpha+1}+(b^{\rho}-x^{\rho})^{\alpha+1}}{b-a}\bigg]\left[\dfrac{1+\alpha}{\alpha\rho}\right]^{\frac{1}{q}};\,x\in[a,b]. \end{align*}

Theorem 2.10. Let \(f:[a^{\rho},b^{\rho}]\rightarrow \mathbb{R}\), \( a,b\geq 0 \), \( a< b \) be a differentiable function on \((a^{\rho},b^{\rho}) \) and \(f^{'} \in L_{1}[a,b]\). If \( \big|f^{'}\big|^{q} \) is \(s\)-Godunova-Levin function of second kind and \(\big|f^{'}(x^{\rho})\big|\leq M\) , \( x \in [a,b] \) \(,q\geq 1,\) then the following inequality for Katugampola fractional integrals holds

Proof. Using Lemma 2.1` and power mean inequality, we have

Since \(\big|f^{'}\big|^{q}\) is \(s\)-Godunova-Levin function of second kind and \(\big|f^{'}(x^{\rho})\big|\leq M\), we get similarly Using (11) and (12) in (10) we can attain (9).

Remarks 2.11. (i) If we put \(\rho=1\) in (9), then we get [4, Theorem 3.3].
(ii) If we put \(\rho=1\) and \(\alpha=1\) in (9), then we get [4, Corollary 3.3].

Corollary 2.12. In Theorem 2.10, if we take \(s=0\), which means that \(\big|f^{'}\big|\) is \(p\)-function, then (9) becomes the following inequality \begin{align*} &\bigg|\left(\dfrac{(x^{\rho}-a^{\rho})^{\alpha}+(b^{\rho}-x^{\rho})^{\alpha}}{b-a}\right)f(x^{\rho})-\dfrac{(\alpha \rho+\rho-1)\Gamma(\alpha)}{\rho^{1-\alpha}(b-a)}\times\nonumber\\&\left[^{\rho}I_{x^{-}}^{\alpha}f(a^{\rho})+\,\,^{\rho}I_{x^{+}}^{\alpha}f(b^{\rho})\right] \nonumber \bigg|\nonumber\\ &\leq \dfrac{M\rho}{(\alpha\rho+\rho)^{1-\frac{1}{q}}}\bigg[\frac{(x^{\rho}-a^{\rho})^{\alpha+1}+(b^{\rho}-x^{\rho})^{\alpha+1}}{b-a}\bigg]\left[\dfrac{2}{\rho(\alpha+1)}\right]^{\frac{1}{q}};\,x\in[a,b]. \end{align*}

Corollary 2.13. In Theorem 2.10, if we take \(s=1\), which means that \(\big|f^{'}\big|\) is Godunova-Levin function, then (9) becomes the following inequality \begin{align*} &\bigg|\left(\dfrac{(x^{\rho}-a^{\rho})^{\alpha}+(b^{\rho}-x^{\rho})^{\alpha}}{b-a}\right)f(x^{\rho})-\dfrac{(\alpha \rho+\rho-1)\Gamma(\alpha)}{\rho^{1-\alpha}(b-a)}\times\nonumber\\&\left[^{\rho}I_{x^{-}}^{\alpha}f(a^{\rho})+\,\,^{\rho}I_{x^{+}}^{\alpha}f(b^{\rho})\right] \nonumber \bigg|\nonumber\\ &\leq \dfrac{M\rho}{(\alpha\rho+\rho)^{1-\frac{1}{q}}}\bigg[\frac{(x^{\rho}-a^{\rho})^{\alpha+1}+(b^{\rho}-x^{\rho})^{\alpha+1}}{b-a}\bigg]\left[\dfrac{1+\alpha}{\alpha\rho}\right]^{\frac{1}{q}};\,x\in[a,b]. \end{align*}

We use the following lemma to establish some new results. Its proof is similar to Lemma 2.1.

Lemma 2.14. Let \(f:[a^{\rho},b^{\rho}] \rightarrow \mathbb{R}\) be a differentiable mapping on \((a^{\rho},b^{\rho})\) with \( a^{\rho}< b^{\rho}\) such that \(f^{'} \in L_{1}[a^{\rho},b^{\rho}]\), where \(\rho>0\). Then we have the following equality

Theorem 2.15. Let \(f:[a^{\rho},b^{\rho}]\rightarrow \mathbb{R}\), \(a,b\geq 0\), \(a< b\) be a differentiable function on \((a^{\rho},b^{\rho})\) and \(f^{'} \in L_{1}[a,b]\). If \(\big|f^{'}\big|\) is \(s\)-Godunova-Levin function of second kind and \(\big|f^{'}(x^{\rho})\big|\leq M\), \(x \in [a,b],\) then the following inequality holds

Proof. Using Corollary 2.8 and \(s\)-Godunova-Levin function of second kind of \(\big|f^{'}\big|\) we proceed as follows \begin{align*} &\bigg|f(x^{\rho})-\dfrac{(\alpha \rho+\rho-1)\Gamma(\alpha)}{\rho^{1-\alpha}}\left[\dfrac{^{\rho}I_{x^{-}}^{\alpha}f(a^{\rho})}{2(x^{\rho}-a^{\rho})^{\alpha}}+\dfrac{^{\rho}I_{x^{+}}^{\alpha}f(b^{\rho})}{2(b^{\rho}-x^{\rho})^{\alpha}}\right] \bigg|\nonumber \\&\leq \frac{\rho(x^{\rho}-a^{\rho})}{2}\int_{0}^{1}t^{\alpha \rho+\rho-1}\big|f^{'}(t^{\rho}x^{\rho}+(1-t^{\rho})a^{\rho})\big|dt \\&+\frac{\rho(b^{\rho}-x^{\rho})}{2}\int_{0} ^{1}t^{\alpha \rho+\rho-1}\big|f^{'}(t^{\rho}x^{\rho}+(1-t^{\rho})b^{\rho})\big|dt \\& \leq \frac{\rho(x^{\rho}-a^{\rho})}{2}\int_{0}^{1}\left[\frac{t^{\alpha \rho+\rho-1}}{(t^{\rho})^{s}}\big|f^{'}(x^{\rho})\big|+\frac{t^{\alpha \rho+\rho-1}}{(1-t^{\rho})^{s}}\big|f^{'}(a^{\rho})\big|\right]dt \\&+\frac{\rho(b^{\rho}-x^{\rho})}{2}\int_{0}^{1}\left[\frac{t^{\alpha \rho+\rho-1}}{(t^{\rho})^{s}}\big|f^{'}(x^{\rho})\big|+\frac{t^{\alpha \rho+\rho-1}}{(1-t^{\rho})^{s}}\big|f^{'}(b^{\rho})\big|\right]dt \\& \leq \frac{M\rho(x^{\rho}-a^{\rho})}{2}\int_{0}^{1}\left[\frac{t^{\alpha \rho+\rho-1}}{(t^{\rho})^{s}}+\frac{t^{\alpha \rho+\rho-1}}{(1-t^{\rho})^{s}}\right]dt \\&+\frac{M\rho(b^{\rho}-x^{\rho})}{2}\int_{0}^{1}\left[\frac{t^{\alpha \rho+\rho-1}}{(t^{\rho})^{s}}+\frac{t^{\alpha \rho+\rho-1}}{(1-t^{\rho})^{s}}\right]dt \\&=M\rho\left[\frac{(x^{\rho}-a^{\rho})+(b^{\rho}-x^{\rho})}{2}\right]\int_{0}^{1}\left[t^{\alpha\rho-\rho s+\rho-1}+t^{\alpha\rho+\rho-1}(1-t^{\rho})^{-s}\right]dt. \\&=M\rho\left[\frac{(x^{\rho}-a^{\rho})+(b^{\rho}-x^{\rho})}{2}\right]\left[\dfrac{1}{\rho(\alpha-s+1)}+\dfrac{^{\rho}\Gamma(\alpha+1)\,\,^{\rho}\Gamma(1-s)}{\rho\,\,^{\rho}\Gamma(\alpha-s+2)}\right] \\&= \frac{M(b^{\rho}-a^{\rho})}{2}\left[\dfrac{1}{\alpha-s+1}+\dfrac{^{\rho}\Gamma(\alpha+1)\,\,^{\rho}\Gamma(1-s)}{^{\rho}\Gamma(\alpha-s+2)}\right]. \end{align*} Here we use (1). The proof is completed.

Corollary 2.16. In Theorem 2.15, if we take \(s=0\), which means that \( \big|f^{'}\big|\) is \(p\)-function, then (14) becomes the following inequality \begin{align*} &\bigg|f(x^{\rho})-\dfrac{(\alpha \rho+\rho-1)\Gamma(\alpha)}{\rho^{1-\alpha}}\left[\dfrac{^{\rho}I_{x^{-}}^{\alpha}f(a^{\rho})}{2(x^{\rho}-a^{\rho})^{\alpha}}+\dfrac{^{\rho}I_{x^{+}}^{\alpha}f(b^{\rho})}{2(b^{\rho}-x^{\rho})^{\alpha}}\right] \bigg|\nonumber\\ &\leq \dfrac{M(b^{\rho}-a^{\rho})}{\alpha+1};\,x\in[a,b]. \end{align*}

Corollary 2.17. In Theorem 2.15, if we take \(s=1\), which means that \( \big|f^{'}\big| \) is Godunova-Levin function, then (14) becomes the following inequality \begin{align*} &\bigg|f(x^{\rho})-\dfrac{(\alpha \rho+\rho-1)\Gamma(\alpha)}{\rho^{1-\alpha}}\left[\dfrac{^{\rho}I_{x^{-}}^{\alpha}f(a^{\rho})}{2(x^{\rho}-a^{\rho})^{\alpha}}+\dfrac{^{\rho}I_{x^{+}}^{\alpha}f(b^{\rho})}{2(b^{\rho}-x^{\rho})^{\alpha}}\right] \bigg|\nonumber\\ &\leq \dfrac{M(\alpha+1)(b^{\rho}-a^{\rho})}{2\alpha};\,x\in[a,b]. \end{align*}

Theorem 2.18. Let \(f:[a^{\rho},b^{\rho}]\rightarrow \mathbb{R}\), \(a,b\geq 0\), \(a< b\) be a differentiable function on \((a^{\rho},b^{\rho})\)) and \(f^{'} \in L_{1}[a,b]\). If \( \big|f^{'}\big|^{q},\) is \(s\)-Godunova-Levin function of second kind and \(\big|f^{'}(x^{\rho})\big|\leq M\), \(x \in [a,b]\) then the following inequality for Katugampola fractional integrals holds

with \(\frac{1}{p}+\frac{1}{q}=1\) where \(q>1\).

Proof. Using Corollary 2.8 and then Holder's inequality, we have

Since \( \big|f^{'}\big|^{q}\) is \(s\)-Godunova-Levin function of second kind and \(\big|f^{'}(x^{\rho})\big|\leq M\), we get similarly We also have Using (17), (18) and (19) in (16) we can get (15).

Corollary 2.19. In Theorem 2.18, if we take \(s=0\), which means that \(\big|f^{'}\big|\) is \(p\)-function, then (15) becomes the following inequality \begin{align*} &\bigg|f(x^{\rho})-\dfrac{(\alpha \rho+\rho-1)\Gamma(\alpha)}{\rho^{1-\alpha}}\left[\dfrac{^{\rho}I_{x^{-}}^{\alpha}f(a^{\rho})}{2(x^{\rho}-a^{\rho})^{\alpha}}+\dfrac{^{\rho}I_{x^{+}}^{\alpha}f(b^{\rho})}{2(b^{\rho}-x^{\rho})^{\alpha}}\right] \bigg|\nonumber\\ &\leq \dfrac{M\rho(b^{\rho}-a^{\rho})}{2(p(\alpha\rho+\rho-1)+1)^{\frac{1}{p}}};\,x\in[a,b]. \end{align*}

Corollary 2.20. In Theorem 2.18, if we take \(s=1\), which means that \(\big|f^{'}\big|\) is Godunova-Levin function, then (15) becomes the following inequality \begin{align*} &\bigg|f(x^{\rho})-\dfrac{(\alpha \rho+\rho-1)\Gamma(\alpha)}{\rho^{1-\alpha}}\left[\dfrac{^{\rho}I_{x^{-}}^{\alpha}f(a^{\rho})}{2(x^{\rho}-a^{\rho})^{\alpha}}+\dfrac{^{\rho}I_{x^{+}}^{\alpha}f(b^{\rho})}{2(b^{\rho}-x^{\rho})^{\alpha}}\right] \bigg|\nonumber\\ &\leq \dfrac{M\rho(b^{\rho}-a^{\rho})}{2(p(\alpha\rho+\rho-1)+1)^{\frac{1}{p}}}\left[\dfrac{1}{1-\rho}\right]^{\frac{1}{q}};\,x\in[a,b]. \end{align*}

Theorem 2.21. Let \(f:[a^{\rho},b^{\rho}]\rightarrow \mathbb{R}\), \(a,b\geq 0\), \(a< b\) be a differentiable function on \((a^{\rho},b^{\rho})\) and \(f^{'} \in L_{1}[a,b]\). If \(\big|f^{'}\big|^{q}\) is \(s\)-Godunova-Levin function of second kind and \(\big|f^{'}(x^{\rho})\big|\leq M\), \( x \in [a,b]\)\(,q\geq 1,\) then the following inequality for Katugampola fractional integrals holds

Proof. Using Corollary 2.8 and power mean inequality, we have

Since \( \big|f^{'}\big|^{q}\) is \(s\)-Godunova-Levin function of second kind on \([a^{\rho},b^{\rho}]\) and \(\big|f^{'}(x^{\rho})\big|\leq M\), we get similarly Using (22) and (23) in (21) we can attain (20).

Corollary 2.22. In Theorem 2.21, if we take \(s=0\), which means that \(\big|f^{'}\big|\) is \(p\)-function, then (20) becomes the following inequality \begin{align*} &\bigg|f(x^{\rho})-\dfrac{(\alpha \rho+\rho-1)\Gamma(\alpha)}{\rho^{1-\alpha}}\left[\dfrac{^{\rho}I_{x^{-}}^{\alpha}f(a^{\rho})}{2(x^{\rho}-a^{\rho})^{\alpha}}+\dfrac{^{\rho}I_{x^{+}}^{\alpha}f(b^{\rho})}{2(b^{\rho}-x^{\rho})^{\alpha}}\right] \bigg|\nonumber\\ &\leq \dfrac{M\rho(b^{\rho}-a^{\rho})}{2(\rho(\alpha+1))^{1-\frac{1}{q}}}\left[\dfrac{2}{\rho(\alpha+1)} \right]^{\frac{1}{q}};\,x\in[a,b]. \end{align*}

Corollary 2.23. In Theorem 2.21, if we take \(s=1\), which means that \(\big|f^{'}\big|\) is Godunova-Levin function, then (20) becomes the following inequality \begin{align*} &\bigg|f(x^{\rho})-\dfrac{(\alpha \rho+\rho-1)\Gamma(\alpha)}{\rho^{1-\alpha}}\left[\dfrac{^{\rho}I_{x^{-}}^{\alpha}f(a^{\rho})}{2(x^{\rho}-a^{\rho})^{\alpha}}+\dfrac{^{\rho}I_{x^{+}}^{\alpha}f(b^{\rho})}{2(b^{\rho}-x^{\rho})^{\alpha}}\right] \bigg|\nonumber\\ &\leq \dfrac{M\rho(b^{\rho}-a^{\rho})}{2(\rho(\alpha+1))^{1-\frac{1}{q}}}\left[\dfrac{1+\alpha}{\rho\alpha} \right]^{\frac{1}{q}};\,x\in[a,b]. \end{align*}

Conclusion

All results proved in this research paper can also be deduced for Hadamard fractional integrals just by taking limits when parameter \(\rho \rightarrow 0^{+} \).

Acknowledgement

The research work of Ghulam Farid is supported by Higher Education Commission of Pakistan under NRPU 2016, Project No. 5421.

Competing Interests

The author(s) do not have any competing interests in the manuscript.

1. Introduction

The Oldroyd-B fluid models is very important among the fluids of rate type due to its special behavior. Also, this model contains the Newtonian fluid model and Maxwell fluid model as special cases. The Oldroyd-B fluid model [1, 2] considered the memory effects and elastic effects exhibited by a large class of fluids such as the biological and polymeric liquids. The motion of a fluid in the neighborhood of a moving body is of great interest for industry. The flow between cylinders or through a rotating cylinder has applications in the food industry and being one of the most important and interesting problem of motion near rotating bodies. Exact solutions for some simple flows of Oldroyd-B fluids were presented by many authors, see for example, Rajagopal and Bhatnagar [3], Hayat et al. [4, 5]. The velocity distriution for different motions of Newtonian fluids through a circular cylinder is given in [6]. Wood [7] has considered the general case of helical flow of an Oldroyd-B fluid, due to the combined action of rotating cylinders(with constant angular velocities) and a constant axial pressure gradient. Accurate solutions regarding motions of Non-Newtonian fluids in cylindrical domains appear to be those of Ting [2], Srivastava [8] and Water and King [9] for second grade, Maxwell and Oldroyd-B fluids respectively. The most general solution corresponding to the helical flow of a second grade fluid seem to be those of Fetecau and Cornia Fetecau [10], in which the cylinder is rotating around its axis and sliding along the same axis with time-dependent velocities. There is a vast literature dealing with such fluids, but we shall recall here only a few of the most recent papers [12, 13, 14, 15, 16]. Most existing solutions in the literature correspond to problems with boundary conditions on the velocity. Though, all above mentioned papers incorporate motion problem in which velocity is given on the boundary. In [16], Renardy explained how well posed boundary value problems can be formulated using boundary conditions on stress. Water and King [17] were among the first specialists who used the shear stress on boundary to find exact solution for motions of rate type fluids. Our goal is to investigate analytical solution for the flow of a generalized Oldroyd-B fluid in a circular cylinder. We considered the boundary conditions on the shear stress. The flow of fluid is due to rotation of the cyliner around its axis, under the action of oscillating shear stress \(\Omega R \sin(\omega t)\) given on boundary. These solutions are obtained by mean of integral transforms. The obtained solution satisfy the all imposed initial and boundary conditions. Finally, solution of the ordinary Oldroyd-B fluid, Maxwell, ordinary Maxwell and Newtonian fluid flows are obtained as particular cases of our general results.

2. Mathematical formulation of the problem

For an Oldroyd-B fluid constitutive equations is $$T=-p \textbf{I}+\textbf{S}\,\,\ ; \,\,\ S+\lambda\bigg({\frac{d \textbf{S}}{d t}}-\textbf{LS}-\textbf{S} \textbf{L}^{T}\bigg),$$ where T (Cauchy stress tensor), -pI (indeterminate spherical stress), S (stress tensor), L (velocity gradient), \(\mu\) (dynamic viscosity), \(\textbf{A}=\textbf{L}+\textbf{L}^{T}\) (first the Rivilin-Erickson tensor,) \(\lambda\) and \(\lambda_{r}\) \((0\leqslant\lambda_{r}< \lambda)\) are relaxation and retardation time. Assume an infinite circular cylinder rotate along z-axis with radius R. Cylinder is filled with an Oldroyd-B fluid which is at rest at time \(t=0\). After \(t=0^{+}\) the cylinder applies an oscillating rotational shear stress \(\Omega R \sin(\omega t)\) to the fluid, where \(\omega\) is the angular frequency. We assumed that velocity field and the extra shear stress are of the form where \(e_{\theta}\) is unit vector in the \(\theta\)-direction of the cylindrical coordinate system.We assume that \(\textbf{S}\) and \(\textbf{V}\) is a function of time and radius only. At \(t=0\) there is no motion in fluid i.e; fluid is at rest then Introducing Eqs.(2) in(1) and using (3) we get \(S_{rr}=S_{rz}=S_{z\theta}=S_{zz}=0\) and the meaning partial differential equation. where \(\tau(r,t)=S_{r\theta}(r,t)\) is non-zero component of extra stress tensor. If we neglect the body force, then due to rotation symmetry the balance of linear momentum leads to the relevant equations. where \(\rho\) is the constant density of the field. In order to obtained the governing equations for shear stress on boundary we eliminate \(\omega(r,t)\) between equation (4) and (5) after elimination \(\omega(r,t)\) we get, where \(\nu=\frac{\mu}{\rho}\) is the Kinematic viscosity of the fluid. The governing model by using fractional derivative is shown as; Where $$D_{t}^{\alpha}f(t)=\frac{1}{\Gamma(1-\alpha)}\frac{d}{dt}\int\limits_0^{t}\frac{f(\tau)}{(t-\tau)^{\alpha}}d\tau \,\,\,\ ; \,\,\,\ 0\leqslant\alpha< 1$$ is Caputo fractional derivative operator and $\Gamma(.)$ is the Euler integral of second kind or Gamma function.
At \(t=0\) fluid is at rest because there is no rotation in the cylinder, at \(t=0^{+}\) cylinder starts its rotation along its axis and boundary of cylinder applies shear stress on fluid and radius of cylinder is \(R\). Appropriate conditions i.e; initial and boundary conditions are, where \(\Omega\) is constant.

3. Calculation Of Shear Stress

We shall use the Laplace transform and Finite Hankel transform to determine the exact analytical solution. Taking the Laplace transform of the equations (9) and (11) we have where \(\overline{\tau}(r,q)\) represent the Laplace transform of the function \(\tau(r,t)\). We can write equation (12) as, Finite Hankel transform of the function \(\overline{\tau}(r,q)\) defined as, identity which we used here is, Multiplying equation (14) by \(rJ_{2}(rr_{n})\) then integrate from \(0\) to R with respect to r, we get $$\int\limits_0^{R} rJ_{2}(rr_{n})\overline{\tau}(r,q)dr=\frac{\nu (1+\lambda_{r}q^{\beta})}{q+\lambda q^{\alpha+1}}\int\limits_0^{R} r\bigg(\frac{\partial^{2}}{\partial r^{2}}+\frac{1}{r}\frac{\partial}{\partial r}-\frac{4}{r^{2}}\bigg)J_{2}(rr_{n})\overline{\tau}(r,q)dr.$$ using equations (13), (15) and (16) we get, $$\overline{\tau}_{H}(r_{n},q)=\frac{\nu (1+\lambda_{r} q^{\beta})}{q+\lambda q^{\alpha +1}}[-Rr_{n}J'_{2}(Rr_{n})\overline{\tau}(R,q)-r^{2}_{n}\overline{\tau}_{H}(r_{n},q)].$$ Now simplification for \(\overline{\tau}_{H}(r_{n},q)\) we get Separating the function in suitable form as, $$\overline{\tau}_{H}(r_{n},q) = \frac{-R^{2}\Omega \omega J'_{2}(Rr_{n})}{(q^{2}+\omega^{2}) r_{n}}+\frac{R^{2}\Omega \omega J'_{2}(Rr_{n})}{(q^{2}+\omega^{2})}\frac{q+\lambda q^{\alpha +1}}{r_{n}(q+\lambda q^{\alpha+1}+\nu r^{2}_{n}+\nu \lambda _{r} q^{\beta} r^{2}_{n})},$$ where, $$\overline{\tau}_{2H}(r_{n},q)=\frac{R^{2}\Omega \omega J'_{2}(Rr_{n})}{r_{n}(q^{2}+\omega^{2})}\frac{q+\lambda q^{\alpha +1}}{(q+\lambda q^{\alpha+1}+\nu r^{2}_{n}+\nu \lambda _{r} q^{\beta} r^{2}_{n})},$$ Using the identity i.e; \begin{align*} &\frac{1}{q+\nu r^{2}_{n}+\lambda q^{\beta+1}+\nu r^{2}_{n}\lambda_{r}q^{\gamma}} \\&=\frac{1}{\lambda}\sum^{\infty}_{k=0} \sum^{k}_{m=0}\frac{k!}{(k-m)!m!}\bigg(\frac{-\nu r^{2}_{n}}{\lambda}\bigg)^{k} \lambda_{r}^{m}\frac{q^{\gamma m-k-1}}{(q^{\beta}+\frac{1}{\lambda})^{k+1}}.\end{align*} Now equation (20) becomes, Taking the inverse Laplace transform of equation (19) Take the inverse Laplace transform and use convolution theorem of equation (21) we get Where \(G_{a,b,c}(.,t)\) is the generalized G-function with \(£^{-1} \{\frac{q^{b}}{(q^{a}-d)^{c}}\} =G_{a,b,c}(d,t)\) , \(Re(ac-b)>0\) , \(Re(q)>0\) , \(\mid\frac{d}{q^{a}}\mid< 1\) and \(G_{a,b,c}(d,t)=\sum^{\infty}_{j=0}\frac{t^{(c+j)a-b-1}}{\Gamma ((c+j)a-b))}\frac{d^{j} \Gamma (c+j)}{\Gamma (c) \Gamma (j+1)}\) Taking inverse Laplace transform of equation (18) and using equation (22) and (23) Apply the inverse Hankel transform to equation (24) and using the known formulae $$H(r^{2})=\int\limits_{0}^{R}r^{3}J_{2}(rr_{n})dr=\frac{-R^{3}J_1{Rr_{n}}}{r_{n}}$$ $$\tau(r,t)=2\sum^{\infty}_{n=1}\frac{J_{2}(rr_{n})}{[J'_{2}(r_{n})]^{2}}\tau_{H}(r_{n},t)$$

4. Calculation for Velocity Field

Rewrite equation (8) and used equation (25) we get the non-integer order differential equation for velocity. $$\rho D_t^{\alpha} \omega(r,t) = \bigg(\frac{\partial}{\partial r}+\frac{2}{r}\bigg)\tau(r,t)$$ where $$\tau(r,t)=\tau_{1}(r,t)+\tau_{2}(r,t)$$ So above equation becomes; where $$\tau_{1}(r,t)=\frac{r^{2}r_{n}\Omega\sin(\omega t)}{R}$$ $$\tau_{2}(r,t)=2R^{2}\Omega\sum^{\infty}_{n=1}\frac{J_{2}(rr_{n})J_{1}(Rr_{n})}{J_{1}^{2}(r_{n})}\frac{1}{\lambda}\sum^{\infty}_{k=0}\sum^{k}_{m=0}\frac{k!}{(k-m)!m!}\lambda_{r}^{m}\bigg(\frac{-\nu r^{2}_{n}}{\lambda}\bigg)^{k}$$ \begin{eqnarray*}\times\bigg[\int\limits_0^{t}\sin\omega(t-s)G_{\alpha,\beta m-k,k+1}(-\lambda^{-1},s)ds\\+\lambda\int\limits_0^{t}\sin\omega(t-s)G_{\alpha,\alpha+\beta m-k,k+1}(-\lambda^{-1},s)ds\bigg]. \end{eqnarray*} So, and Use equation (27) and (28) in equation (26) we get, The Laplace transform of equation (29) is Now Apply inverse Laplace transform to equation (30) and using the Convolution theorem

5. Limiting case

5.1. Ordinary Oldroyd-B fluid

Letting \(\alpha\rightarrow1\) , \(\beta\rightarrow1\) into equations (25) and (31)we get the result of shear stress and velocity field respectively for ordinary Oldroyd-B fluid.

5.2. Generalized Maxwell Fluid

By placing \(\lambda_{r}\rightarrow0\) , \(\beta\rightarrow0\) into equations (25) and (31)we get the results of shear stress and velocity field for generalized Maxwell fluid respectively.

5.3. Ordinary Maxwell Fluid

By putting \(\alpha\rightarrow1\) in equation (34) and (35) we get expressing for the shear stress and velocity field for ordinary Maxwell fluid respectively.

5.4. Newtonian Fluid

Let \(\lambda_{r}\rightarrow0\) and using the limit, i.e; $${\lim_{\lambda\rightarrow 0}}\frac{1}{\lambda^{k}}G_{1,b,k}\bigg(\frac{-1}{\lambda},t\bigg)=\frac{t^{-b-1}}{\Gamma(-b)},$$ \(b< 0\) in (25) and (31) we can get results for Newtonian fluid as,

6. Conclusion

The idea presented in this paper is to find a formula which is useful to find out exact solutions for shear stress and velocity field of any Oldroyd-B fluid which is present in rotationally oscillating cylinders. We used two transformation i.e; Hankel transform and Laplace transform. At time \(t=0^{+}\), cylinder starts its rotation about its axis. To obtain the solutions we used the finite Hankel and Laplace transforms. We express our results in the form of generalized G-function, which satisfy the governing equations and fulfilled all imposed initial and boundary conditions. Furthermore, the results for Ordinary Oldroyd-B fluid, fractional Maxwell fluid, ordinary Maxwell fluid and classical Newtonian fluid are obtained as limiting cases.

Competing Interests

The author(s) do not have any competing interests in the manuscript.

1. Introduction

The Kauffman bracket was introduced by L. H. Kauffman in 1987 in [1]. The Kauffman bracket (polynomial) is actually not a knot invariant because it is not invariant under the first Reidemeister move. However, it has many applications and it can be extended to the popular Jones polynomial, which is invariant under all three Reidemeister moves. In the present work we shall confine ourselves to the Kauffman bracket to avoid from unnecessary length and to leave it for applications. In [2] Nizami et al, computed Khavanov Homology of Braid Links and in [3] gave recursive form of Kauffman Bracket. This paper is organized as follows: In Section 2 we shall give the basic ideas about knots, braids, and the Kauffman bracket. In Section 3 we shall present the main results.

2. Preliminary Notions

2.1. Links. A link is a disjoint union of circles embedded in \(\mathbb{R}^{3}\). A one-component link is called a knot. Links are usually studied via projecting them on a plan; a projection with extra information of overcrossing and undercrossing is called the link diagram.

Two links are isotopic iff one of them can be transformed to the other by a diffeomorphism of the ambient space onto itself. A fundamental result by Reidemeister [4] about the isotopic link diagrams is:
Two unoriented links \(L_{1}\) and \(L_{2}\) are equivalent if and only if a diagram of \(L_{1}\) can be transformed into a diagram of \(L_{2}\) by a finite sequence of ambient isotopies of the plane and the local (Reidemeister) moves of the following three types:

The set of all links that are equivalent to a link \(L\) is called a class of \(L\). By a link \(L\) we shall always mean the class of \(L\). The main question of knot theory is Which two links are equivalent and which are not? To address this question one needs a knot invariant, a function that gives one value on all links that belong to a single class and gives different values (but not always) on knots that belong to different classes. The present work is basically concerned with this question.
2.2 Braids. Braids were first studied by Emil Artin in 1925 [5, 6], which play an important role in knot theory, see [7, 8] for detail. An \(n\)-strand braid is a set of \(n\) non intersecting smooth paths connecting \(n\) points on a horizontal plane to \(n\) points exactly below them on another horizontal plane in an arbitrary order. The smooth paths are called strands of the braid.

The product \(ab\) of two \(n\)-strand braids is defined by putting \(b\) above \(a\) and gluing their end points. A braid with only one crossing is called elementary braid. The ith elementary braid \(x_{i}\) on \(n\) strands is:

A useful property of elementary braids is that every braid can be written as a product of elementary braids. For instance, the above 2-strand braid is \(x_{i}^{-3}=(x_{i}^{-1})(x_{i}^{-1})(x_{i}^{-1})\). The closure of a braid \(b\) is the link \(\widehat{b}\) obtained by connecting the lower ends of \(b\) with the corresponding upper ends.

An important result by Alexander [9] connecting knots and braids is: Each link can be represented as the closure of a braid.

Remark 2.1. In the last section we shall present all the links as closures of products of elementary braids.

2.3. The Kauffman Bracket The Kauffman bracket was introduced by Kauffman in [10]. Before the definition it is better to understand the two types of splitting of a crossing, the \(A\)-type and the \(B\)-type splittings:

In the following, the symbols \(\bigcirc\) and \(\bigsqcup\) represent respectively the unknot and the disconnected sum.

Definition 2.1 The Kauffman bracket is the function \(\langle \cdot\rangle: \mbox{Links}\rightarrow \mathbb{Z}[a,a^{-1}]\) defined by the axioms: \begin{eqnarray*} % \nonumber to remove numbering (before each equation) \langle L\rangle &=& a\langle L_{A} \rangle + a^{-1} \langle L_{B} \rangle \\ \langle L \sqcup \bigcirc\rangle &=& (-a^{2} - a^{-2} )\langle L \rangle \\ \langle \bigcirc\rangle &=& 1 \end{eqnarray*} Here \(L\), \(L_{A}\), and \(L_{B}\) are three links which are isotopic everywhere except at one crossing where the look as in the figure:

Proposition 2.2. The Kauffman polynomial is invariant under second and third Reidemeister moves but not under the first Reidemeister move.

3. Main Results

In this section we shall give the Kauffman bracket of the links \(\widehat{x_{1}^{n}}\) and \(\widehat{x_{1}^{b}x_{2}^{m}}\), and show that \(\langle\widehat{x_{1}^{b}x_{2}^{m}}\rangle=\langle\widehat{x_{1}^{b}}\rangle\langle\widehat{x_{2}^{m}}\rangle\).

The links \(\widehat{x_{1}^{n}}\) fall into two categories, the two-component links when \(n\) is even and the one-component links (means knots) when \(n\) is odd. When \(n\) is even, we have:

Proposition 3.1. The Kauffman bracket of the link \(\widehat{x_{1}^{n}}\), when \(n\geq2\) is even, is

Proof. We prove it by induction on \(n\).
When \(n = 2\),

Similarly, we have and We now assume the result holds for \(n = k\), that is Now for \(n=k+1\), we, following Equations (3) and (4), write \begin{eqnarray*} % \nonumber to remove numbering (before each equation) \langle \widehat{x_{1}^{k+2}} \rangle &=& -a^{3(k+2)-2} + a^{3(k+2)-6} + a^{-2}\langle \widehat{x_{1}^{k}} \rangle \\ &=& -a^{3(k+2)-2} + a^{3(k+2)-6}+ a^{-2}\Big[-a^{3k-2}+a^{3k-6}-a^{3k-10}\\ && +a^{3k-14}-\cdots-a^{6-k}-a^{-k-2}\Big]\\ &=& -a^{3(k+2)-2} + a^{3(k+2)-6}-a^{3k-4}+a^{3k-8}-a^{3k-12}+a^{3k-16}\\ &&-\cdots-a^{4-k}-a^{-k-4}\\ &=& -a^{3(k+2)-2} + a^{3(k+2)-6}-a^{3(k+2)-10}+a^{3(k+2)-14}-a^{3(k+2)-18}\\ &&+a^{3(k+2)-22}-\cdots-a^{6-(k+2)}-a^{-(k+2)-2} \end{eqnarray*} This completes the proof.

Proposition 3.2. The Kauffman bracket of the knots \(\widehat{x_{1}^{n}}\), when \(n\) is odd, is

Proof. Similar to the proof of Proposition 3.1.

Proposition 3.3. The Kauffman bracket of the braid link \(\widehat{x_{1}^{b} x_{2}^{b}}\), when \(b\) is even, is \begin{eqnarray*} % \nonumber to remove numbering (before each equation) \langle \widehat{x_{1}^{b} x_{2}^{b}} \rangle &=& \sum_{i=1}^{b-1}i(-1)^{i+1}a^{6b-4i}+\sum_{i=1}^{b}(-1)^{i+1}(b-i)a^{2b-4i}-(b-2)a^{2b}\\ &&+a^{4-2b}+a^{-2b-4}. \end{eqnarray*}

Proof. We prove it by induction on \(b\).
When \(b = 2\), we have

as required.
Similarly, we get In order to manage the proof, we reorganize (7): Similarly, Deducting from Equations (9) and (10), we can write \begin{eqnarray} % \nonumber to remove numbering (before each equation) \nonumber\langle \widehat{x_{1}^{b} x_{2}^{b}} \rangle &=& a^{-4}\big[\langle \widehat{x_{1}^{b-2} x_{2}^{b-2}} \rangle\big]-\sum_{i=1}^{b-3}i(-1)^{i+1}a^{6b-4i-16}+\sum_{i=1}^{b-3}i(-1)^{i+1}a^{6b-4i}\\ \nonumber&&-2a^{2b-8}+\sum_{i=b-2}^{b-1}i(-1)^{i+1}a^{6b-4i}-(b-2)a^{2b}+(b-1)a^{2b-4}. \end{eqnarray} We now assume the result holds for \(b=k\), that is Now for \(b=k+2\), we have \begin{eqnarray*} % \nonumber to remove numbering (before each equation) \langle \widehat{x_{1}^{k+2} x_{2}^{k+2}} \rangle &=& a^{-4}\big[\langle \widehat{x_{1}^{k} x_{2}^{k}} \rangle\big]-\sum_{i=1}^{k-1}i(-1)^{i+1}a^{6k-4i-4}+\sum_{i=1}^{k-1}i(-1)^{i+1}a^{6k-4i+12}\\ &&-2a^{2k-4}+\sum_{i=k}^{k+1}i(-1)^{i+1}a^{6k-4i+12}-ka^{2k+4}+(k+1)a^{2k}\\ &=&\sum_{i=1}^{k-1}i(-1)^{i+1}a^{6k-4i-4}+\sum_{i=1}^{k}(-1)^{i+1}(k-i)a^{2k-4i-4}\\ &&-(k-2)a^{2k-4}+a^{-2k}+a^{-2k-8}\\ &&-\sum_{i=1}^{k-1}i(-1)^{i+1}a^{6k-4i-4}+\sum_{i=1}^{k-1}i(-1)^{i+1}a^{6k-4i+12}-2a^{2k-4}\\ &&+\sum_{i=k}^{k+1}i(-1)^{i+1}a^{6k-4i+12}-ka^{2k+4}+(k+1)a^{2k}\\ &=&\sum_{i=1}^{k+1}i(-1)^{i+1}a^{6k-4i+12}+\sum_{i=-1}^{k}(-1)^{i+1}(k-i)a^{2k-4i-4}\\ &&+a^{-2k}+a^{-2k-8}-ka^{2k+4}\\ &=&\sum_{i=1}^{k+1}i(-1)^{i+1}a^{6k-4i+12}+\sum_{i=1}^{k+2}(-1)^{i+1}(k+2-i)a^{2k-4i+4}\\ &&+a^{-2k}+a^{-2k-8}-ka^{2k+4}, \end{eqnarray*} and the induction is completed.

Proposition 3.4. The Kauffman bracket of the braid link \(\widehat{x_{1}^{b} x_{2}^{b}}\), when \(b\) is odd, is \begin{eqnarray*} % \nonumber to remove numbering (before each equation) \langle \widehat{x_{1}^{b} x_{2}^{b}} \rangle &=& \sum_{i=1}^{b-1}i(-1)^{i+1}a^{6b-4i}+\sum_{i=1}^{b}(-1)^{i}(b-i)a^{2b-4i}+(b-2)a^{2b}\\ &&+a^{4-2b}+a^{-2b-4}. \end{eqnarray*}

Proof. Similar to the proof of proposition 3.3.

Proposition 3.5. The Kauffman bracket of \(\widehat{x_{1}^{b} x_{2}^{m}}\), when \(b>m\geq2\), is \begin{eqnarray*} % \nonumber to remove numbering (before each equation) \langle \widehat{x_{1}^{b} x_{2}^{m}} \rangle &=& \sum_{i=1}^{m-1}(-1)^{b+m+1-i}(i)a^{3(b+m)-4i}+(-1)^{b+1}(m-1)a^{3b-m}\\ &&+m\sum_{i=1}^{b-m-1}(-1)^{b+1-i}a^{3b-m-4i}+(-1)^{m+1}(m-1)a^{-b+3m}\\ &&+\sum_{i=1}^{m-2}(-1)^{m+1-i}(m-i)a^{-b+3m-4i}+2a^{-b-m+4}+a^{-b-m-4}. \end{eqnarray*}

Proof. We first verify the result for arbitrary \(b\) and \(m=2\):
Resolving all \(2^{3+2}\) crossings as were resolved for \(\langle \widehat{x_{1}^{2} x_{2}^{2}} \rangle\) in Proposition 3.3, we get \begin{eqnarray*} % \nonumber to remove numbering (before each equation) \langle \widehat{x_{1}^{3} x_{2}^{2}} \rangle &=& -a^{11}+a^{7}-a^{3}+2a^{-1}+a^{-9} \end{eqnarray*} Similarly, we get

It follows from (11), (12), and (13) that \begin{eqnarray*} % \nonumber to remove numbering (before each equation) \nonumber \langle \widehat{x_{1}^{b} x_{2}^{2}} \rangle &=& -a^{3}\langle\widehat{x_{1}^{b-1} x_{2}^{2}}\rangle+a^{-b+10}+a^{-b+6}+2a^{-b+2}+a^{-b-2}+a^{-b-6}. \end{eqnarray*} Now suppose the result is true for \(b=t\) and \(m=2\), that is For \(b=t+1\), we have \begin{eqnarray*} % \nonumber to remove numbering (before each equation) \langle \widehat{x_{1}^{t+1} x_{2}^{2}} \rangle &=& -a^{3}\langle\widehat{x_{1}^{t} x_{2}^{2}}\rangle+a^{-t+9}+a^{-t+5}+2a^{-t+1}+a^{-t-3}+a^{-t-7}\\ &=&-a^{3}\Big[(-1)^{-t+2}a^{3t+2}+(-1)^{t+1}a^{3t-2}+2\sum_{i=1}^{t-3}(-1)^{t+1-i}a^{3t-2-4i}\\ &&-a^{-t+6}+2a^{-t+2}+a^{-t-6}\Big]+a^{-t+9}+a^{-t+5}+2a^{-t+1}\\ &&+a^{-t-3}+a^{-t-7}\\ &=&(-1)^{t+3}a^{3t+5}+(-1)^{t+2}a^{3t+1}+2\sum_{i=1}^{t-3}(-1)^{t+2-i}a^{3t+1-4i}\\ &&+a^{-t+9}-2a^{-t+5}-a^{-t-3}+a^{-t+9}+a^{-t+5}+2a^{-t+1}\\ &&+a^{-t-3}+a^{-t-7}\\ &=&(-1)^{t+3}a^{3t+5}+(-1)^{t+2}a^{3t+1}\\ &&+\Big[2\sum_{i=1}^{t-3}(-1)^{t+2-i}a^{3t+1-4i}+2a^{-t+9}\Big]\\ &&-a^{-t+5}+2a^{-t+1}+a^{-t-7}\\ &=&(-1)^{(t+1)+2}a^{3(t+1)+2}+(-1)^{(t+1)+1}a^{3(t+1)-2}\\ &&+2\sum_{i=1}^{(t+1)-3}(-1)^{(t+1)+1-i}a^{3(t+1)-2-4i}\\ &&-a^{-(t+1)+6}+2a^{-(t+1)+2}+a^{-(t+1)-6}. \end{eqnarray*} Similarly, we get \begin{eqnarray*} % \nonumber to remove numbering (before each equation) \langle \widehat{x_{1}^{b} x_{2}^{3}} \rangle &=& \sum_{i=1}^{2}(-1)^{b+4-i}(i)a^{3b+9-4i}+(-1)^{b+1}2a^{3b-3}\\ &&+3\sum_{i=1}^{b-4}(-1)^{b+1-i}a^{3b-3-4i}+2a^{-b+9}\\ &&-a^{-b+5}+2a^{-b+1}+a^{-b-7} \end{eqnarray*} and \begin{eqnarray*} % \nonumber to remove numbering (before each equation) \langle \widehat{x_{1}^{b} x_{2}^{4}} \rangle &=& \sum_{i=1}^{3}(-1)^{b+5-i}(i)a^{3b+12-4i}+(-1)^{b+1}3a^{3b-4}\\ &&+4\sum_{i=1}^{b-5}(-1)^{b+1-i}a^{3b-4-4i}-3a^{-b+12}\\ &&+\sum_{i=1}^{2}(-1)^{5-i}(4-i)a^{-b+12-4i}+2a^{-b}+a^{-b-8}. \end{eqnarray*} Now with the assumption that the result is true for an arbitrary \(m\), we have \begin{eqnarray*} &&\\ &&\langle \widehat{x_{1}^{b} x_{2}^{m+1}} \rangle\\ =&& -a^3\langle \widehat{x_{1}^{b} x_{2}^{m}}\rangle+(-1)^{b}a^{3b-(m+1)+4}+\sum_{i=1}^{b-3}(-1)^{b+1-i}(i)a^{3b-(m+1)-4i}\\ &&+2a^{-b-(m+1)+4}+a^{-b-(m+1)}+a^{-b-(m+1)-4} \end{eqnarray*} \begin{eqnarray*}=&&\sum_{i=1}^{m-1}(-1)^{b+m+2-i}(i)a^{3(b+m)+3-4i}+(-1)^{b+2}(m-1)a^{3b-m+3}\\ &&+m\sum_{i=1}^{b-m-1}(-1)^{b+2-i}a^{3b-m+3-4i}+(-1)^{m+2}(m-1)a^{-b+3m+3}\\ &&+\sum_{i=1}^{m-2}(-1)^{m+2-i}(m-i)a^{-b+3m+3-4i}-2a^{-b-m+7}-a^{-b-m-1} \end{eqnarray*} \begin{eqnarray*}&&+(-1)^{b}a^{3b-(m+1)+4}+\sum_{i=1}^{b-3}(-1)^{b+1-i}(i)a^{3b-(m+1)-4i}\\ &&+2a^{-b-(m+1)+4}+a^{-b-(m+1)}+a^{-b-(m+1)-4} \end{eqnarray*} \begin{eqnarray*}=&&\sum_{i=1}^{(m+1)-1}(-1)^{b+(m+1)+1-i}(i)a^{3(b+(m+1))-4i}\\ &&+m\sum_{i=1}^{b-m-1}(-1)^{b+2-i}a^{3b-m+3-4i}+(-1)^{m+2}(m-1)a^{-b+3m+3}\\ &&+\sum_{i=1}^{m-2}(-1)^{m+2-i}(m-i)a^{-b+3m+3-4i}-2a^{-b-m+7} \end{eqnarray*} \begin{eqnarray*}&&+\sum_{i=1}^{b-3}(-1)^{b+1-i}(i)a^{3b-(m+1)-4i}+2a^{-b-(m+1)+4}+a^{-b-(m+1)-4} \end{eqnarray*} \begin{eqnarray*}=&&\sum_{i=1}^{(m+1)-1}(-1)^{b+(m+1)+1-i}(i)a^{3(b+(m+1))-4i}\\ &&+\Big[(-1)^{b+1}m a^{3b-m-1}+(-1)^{b}m a^{3b-m-5}+(-1)^{b-1}m a^{3b-m-9}\\ &&+\cdots+(-1)^{m+3}m a^{-b+3m+7}+\Big]+(-1)^{m+2}(m-1)a^{-b+3m+3}\\ &&+\Big[(-1)^{m+1}(m-1) a^{-b+3m-1}+(-1)^{m}(m-1) a^{-b+3m-5}\\ &&+(-1)^{m-1}(m-3) a^{-b+3m-9}+\cdots+(-1)^{4}2 a^{-b-m+11}+\Big]\\ &&-2a^{-b-m+7} \end{eqnarray*} \begin{eqnarray*}&&+\Big[\Big((-1)^{b}a^{3b-m-5}+(-1)^{b-1}a^{3b-m-9}+\cdots+(-1)^{m+3}a^{-b+3m+7}\Big)\\ &&+\Big((-1)^{m+2}a^{-b+3m+3}+(-1)^{m+1}a^{-b+3m-1}+(-1)^{m}a^{-b+3m-5}\\ &&+\cdots+(-1)^{4}a^{-b-m+11}\Big)\Big]+2a^{-b-(m+1)+4}+a^{-b-(m+1)-4} \end{eqnarray*} Now collecting terms of same exponents, we get \begin{eqnarray*}=&&\sum_{i=1}^{(m+1)-1}(-1)^{b+(m+1)+1-i}(i)a^{3(b+(m+1))-4i}+(-1)^{b+1}m a^{3b-m-1}\\ &&+\Big[+(-1)^{b}(m+1) a^{3b-m-5}+(-1)^{b-1}(m+1)a^{3b-m-9}\\ &&+\cdots+(-1)^{m+3}(m+1)a^{-b+3m+7}+\Big]+(-1)^{m+2}(m)a^{-b+3m+3}\\ &&+\Big[(-1)^{m+1}(m)a^{-b+3m-1}+(-1)^{m}(m-1)a^{-b+3m-5}\\ &&+\cdots+(-1)^{4}3 a^{-b-m+11}-2a^{-b-m+7}\Big]\\ &&+2a^{-b-(m+1)+4}+a^{-b-(m+1)-4} \end{eqnarray*} which finally, in terms of summation form, is the required result.

Theorem 3.6. For any \(b,m\geq2\), $$\langle \widehat{x_{1}^{b} x_{2}^{m}} \rangle = \langle \widehat{x_{1}^{b}} \rangle \langle \widehat{x_{1}^{m}} \rangle.$$

Proof. Depending on \(b\) and \(m\), the proof is divided into three cases: when \(b,m\) are even and equal, when \(b,m\) are odd and equal, and when \(b,m\) are distinct.
Case I. (When \(b\) and \(m\) are even and equal.)
In this case, letting \(m=b\), we show that \(\langle \widehat{x_{1}^{b} x_{2}^{m}} \rangle = \langle \widehat{x_{1}^{b}} \rangle \langle \widehat{x_{1}^{m}} \rangle.\) So, we proceed as follows: \begin{eqnarray*} % \nonumber to remove numbering (before each equation) \langle \widehat{x_{1}^{2} x_{2}^{2}} \rangle &=& a^{8} + 2 + a^{-8}= (-a^{4}-a^{-4})(-a^{4}-a^{-4})=\langle \widehat{x_{1}^{2}} \rangle \langle \widehat{x_{1}^{2}} \rangle. \end{eqnarray*} Also, we have \begin{eqnarray*} % \nonumber to remove numbering (before each equation) \langle \widehat{x_{1}^{4} x_{2}^{4}} \rangle &=& a^{20}-2a^{16}+3a^{12}-2a^{8}+3a^{4}-2+2a^{-4}+a^{-12} \\ &=& (-a^{10}+a^{6}-a^{2}-a^{-6})(-a^{10}+a^{6}-a^{2}-a^{-6})\\ &=& \langle \widehat{x_{1}^{4}} \rangle \langle \widehat{x_{1}^{4}} \rangle \end{eqnarray*} and \begin{eqnarray*} % \nonumber to remove numbering (before each equation) \langle \widehat{x_{1}^{6} x_{2}^{6}} \rangle &=& a^{32}-2a^{28}+3a^{24}-4a^{20}+5a^{16}-4a^{12}+5a^{8}-4a^{4}\\ &&+3-2a^{-4}+2a^{-8}+a^{-16} \\ &=& (-a^{16}+a^{12}-a^{8}+a^{4}-a^{0}-a^{-8})(-a^{16}+a^{12}-a^{8}\\ &&+a^{4}-a^{0}-a^{-8})=\langle \widehat{x_{1}^{6}} \rangle \langle \widehat{x_{1}^{6}} \rangle. \end{eqnarray*} Now we assume that the result is true for \(b=k\), that is $$\langle \widehat{x_{1}^{k} x_{2}^{k}} \rangle = \langle \widehat{x_{1}^{k}} \rangle \langle \widehat{x_{1}^{k}} \rangle.$$ Since \(\langle \widehat{x_{1}^{n}} \rangle = -a^{3(n)-2} + a^{3(n)-6} + a^{-2}(\langle \widehat{x_{1}^{n-2}} \rangle)\), we have

Also The result now follows from (15) and (16).
Case II. (When \(b\) and \(m\) are odd and equal.) Similar to Case I.
Case III. (When \(b\) and \(m\) are distinct.)
In order to prove this part let us agree on the terminology: \begin{eqnarray*} % \nonumber to remove numbering (before each equation) \overline{x}_{n} &=& (-1)^{m+n}a^{3m-(4n-2)}, n=1,2,\ldots,m-1, \overline{x}_{m}=-a^{-m-2}\\ \overline{y}_{l} &=& (-1)^{b+l}a^{3b-(4l-2)}, l=1,2,\ldots,b-1, \overline{y}_{b}=-a^{-b-2}\\ i&=&1,2,\ldots,m, j=1,2,\ldots,b; b\geq2\\ \end{eqnarray*} \begin{eqnarray*} % \nonumber to remove numbering (before each equation) &&\langle \widehat{x_{1}^{b}} \rangle \langle \widehat{x_{1}^{m}}\rangle \\ &=&\sum_{i+j=2}^{m}\overline{x}_{i}\overline{y}_{j}+\sum_{i+j=m+1, i\neq m}\overline{x}_{i}\overline{y}_{j}+\Big[\sum_{i+j=m+2, i\neq m}\overline{x}_{i}\overline{y}_{j}+\overline{x}_{m}\overline{y}_{1}\\ &&+\sum_{i+j=m+3, i\neq m}\overline{x}_{i}\overline{y}_{j}+\overline{x}_{m}\overline{y}_{2}+\cdots+\sum_{i+j=b, i\neq m}\overline{x}_{i}\overline{y}_{j}+\overline{x}_{m}\overline{y}_{b-m-1}\Big]\\ &&+\Big[\sum_{i+j=b+1, i\neq 1,m}\overline{x}_{i}\overline{y}_{j}+\overline{x}_{m}\overline{y}_{b-m}\Big]+ \Big[\Big(\sum_{i+j=b+2, i\neq 2,m}\overline{x}_{i}\overline{y}_{j}+\overline{x}_{m}\overline{y}_{b-m+1}+\overline{x}_{1}\overline{y}_{b}\Big)\\ &&+\Big(\sum_{i+j=b+3, i\neq 3,m}\overline{x}_{i}\overline{y}_{j}+\overline{x}_{m}\overline{y}_{b-m+2}+\overline{x}_{2}\overline{y}_{b}\Big)+\cdots\\ &&+\Big(\sum_{i+j=b+m-3, i\neq m-3,m}\overline{x}_{i}\overline{y}_{j}+\overline{x}_{m}\overline{y}_{b-4}+\overline{x}_{m-4}\overline{y}_{b}\Big)\\ &&+\Big(\overline{x}_{m-1}\overline{y}_{b-1}+\overline{x}_{m}\overline{y}_{b-3}+\overline{x}_{m-3}\overline{y}_{b}\Big) +\Big(\overline{x}_{m}\overline{y}_{b-2}+\overline{x}_{m-2}\overline{y}_{b}\Big)\Big]\\ &&+\Big(\overline{x}_{m}\overline{y}_{b-1}+\overline{x}_{m-1}\overline{y}_{b}\Big)+\overline{x}_{m}\overline{y}_{b} \end{eqnarray*} Since this agrees with the result of Proposition 3.5, the proof is finished.

Competing Interests

The author(s) do not have any competing interests in the manuscript.

1. Introduction and Preliminaries

Mathematical chemistry is the branch of theoretical chemistry in which we discuss and predict the behavior of mathematical structure by using mathematical tools. There is lot of research which is done in this area in the last few decades. This theory contributes a major role in the field of chemical sciences.

Let \(G\) be the molecular graph in which \(V(G)\) represents the set of vertices corresponds the atoms and \(E(G)\) the set of edges to the chemical bonds. A line graph \(L(G)\) of a simple graph \(G\) is obtained by associating a vertex with each edge of the graph and connecting two vertices with an edge if and only if the corresponding edges of \(G\) have a vertex in common.

The very first topological Index Randić index introduced by Milan Randić in 1975 (see [1]). and is defined as: \[R(G)=\sum_{uv \in E(G)} \frac{1}{\sqrt{d_ud_v}}\].

Later, this index was generalized by Bollobás and Erdős( see [2]) to the following form for any real number \(\alpha\), and named the general Randić index: \begin{equation*} R(G)=\sum_{uv \in E(G)} [{d_ud_v}]^\alpha . \end{equation*} The Zagreb indices were first introduced by Gutman in [3], they are important molecular descriptors and have been closely correlated with many chemical properties (see [4]) and defined as: \begin{align*}\label{F1} M_1(G) &=\sum_{u \in V(G)} d_u^2 \hspace{10mm} \mbox{and} \\ M_2(G) &= \sum_{uv \in E(G)} d_{u}d_{v}. \end{align*} The third Zagreb index, introduced by Fath-Tabar in [5]. This index is defined as follows: \begin{eqnarray*} M_3(G) = \sum_{uv \in E(G)} |d_u- d_v|. \end{eqnarray*} The hyper-Zagreb index was first introduced in [6]. This index is defined as follows: \begin{eqnarray*} HM(G) &=& \sum_{uv \in E(G)} (d_u+d_v)^{2}. \end{eqnarray*} The Atom-Bond Connectivity index (ABC), introduced by Estrda et al. in [7] and applied up until now to study the stability of alkanes and the strain energy of cycloalkanes. The ABC index of \(G\) is defined as: \begin{equation*}\label{f4*} ABC(G)=\sum_{uv \in E(G)} \sqrt{\frac{d_u+d_v-2}{d_ud_v}}. \end{equation*} For more details see the article [8]. In 2010, the general sum-connectivity index \(\chi(G)\) has been introduced in [9]. For more detail on sum connectivity we refer the articles [9, 10]. This index is defined as follows: \begin{equation*} \chi(G)=\sum_{uv \in E(G)} \frac{1}{\sqrt{d_u+d_v}}. \end{equation*} Vukicevic and Furtula introduced the geometric arithmetic (GA) index in [11]. The GA index for \(G\) is defined by \begin{equation*} GA(G)=\sum_{uv \in E(G)} \frac{2\sqrt{d_ud_v}}{d_u+d_v}. \end{equation*} Inspired by the work on the ABC index, Furtula et al. proposed the following modified version of the ABC index and called it as Zagreb index (AZI) in [12]. This index is defined as follows: \begin{equation*}\label{F2} AZI(G)=\sum_{uv \in E(G)} (\frac{d_ud_v}{d_u+d_v-2})^3. \end{equation*} The hexagonal and honeycomb networks have also been recognized as crucial evolutionary biology, in particular for the evolution of cooperation, where the overlapping triangles are vital for the propagation of cooperation in social dilemmas. Relevant research that applies this theory and which could benefit further from the insights of the new research in (see [13]). The following lemma is helpful for computing the degree of a vertex of line graph.

Lemma 1.1. Let \(G\) be a graph with \(u,v\in V(G)\) and \(e=uv \in E(G).\) Then: $$d_e=d_u+d_v-2.$$

Lemma 1.2.[14] Let \(G\) be a graph of order \(p\) and size \(q\), then the line graph \(L(G)\) of \(G\) is a graph of order \(p\) and size \(\frac{1}{2}M_{1}(G) -q\).

2. Topological indices of Hex board

In this section we will compute the topological indices of Hex board.

Theorem 2.1. Let \(G\) be the Hex board \(H_n\). Then

1. \(M_{1}(G) = 36n^2-80n+42;\)
2. \(M_{2}(G) = 108n^2-320n+232;\)
3. \(M_{3}(G) = -16n+22;\)
4. \(HM(G) = 432n^2-1248n+886;\)
5. \(R(G) = \frac{1}{2}n^2+\frac{1}{12}(8n-20)\sqrt{6}-\frac{5}{3}n+\frac{4}{3}\sqrt{2}+\frac{2}{3}\sqrt{3}+1\);
6. \(ABC(G)=\frac{1}{6}(3n^2-16n+21)\sqrt{10}+\frac{1}{3}(8n-20)\sqrt{3}+\frac{1}{4}(4n-10)\sqrt{6}+\frac{1}{3}\sqrt{14}+\frac{2}{3}\sqrt{15}+2\sqrt{2} ;\)
7. \(\chi(G) = \frac{1}{6}(3n^2-16n+21)\sqrt{3}+\frac{1}{10}(8n-20)\sqrt{10}+\frac{1}{4}(4n-10)\sqrt{2}+\frac{2}{3}\sqrt{6}+\frac{4}{7}\sqrt{7}+\frac{2}{3}\);
8. \(GA(G) = 3n^2+\frac{2}{5}(8n-20)\sqrt{6}-12n+4\sqrt{2}+\frac{16}{7}\sqrt{3}+11\);
9. \(AZI(G) = \frac{17496}{125}n^2-\frac{1534424}{3375}n+\frac{429997724}{1157625}\).

Proof. The graph \(G\) for \(n=4\) is shown in Figure 1. It is easy to see that the order of \(G\) is \(n^2\) out of which \(2\) vertices are of degree \(2\), \(2\) vertices are of degree \(3\), \(4(n-2)\) vertices are of degree \(4\) and \(n^2-4(n-1)\) vertices are of degree \(6\) and \(G\) has size \(3n^2-4n+1\). We partition the size of \(G\) into edges of the type \(E_{(d_u,d_v)}\) where \(uv\) is an edge. In \(G\), we get edges of the type \(E_{(2,4)}\), \(E_{(3,4)}\), \(E_{(3,6)}\), \(E_{(4,4)}\), \(E_{(4,6)}\) and \(E_{(6,6)}\). The number of edges of these types are given in the Table 1. Then we obtain the required results by using Table 1 as follows:

1. \begin{eqnarray*} M_1(G) &=& \sum_{uv \in E(G)} \big[ d_u +d_v \big]\\ &=& 4(2+4)+4(3+4)+2(3+6)+(4n-10)(4+4)+(8n-20)(4+6)+(3n^2-16n+21)(6+6)\\ &=& 36n^2-80n+42. \end{eqnarray*} 2.\begin{eqnarray*} M_2(G)&=& \sum_{uv \in E(G)} d_{u}d_{v}\\ &=& 4(2\times4)+4(3\times4)+2(3\times6)+(4n-10)(4\times4)+(8n-20)(4\times6)+(3n^2-16n+21)(6\times6)\\ &=& 108n^2-320n+232. \end{eqnarray*} 3.\begin{eqnarray*} M_3(G) &=& \sum_{uv \in E(G)} |d_u- d_v|\\ &=& 4(2-4)+4(3-4)+2(3-6)+(4n-10)(4-4)+(8n-20)(4-6)+(3n^2-16n+21)(6-6).\\ &=& -16n+22. \end{eqnarray*} 4. \begin{eqnarray*} HM(G) &=& \sum_{uv \in E(G)} (d_u+d_v)^{2}\\ &=& 4(2+4)^2+4(3+4)^2+2(3+6)^2+(4n-10)(4+4)^2+(8n-20)(4+6)^2+(3n^2-16n+21)(6+6)^2.\\ &=& 432n^2-1248n+886. \end{eqnarray*} 5. \begin{eqnarray*} R(G)&=& \sum_{uv \in E(G)} \frac{1}{\sqrt{d_ud_v}}$\\ &=& 4(\frac{1}{\sqrt{2\times4}})+4(\frac{1}{\sqrt{3\times4}})+2(\frac{1}{\sqrt{3\times6}})\\&&+(4n-10)(\frac{1}{\sqrt{4\times4}})+(8n-20)(\frac{1}{\sqrt{4\times6}})+(3n^2-16n+21)(\frac{1}{\sqrt{6\times6}}).\\ &=&\frac{1}{2}n^2+\frac{1}{12}(8n-20)\sqrt{6}-\frac{5}{3}n+\frac{4}{3}\sqrt{2}+\frac{2}{3}\sqrt{3}+1. \end{eqnarray*} 6. \begin{eqnarray*} ABC(G)&=& \sum_{uv \in E(G)} \sqrt{\frac{d_u+d_v-2}{d_ud_v}}\\ &=& 4\sqrt{\frac{2+4-2}{2\times4}} +4\sqrt{\frac{3+4-2}{3\times4}}+2\sqrt{\frac{3+6-2}{3\times6}}+(4n-10)\sqrt{\frac{4+4-2}{4\times4}}\\ &&+(8n-20)\sqrt{\frac{4+6-2}{4\times6}}+(3n^2-16n+21)\sqrt{\frac{6+6-2}{6\times6}}.\\ &=&\frac{1}{6}(3n^2-16n+21)\sqrt{10}+\frac{1}{3}(8n-20)\sqrt{3}+\frac{1}{4}(4n-10)\sqrt{6}+\frac{1}{3}\sqrt{14}+\frac{2}{3}\sqrt{15}+2\sqrt{2}. \end{eqnarray*} 7. \begin{eqnarray*} \chi(G)&=& \sum_{uv \in E(G)} \frac{1}{\sqrt{d_u+d_v}}\\ &=& 4(\frac{1}{\sqrt{2+4}})+4(\frac{1}{\sqrt{3+4}})+2(\frac{1}{\sqrt{3+6}})+(4n-10)(\frac{1}{\sqrt{4+4}})\\ &&+(8n-20)(\frac{1}{\sqrt{4+6}})+(3n^2-16n+21)(\frac{1}{\sqrt{6+6}}).\\ &=& \frac{1}{6}(3n^2-16n+21)\sqrt{3}+\frac{1}{10}(8n-20)\sqrt{10}+\frac{1}{4}(4n-10)\sqrt{2}+\frac{2}{3}\sqrt{6}+\frac{4}{7}\sqrt{7}+\frac{2}{3}. \end{eqnarray*} 8. \begin{eqnarray*} GA(G)&=& \sum_{uv \in E(G)} \frac{2\sqrt{d_ud_v}}{d_u+d_v}\\ &=& 4(\frac{2\sqrt{2\times4}}{2+4})+4(\frac{2\sqrt{3\times4}}{3+4})+2(\frac{2\sqrt{3\times6}}{3+6})\\ &&+(4n-10)(\frac{2\sqrt{4\times4}}{4+4})+(8n-20)(\frac{2\sqrt{4\times6}}{4+6})+(3n^2-16n+21)(\frac{2\sqrt{6\times6}}{6+6}).\\ &=& 3n^2+\frac{2}{5}(8n-20)\sqrt{6}-12n+4\sqrt{2}+\frac{16}{7}\sqrt{3}+11. \end{eqnarray*} 9. \begin{eqnarray*} AZI(G)&=& \sum_{uv \in E(G)} \left(\frac{d_ud_v}{d_u+d_v-2}\right)^3\\ &=&4(\frac{2\times4}{2+4-2})^3+4(\frac{3\times4}{3+4-2})^3+2(\frac{3\times6}{3+6-2})^3\\ &&+(4n-10)(\frac{4\times4}{4+4-2})^3+(8n-20)(\frac{4\times6}{4+6-2})^3+(3n^2-16n+21)(\frac{6\times6}{6+6-2})^3.\\ &=& \frac{17496}{125}n^2-\frac{1534424}{3375}n+\frac{429997724}{1157625}. \end{eqnarray*}

Table 1. The size partition of \(G\)

\((d_u,d_v)\) where \(uv\in E(G)\)	(2,4)	(3,4)	(3,6)
Number of edges	4	4	2
\((d_u,d_v)\) where \(uv\in E(G)\)	(4, 4)	(4, 6)	(6, 6)
Number of edges	4n 10	8n 20	\(3n^2-16n+21\)

3. Topological indices of line graph of Hex board

In this section we will compute the topological indices of the line graph of Hex board.

Theorem 3.1. Let \(G\) be the line graph of the Hex board \(H_n\). Then

1. \(M_{1}(G) = 300n^2-944n+722;\)
2. \(M_{2}(G) = 1500n^2-5616n+5194;\)
3. \(M_{3}(G) = -96n+206;\)
4. \(HM(G) = 6000n^2-22272n+20430;\)
5. \(R(G) = \frac{3}{2}n^2-\frac{119}{15}n+\frac{1}{20}(32n-100)\sqrt{5}+\frac{3}{35}\sqrt{100}+\frac{1}{7}\sqrt{14}+\frac{1}{12}(16n-48)\sqrt{3}+\frac{2}{5}\sqrt{10}+\frac{4}{35}\sqrt{35}+\frac{2}{15}\sqrt{30}+\frac{1}{2}\sqrt{2}+\frac{2}{3}\sqrt{6}+\frac{247}{20}\);
6. \(ABC(G) =\frac{3}{10}(15n^2-96n+152)\sqrt{2}+\frac{1}{5}(32n-100)\sqrt{5}+\frac{1}{6}(4n-12)\sqrt{10}+8n+\frac{1}{8}(8n-14)\sqrt{14}+\frac{3}{7}\sqrt{42}+\frac{1}{7}\sqrt{82}+\frac{2}{5}\sqrt{110}+\frac{4}{7}\sqrt{14}+\frac{2}{5}\sqrt{30}+\frac{4}{5}\sqrt{2}+\sqrt{5}+\frac{8}{3}\sqrt{3}+\frac{1}{2}\sqrt{6}-24;\)
7. \(\chi(G) = \frac{1}{10}(15n^2-96n+152)\sqrt{5}+\frac{1}{6}(32n-100)\sqrt{2}+\frac{1}{14}(16n-48)\sqrt{14}+\frac{1}{6}(4n-12)\sqrt{3}+2n-\frac{7}{2}+\frac{6}{17}\sqrt{17}+\frac{4}{15}\sqrt{15}+\frac{8}{13}\sqrt{13}+\frac{4}{11}\sqrt{11}+\frac{4}{3}\sqrt{3}+\sqrt{10}+\frac{1}{2}\sqrt{2}\);
8. \(GA(G) = 15n^2+\frac{4}{9}(32n-100)\sqrt{5}+\frac{4}{7}(16n-48)\sqrt{3}-84n+\frac{12}{17}\sqrt{70}+\frac{16}{15}\sqrt{14}+\frac{32}{13}\sqrt{10}+\frac{2}{3}\sqrt{35}+\frac{8}{11}\sqrt{30}+\frac{8}{3}\sqrt{2}+\frac{16}{5}\sqrt{6}+130\);
9. \(AZI(G) = \frac{625000}{243}n^2-\frac{109249838528}{10418625}n+\frac{1427097955303080667}{133703165756160}\).

Proof. The graph \(G\) for \(n=4\) is shown in Figure 2. By using Lemma 1.1, It is easy to see that the order of \(G\) is \(3n^2-4n+1\) out of which \(4\) vertices are of degree \(4\), \(4\) vertices are of degree \(5\), \(2\) vertices are of degree \(7\), \(4n-10\) vertices are of degree \(6\), \(8n-20\) vertices are of degree \(8\) and \(3n^2-16n+21\) vertices are of degree \(10\). Therefore by using Lemma 1.2, \(G\) has size \(15n^2-36n+20\). We partition the size of \(G\) into edges of the type \(E_{(d_u,d_v)}\) where \(uv\) is an edge. In \(G\), we get edges of the type \(E_{(4,4)}\), \(E_{(4,6)}\), \(E_{(4,8)}\), \(E_{(5,5)}\), \(E_{(5,6)}\), \(E_{(5,7)}\), \(E_{(5,8)}\), \(E_{(6,6)}\), \(E_{(6,8)}\), \(E_{(7,8)}\), \(E_{(7,10)}\), \(E_{(8,8)}\), \(E_{(8,10)}\) and \(E_{(10,10)}\). The number of edges of these types are given in the Table 2. Then we obtain the required results by using Table 2 as follows:

1. \begin{eqnarray*} M_1(G)&=&\sum_{uv \in E(G)} \big[ d_u +d_v \big]\\ &=&2(4+4)+8(4+6)+4(4+8)+2(5+5)+4(5+6)+4(5+7)+8(5+8)+(4n-12)(6+6)\\ &&+(16n-48)(6+8)+4(7+8)+6(7+10)+(8n-14)(8+8)+(32n-100)(8+10)+(15n^2-96n+152)(10+10).\\ &=& 300n^2-944n+722. \end{eqnarray*} 2. \begin{eqnarray*} M_2(G) &=&\sum_{uv \in E(G)} d_{u}d_{v}\\ &=& 2(4\times4)+8(4\times6)+4(4\times8)+2(5\times5)+4(5\times6)+4(5\times7)+8(5\times8)+(4n-12)(6\times6)\\ && +(16n-48)(6\times8)+4(7\times8)+6(7\times10)+(8n-14)(8\times8)+(32n-100)(8\times10)+(15n^2-96n+152)(10\times10).\\ &=& 1500n^2-5616n+5194. \end{eqnarray*} 3. \begin{eqnarray*} M_3(G) &=& \sum_{uv \in E(G)} |d_u- d_v|\\ &=& 2(4-4)+8(4-6)+4(4-8)+2(5-5)+4(5-6)+4(5-7)+8(5-8)+(4n-12)(6-6)\\ &&+(16n-48)(6-8)+4(7-8)+6(7-10)+(8n-14)(8-8)+(32n-100)(8-10)+(15n^2-96n+152)(10-10).\\ &=& -96n+206. \end{eqnarray*} 4. \begin{eqnarray*} HM(G)&=& \sum_{uv \in E(G)} (d_u+d_v)^{2}\\ &=& 2(4+4)^2+8(4+6)^2+4(4+8)^2+2(5+5)^2+4(5+6)^2+4(5+7)^2+8(5+8)^2+(4n-12)(6+6)^2+(16n-48)(6+8)^2\\ &&+4(7+8)^2+6(7+10)^2+(8n-14)(8+8)^2+(32n-100)(8+10)^2+(15n^2-96n+152)(10+10)^2.\\ &=& 6000n^2-22272n+20430. \end{eqnarray*} 5. \begin{eqnarray*} R(G)&=& \sum_{uv \in E(G)} \frac{1}{\sqrt{d_ud_v}}\\ &=& 2(\frac{1}{\sqrt{4\times4}})+8(\frac{1}{\sqrt{4\times6}})+4(\frac{1}{\sqrt{4\times8}})+2(\frac{1}{\sqrt{5\times5}})\\ && +4(\frac{1}{\sqrt{5\times6}})+4(\frac{1}{\sqrt{5\times7}})+8(\frac{1}{\sqrt{5\times8}})+(4n-12)(\frac{1}{\sqrt{6\times6}})\\ && +(16n-48)(\frac{1}{\sqrt{6\times8}})+4(\frac{1}{\sqrt{7\times8}})+6(\frac{1}{\sqrt{7\times10}})+(8n-14)(\frac{1}{\sqrt{8\times8}})\\ && +(32n-100)(\frac{1}{\sqrt{8\times10}})+(15n^2-96n+152)(\frac{1}{\sqrt{10\times10}}).\\ &=& \frac{3}{2}n^2-\frac{119}{15}n+\frac{1}{20}(32n-100)\sqrt{5}+\frac{3}{35}\sqrt{100}+\frac{1}{7}\sqrt{14}\\ &&+ \frac{1}{12}(16n-48)\sqrt{3}+\frac{2}{5}\sqrt{10}+\frac{4}{35}\sqrt{35}+\frac{2}{15}\sqrt{30}\\ &&+ \frac{1}{2}\sqrt{2}+\frac{2}{3}\sqrt{6}+\frac{247}{20}. \end{eqnarray*} 6. \begin{eqnarray*} ABC(G)&=&\sum_{uv \in E(G)} \sqrt{\frac{d_u+d_v-2}{d_ud_v}}\\ &=& 2(\sqrt{\frac{4+4-2}{4\times4}})+8(\sqrt{\frac{4+6-2}{4\times6}})+4(\sqrt{\frac{4+8-2}{4\times8}})\\ && +2(\sqrt{\frac{5+5-2}{5\times5}})+4(\sqrt{\frac{5+6-2}{5\times6}})+4(\sqrt{\frac{5+7-2}{5\times7}})\\ && +8(\sqrt{\frac{5+8-2}{5\times8}})+(4n-12)(\sqrt{\frac{6+6-2}{6\times6}})+(16n-48)(\sqrt{\frac{6+8-2}{6\times8}})\\ && +4(\sqrt{\frac{7+8-2}{7\times8}})+6(\sqrt{\frac{7+10-2}{7\times10}})+(8n-14)(\sqrt{\frac{8+18-2}{8\times8}})\\ && +(32n-100)(\sqrt{\frac{8+10-2}{8\times10}})+(15n^2-96n+152)(\sqrt{\frac{10+10-2}{10\times10}}).\\ &=& \frac{3}{10}(15n^2-96n+152)\sqrt{2}+\frac{1}{5}(32n-100)\sqrt{5}+\frac{1}{6}(4n-12)\sqrt{10}+8n\\ && +\frac{1}{8}(8n-14)\sqrt{14}+\frac{3}{7}\sqrt{42}+\frac{1}{7}\sqrt{82}+\frac{2}{5}\sqrt{110}+\frac{4}{7}\sqrt{14}\\ && +\frac{2}{5}\sqrt{30}+\frac{4}{5}\sqrt{2}+\sqrt{5}+\frac{8}{3}\sqrt{3}+\frac{1}{2}\sqrt{6}-24; \end{eqnarray*} 7. \begin{eqnarray*} \chi(G)&=& \sum_{uv \in E(G)} \frac{1}{\sqrt{d_u+d_v}}\\ &=& 2(\frac{1}{\sqrt{4+4}})+8(\frac{1}{\sqrt{4+6}})+4(\frac{1}{\sqrt{4+8}})+2(\frac{1}{\sqrt{5+5}})\\ && +4(\frac{1}{\sqrt{5+6}})+4(\frac{1}{\sqrt{5+7}})+8(\frac{1}{\sqrt{5+8}})+(4n-12)(\frac{1}{\sqrt{6+6}})\\ && +(16n-48)(\frac{1}{\sqrt{6+8}})+4(\frac{1}{\sqrt{7+8}})+6(\frac{1}{\sqrt{7+10}})+(8n-14)(\frac{1}{\sqrt{8+8}})\\ && +(32n-100)(\frac{1}{\sqrt{8+10}})+(15n^2-96n+152)(\frac{1}{\sqrt{10+10}}).\\ &=&\frac{1}{10}(15n^2-96n+152)\sqrt{5}+\frac{1}{6}(32n-100)\sqrt{2}+\frac{1}{14}(16n-48)\sqrt{14}\\ && +\frac{1}{6}(4n-12)\sqrt{3}+2n-\frac{7}{2}+\frac{6}{17}\sqrt{17}+\frac{4}{15}\sqrt{15}+\frac{8}{13}\sqrt{13}\\ && +\frac{4}{11}\sqrt{11}+\frac{4}{3}\sqrt{3}+\sqrt{10}+\frac{1}{2}\sqrt{2}. \end{eqnarray*} 8. \begin{eqnarray*} GA(G)&=& \sum_{uv \in E(G)} \frac{2\sqrt{d_ud_v}}{d_u+d_v}\\ &=& 2 \frac{2\sqrt{4\times4}}{4+4}+8 \frac{2\sqrt{4\times6}}{4+6}+4 \frac{2\sqrt{4\times8}}{4+8}\\ && +2 \frac{2\sqrt{5\times5}}{5+5}+4 \frac{2\sqrt{5\times6}}{5+6}+4 \frac{2\sqrt{5\times7}}{5+7}+8 \frac{2\sqrt{5\times8}}{5+8}\\ && +(4n-12) \frac{2\sqrt{6\times6}}{6+6}+(16n-48) \frac{2\sqrt{6\times8}}{6+8}+4 \frac{2\sqrt{7\times8}}{7+8}\\ &&+6 \frac{2\sqrt{7\times10}}{7+10}+(8n-14) \frac{2\sqrt{8\times8}}{8+8}+(32n-100) \frac{2\sqrt{8\times10}}{8+10}+(15n^2-96n+152) \frac{2\sqrt{10\times4}}{10+10}.\\ &=& 15n^2+\frac{4}{9}(32n-100)\sqrt{5}+\frac{4}{7}(16n-48)\sqrt{3}-84n+\frac{12}{17}\sqrt{70}\\ && +\frac{16}{15}\sqrt{14}+\frac{32}{13}\sqrt{10}+\frac{2}{3}\sqrt{35}+\frac{8}{11}\sqrt{30}+\frac{8}{3}\sqrt{2}+\frac{16}{5}\sqrt{6}+130. \end{eqnarray*} 9. \begin{eqnarray*} AZI(G)&=& \sum_{uv \in E(G)} (\frac{d_ud_v}{d_u+d_v-2})^3\\ &=& 2(\frac{4\times4}{4+4-2})^3+8(\frac{4\times6}{4+6-2})^3+4(\frac{4\times8}{4+8-2})^3\\ &&+2(\frac{5\times5}{5+5-2})^3+4(\frac{5\times6}{5+6-2})^3+4(\frac{5\times7}{5+7-2})^3+ 8(\frac{5\times8}{5+8-2})^3+(4n-12)(\frac{6\times6}{6+6-2})^3+(16n-48)(\frac{6\times8}{6+8-2})^3\\ && +4(\frac{7\times8}{7+8-2})^3+6(\frac{7\times10}{7+10-2})^3+(8n-14)(\frac{8\times8}{8+8-2})^3\\ && +(32n-100)(\frac{8\times10}{8+10-2})^3+(15n^2-96n+152)(\frac{10\times10}{10+10-2})^3$\\ &=& \frac{625000}{243}n^2-\frac{109249838528}{10418625}n+\frac{1427097955303080667}{133703165756160}$. \end{eqnarray*}

Table 2. The size partition of \(G\)

\((d_u,d_v)\) where \(uv\in E(G)\)	(4, 4)	(4, 6)	(4, 8)	(5,5)	(5, 6)
Number of edges	2	8	4	2	4
\((d_u,d_v)\) where \(uv\in E(G)\)	(5, 7)	(5, 8)	(6, 6)	(6, 8)	(7, 8)
Number of edges	4	8	4n - 12	16n - 48	4
\((d_u,d_v)\) where \(uv\in E(G)\)	(7, 10)	(8, 8)	(8, 10)	(10, 10)
Number of edges	6	8n - 14	32n - 100	15\(n^2\) - 96n + 152

Competing Interests

The author(s) do not have any competing interests in the manuscript.

1. Introduction

Fluid has necessarily become part and parcel of daily human life. Take a look at any material of some importance in your life and you will eventually encounter fluid of some type. Air, gases, water, and liquids of various types all fall under the definition of a fluid. Consequently, basic concepts and principles of fluid mechanics are essentially important for us. Any system in which fluid is a working medium will be subject to analysis through the help of fluid mechanics. Principles of fluid mechanics are applied on the designs of almost all means of transportation. We can safely include the subsonic and supersonic aircrafts, hovercrafts, surface ships, submarines and other automobiles. Aerodynamic designs which had previously been confined for racing cars and boats only are now a subject of importance for all automobile manufacturers. Rockets, space flights and others having propulsion systems are based on the fluid mechanics principles. It is a common practice now to undertake prior studies on the aerodynamic forces around buildings and structures. The designs of some devices like pumps, blowers, compressors and turbines require the basic knowledge of fluid mechanics. Heating/cooling and ventilating of large underground tunnels, mines, pipeline systems, large office buildings and even private rooms require knowledge of fluid mechanics. Basic principles of the fluid mechanics are even applicable in the designs of artificial hearts, heart-ling machines, breathing aids and other such devices. All we want to do is to establish the fact that knowledge and principles of fluid mechanics are used in industry, manufacturing, and even in daily human life.

Having said this, we can now safely say that governing and studying the movement of the fluid flow in rotating or sliding cylinders is of much importance for industrial point of view.

Movement of a typical viscoelastic fluid i.e. fractional Maxwell fluid in rotating cylinder has been an area of keen interest for the theorists, researchers and mathematicians working on the fluid mechanics [1, 2]. When analyzing fluid motion, researchers are interested in use of rheological constitutive equations with fractional derivatives, in addition to the classical rheological constitutive equations. Time ordinary derivative are changed to fractional order derivative to find these equations, for fractional calculus [3]. Initially Maxwell model could not provide reasonable fit of data to be experimented for complete range of frequencies and this gave birth to Fractional Maxwell model. In comparison to the classical Maxwell model, the Fractional Maxwell fluid model can demonstrate better agreement of the data to be experimented.

Recently, among the various flow of fluids the oscillating and rotatory flows of fluid has attained significant attention. For example in [4] authors considered Exact Solutions for rotating flows of a Generalized Burgers's Fluid in Cylindrical domains where as in [5] same has been done for Oldroyd-B fluid in circular cylinder. Mainly the reason for this is that oscillating flows are more common in practical processes like towing operation, oil drilling, mixing and bioengineering. The reason for this is the flow of blood in veins due to periodic pressure gradient. In 2005 Yin and Zhu has exhibited research paper on oscillating issues of Fractional Maxwell fluid [6].

In 2012, Karam Rahaman discussed oscillating flow of upper convected Maxwell fluid and this was done by a cylinder [7]. The precise solutions for sinusoidal motion of visco- elastic non-Newtonian fluids were discovered by Mahmood et. al [8, 9, 10, 11, 12]. Various number of researchers are doing research since last decade to find the exact solutions of non Newtonian fluids, particulary the ones having boundary conditions of shear stress. The first ones to develop and come up with the solutions were Water et. al [13]. The exact solution for the Fractional Maxwell fluid which have shear stress boundary condition and flow rotationally in circular cylinder was given by Siddique [14]. Moreover, Rajagopal came up with two very firm yet simple solutions in regard to the motion of second grade fluid which had the torsional and longitudinal oscillation of unbounded rod. Bandelli et. al discus unsteady motions of second grade fluid [15]. The Erdagon's work was extended by Fetecau which had association to non-Newtonian fluid to cosine and sine oscillations of the flat plate [15, 16]. The exact solution for the Maxwell fluid which had oscillatory flow in cylinder was given by Vieru. et. al so considering all of this Laplace and Hankel transforms are practiced in this thesis to examine torsional sine oscillations of Fractional Maxwell Fluid which endured shear stress in an infinite circular cylinder [18].

According to the best knowledge of the us no attempt has so far been made to find the solutions for the movement of fractional Maxwell fluid present within two coaxial cylinders of infinite lengths and oscillating within. In this article we find the exact solutions for the velocity field \(v(r,t)\) and shear stress \(\tau(r,t)\) corresponding to the motion of a Maxwell fluid between two cylinders, which are infinite and coaxial with inner cylinder is at rest and outer one is oscillating. Laplace transform and finite Hankel transform are used to obtain their solutions. The solutions are presented under series form in term of generalized G-functions. These will satisfy governing equations and all imposed boundary and initial conditions. Furthermore the corresponding solutions for the Maxwell flow of Newtonian fluid are also obtained in term of limiting cases.

2. Mathematical Formulation and Governing Equation of Problem

The constitutive equation for the Maxwell Fluid is defined by \(S\) denotes the extra stress tensor and \(p\) is the pressure, \(I\) denotes unit tensor, \(\lambda\) is called relaxation time, \(\mu\) is called dynamic viscosity and \(T\) stands for Cauchy stress tensor stands for hydrostatic pressure. Assume that the Maxwell fluid is in the annulus of coaxial circular cylinders whose radii \(R_{1}\) and \(R_{2}\) respectively, where \(R_{1}\) is less than \(R_{2}\) and lengths are infinite. when \(t=0\), the fluid and two cylinders are fixed. when \(t>0\), the outer cylinder starts oscillating around their axis with velocity \(W\sin(\omega~t)\) and \(\omega\) is the angular velocity of the outer cylinder. Then the velocity field \(\textbf{v}\) and the extra stress \(\textbf{S}\) is of the form where \(e_{z}\) is called the unit vector in the \(z- direction\).
In start, when the fluid is fixed, we have The Maxwell fluid have the governing equations, The non-trivial shear stress is denoted by \(\tau(r,t)\) , \(\lambda\) is the material constant, \(\mu\) is called the dynamic viscosity and kinematic viscosity is denoted by \(\nu = \frac{\mu}{\rho}\), where \(\rho\) is constant density of the Maxwell fluid.
Furthermore, the fractional differential operator \(D^{\beta}_{t}\) of the Maxwell fluid is defined by \cite{10} where \(\Gamma(.)\) is the Gamma function which can be defined as

3. Calculation of the Velocity Field

Now we apply Laplace transformation to equation, we have Here \(q\) denotes the parameter of transformation. We can define, Hankel Transform of \(\bar{v}(r,q)\) as $$\bar{V}_H(r_n,q)=\int^{R_2}_{R_1} r\bar{v}(r,q)B_0(rr_n)dr,$$ where Here, \(r_n\) are positive roots of \(B_0(R_1r)=0\), and \(J_0(.)\) and \(Y_0(.)\) represent the Bessel functions whose order is zero of first and second type, respectively. Now consider R.H.S of the equation (12), we have, Again, from equation (12), we have the following result Then, Now, simplification for \(\bar{V}_H(r_n,q)\) we can write the above equation as, or we can say as, $$\bar{V}_H(r_n,q)=\bar{V}_{1H}(r_n,q)-\bar{V}_{2H}(r_n,q),$$ where $$\bar{V}_{1H}(r_n,q)=\frac{2}{\pi{r_n^2}}\frac{V\Omega}{(q^2+\Omega^2)},$$ and $$\bar{V}_{2H}(r_n,q)=\frac{2V\Omega}{\pi{r_n^2}(q^2+\Omega^2)}\frac{(q+\lambda{q^{\beta+1}})}{(q+\lambda{q^{\beta+1}}+\nu{r_n^{2}})}.$$ Now, going to define Inverse Hankel Transform, Inverse Hankel Transform or \(\bar{V}_{1H}(r_n,q)\) and \(\bar{V}_{2H}(r_n,q)\) are $$\bar{v}_1(r,q)=\frac{\ln(r/R_1)}{\ln(R_2/R_1)}\frac{V\Omega}{q^2+\Omega^2}.$$ $$\bar{v}_2(r,q)=\frac{\pi^2}{2}\sum^{\infty}_{n=1}\frac{r_n^2J^{2}_{0}(R_1r_n)B_0(rr_n)}{J^{2}_{0}(R_1r_n)-J^{2}_{0}(R_2r_n)}\bar{V}_{2H}(r_n,q).$$ Now, or,equivalently Now by using the identity, as, Taking inverse Laplace transformation and using the formula and also by using the convolution theorem we get,

4. Calculation of the shear Stress

Now taking Laplace transformation of (7)
\(\Big[( 1 + \lambda D^{\beta}_{t})\tau(r,t) \Big] = £\Big[\mu\frac{\partial{v(r,t)}}{\partial r}\Big] \bar{\tau}(r,q) + \lambda q^{\beta} \bar{\tau}(r,q) = \mu\frac{\partial\bar{v}(r,q)}{\partial r} \bar{\tau}(r,q)\Big[1 + \lambda q^{\beta}\Big] = \mu \frac{\partial\bar{v}(r,q)}{\partial r}, \) we have

5. Limiting Cases

5.1. Ordinary Maxwell Fluid

By putting \(\alpha\rightarrow1\),\(\beta\rightarrow1\) in the above results,
Velocity Field: Shear Stress:

5.2. Newtonian Fluid

By placing \(\lambda\rightarrow0\), \(\lambda_{r}\rightarrow0\) in the last consequences of ordinary Maxwell fluid, we can get the results for newtonian fluid. Above result is similar to the result which already established in [1].

6. Conclusion

The purpose of this work is to find exact solutions for velocity field and adequate shear stress corresponding to the flow of Maxwell fluid between two longitudinally oscillating circular cylinders, whose lengths are infinite, with fractional derivatives. The motion of the fluid is produced by outer cylinder where inner cylinder is fixed, at time \(t=0^+\). It is worthwile to note that results obtained in [1] are special cases of our results. Finally we recover corresponding solutions for ordinary Maxwell and Newtonian fluids are as limiting cases.

Appendix

Followings are some expressions used in the text:
(A1).~~~The finite Hankel transform of the funcion \begin{equation*} a(r)=\frac{C_1\ln(R_2/r)+ C_2\ln(r/R_1)}{\ln(R_2/R_1)} ~~~~~. \end{equation*} satisfying \(a(R_1)=C_1\) and \(a(R_2)=C_2\) is \begin{equation*} a_n(r)=\int_{R_1}^{R_2}r a(r) B_0(rr_n) dr=\frac{2C_2}{\pi r_n^2}-\frac{2C_1}{\pi r_n^2}\frac{J_0(R_2 r_n)}{J_0(R_1 r_n)}. \end{equation*}

Competing Interests

The author do not have any competing interests in the manuscript.

1. Introduction

Let \(G=(V,E)\) be a simple connected graph of finite order \(n=|V|=|V(G)|\) with the set of vertices and the set of edges \(E=E(G)\). We denote by \(d_v\), the degree of a vertex \(v\) of \(G\) which is defined as the number of edges incident to \(v\). A general reference for the notation in graph theory is [1]. A molecular graph is a simple finite graph such that its vertices correspond to the atoms and the edges to the bonds.

In chemistry, graph invariants are known as topological indices. In graph theory, we have many different topological indices of arbitrary graph \(G\). A topological index of a graph is a number related to a graph which is invariant under graph automorphisms. Obviously, every topological index defines a counting polynomial and vice versa.

The Wiener index \(W(G)\) is the oldest based structure descriptor [2, 3, 4], introduced by Harold Wiener in 1947, is the first topological index in chemistry. The Wiener index of \(G\) is defined as the sum of distances between all pairs of vertices of \(G\) and is equal as follow: \begin{eqnarray*}\label{F1} W(G) &=& \frac{1}{2} \sum \limits_{(u,v)} d(u,v). \end{eqnarray*} where \((u,v)\) is any ordered pair of vertices in \(G\) and \(d(u,v)\) is \(u-v\) geodesic.

The Zagreb index was defined about forty years ago by I. Gutman and N. Trinajstic in 1972 [5] (or, more precisely, the First Zagreb index, because there exists also a Second Zagreb index [4]). The First Zagreb index of \(G\) is defined as the sum of the squares of the degrees of all vertices of \(G\) [6,7]. The first and second Zagreb indices of \(G\) are denoted by \(M_1(G)\) and \(M_2(G)\), respectively and defined as follows: \begin{eqnarray*} M_1(G) &=& \sum_{u \in V(G)} d_u^2 \hspace{10mm} \mbox{and} \hspace{10mm}\\ M_2(G) &=& \sum_{uv \in E(G)} d_{u}d_{v}. \end{eqnarray*} In fact, one can rewrite the first Zagreb index as $$M_1(G)=\sum_{uv \in E(G)} \big[ d_u+d_v \big].$$ where \(d_u\) and \(d_v\) are the degrees of \(u\) and \(v\), respectively. In 2011, M. Azari and A. Iranmanesh [8] introduced the generalized Zagreb index of a connected graph \(G\), based on degree of vertices of \(G\). The Generalized Zagreb index of \(G\) is defined for arbitrary non-negative integer \(r\) and \(s\) as follows: $$M_{(r,s)}(G)=\sum_{uv \in E(G)} \big[ d_u^rd_v^s+d_u^sd_v^r \big].$$

In [8], A. Iranmanesh and M. Azari expressed some of the obvious properties of the Generalized Zagreb index of a graph \(G\).

Theorem 1.1. [8] Let \(G\) be a graph with the vertex and edge sets \(V(G)\) and \(E(G)\). Some of the properties of the generalized Zagreb index of \(G\) are as

1. \(M_{(0,0)}(G)=2\sum_{v \in V(G)}=2|E(G)|\).
2. \(M_{(1,0)}(G)=M_1(G)\).
3. \(M_{(r-1,0)}(G)=\sum_{v \in V(G)}dv^r\).
4. \(M_{(1,1)}(G)=2M_2(G)\).
5. \(M_{(r,r)}(G)=2 \sum_{uv \in E(G)}(d_u\times d_v)^r\).

2. What is the Capra Operation?

A mapping is a new drawing of an arbitrary planar graph \(G\) on the plane. In graph theory, there are many different mappings (or drawing); one of them is Capra operation. This method enables one to build a new structure of a planar graph \(G\).

Let \(G\) be a cyclic planar graph. Capra map operation is achieved as follows:

1. insert two vertices on every edge of \(G\);
2. add pendant vertices to the above inserted ones and
3. connect the pendant vertices in order \((-1,+3)\) around the boundary of a face of \(G\). By runing these steps for every face/cycle of \(G\), one obtains the Capra-transform of \(G\) \(Ca(G)\), see Figure 1.

By iterating the Capra-operation on the hexagon (i.e. benzene graph \(C_6\)) and its \(Ca-\)transforms, a benzenoid series (Figures 2 and 3) can be designed. We will use the Capra-designed benzene series to calculate some connectivity indices (see below).

This method was introduced by M.V. Diudea and used in many papers [9, 10, 11, 12, 13, 14, 15, 16]. Since Capra of planar benzenoid series has a very remarkable structure, we lionize it.

We denote Capra operation by \(Ca\), in this paper, as originally Diudea did. Thus, Capra operation of arbitrary graph \(G\) is \(Ca(G)\), iteration of Capra will be denoted by \(CaCa(G)\) (or we denote \(Ca_2(G))\) (Figures 2 and 3).

The benzene molecule is a usual molecule in chemistry, physics and nano sciences. This molecule is very useful to synthesize aromatic compounds. We use the Capra operation to generate new structures of molecular graph benzene series.

In this paper, we focus on the the structure of Capra-designed planar benzenoid series \(Ca_k(C_6)\), \(k\geq0\) and compute its generalized Zagreb index.

The aim of this paper is to compute the generalized Zagreb index \(M_{(r,s)}\)(G) of Capra-designed planar benzenoid series \(Ca_k(C_6)\).

3. Main Results

Capra transforms of a planar benzenoid series is a family of molecular graphs which are generalizations of benzene molecule \(C_6\). In other words, we consider the base member of this family is the planar benzene, denoted here \(Ca_0(C_6)=C_6=\) benzene. It is easy to see that \(Ca_k(C_6)=Ca(Ca_{k-1}(C_6))\) (Figures 2 and 3) ([9, 10, 11, 12, 13, 14, 15, 16]). In addition, we need the following notions [17]. Let \(G\) be a molecular graph and \(d_v\) is the degree of vertex \(v\in V(G)\). We divide vertex set \(V(G)\) and edge set \(E(G)\) of graph \(G\) to several partitions, as follow:

\(\forall i\), \(\delta < i< \Delta\), \(V_i=\{v \in V(G) |d_v=i\}\),
and \(\forall k, \delta^2\leq k\leq \Delta^2, E^*_k=\{e=uv\in E(G)|d_v\times d_u=k\}\).
Obviously, \(1\leq\delta\leq d_v\leq\Delta\leq n-1\) such that \(\delta=Min\{d_v|v\in V(G)\}\) and \(\Delta=Max\{d_v|v\in V(G)\}\).

Theorem 3.1. Consider the graph \(G=Ca_k(C_6)\) as the iterative Capra of planar benzenoid series. Then: $$ M_{(r,s)}=3(7^k-2(3^{k-1})-1)\times \big[ 2(3^{r+s}) \big]+4(3^k) \times\big[ 2^r3^s+2^s3^r \big]+3(3^{k-1}+1)\times \big[2(2^{r+s}) \big]. $$

Proof. Consider the Capra of planar benzenoid series \(G=Ca_k(C_6)\) \((k\geq1)\). By construction, the structure \(Ca_k(C_6)\) collects seven times of structure \(Ca_k-1(C_6)\) (we call "flower" the substructure \(Ca_{k-1}(C_6)\) in the graph \(Ca_k(C_6))\). Therefore, by simple induction on \(k\), the vertex set of \(Ca_k(C_6)\) will have \(7\times|V(Ca_k(C_6))|-6(2\times3^{k-1}+1)\) members. Because, there are \(3^{k-1}+1\) and \(3^{k-1}\) common vertices between seven flowers \(Ca_{k-1}(C_6)\) in \(Ca_k(C_6)\), marked by full black color in the above figures. Similarly, the edge set \(E(Ca_k(C6))\) have \(7\times|E(Ca_k(C_6))|-6(2×3^{k-1}+1)\) members. Since, there are \(3^{k-1}\) and \(3^{k-1}\) common edges (full black color in these figures).
Now, we solve the recursive sequences \(|V(Ca_k(C_6))|\) and \(|E(Ca_k(C_6))|\). First, suppose \(n_k=|V(Ca_k(C_6))|\) and \(e_k=|E(Ca_k(C_6))|\)
so \(n_k=7n_{k-1}-4(3^k)-6\) and \(e_k=7e_{k-1}-4(3^k)\). Thus, we have
\(n_k=7n_{k-1}-4ò_k-6\)
\(=7(7n_{k-2}-4ò_{k-1}-6)-4ò_k-6\)
\(=7^2n_{k-2}-7(4ò_{k-1}+6)-(4ò_k+6)\)
\(=7^3n_{k-3}-7^2(4ò_{k-2}+6)-7(4ò_{k-1}+6)-(4ò_k+6)\)
.
.
.
\(=7^in_{k-i}-7^{i-1}(4ò_{k-(i-1)}+6)-...-7(4ò_{k-1}+6)-(4ò_k+6)\)
\(=7^in_{k-i}-\sum^{i-1}_{j=0}7^j(4ò_{k-j}+6)\)
.
.
.
\(=7^kn_{k-k}-\sum^{k-1}_{i=0}7^i(4ò_{k-i}+6)\)

where \(n_0=6\) is the number of vertices in benzene \(C_6\) (Figure 2) and \(6\sum^{k-1}_{i=0}7^i\) is equal to \(\frac{6(7^k-1)}{7-1}=7^k-1\). On the other hand, since
\((\alpha-\beta)\sum^n_{i=0}\alpha^i\beta^{n-1}=(\alpha-\beta)(\alpha^0\beta^n+\alpha^1\beta^{n-1}+...+\alpha^{n-1}\beta^1+\alpha^n\beta^0)=(\alpha^{n+1}-\beta^{n+1})\) Hence
\(\sum^{k-1}_{i=0}7^i3^{k-i}=(7^03^k+7^13^{k-1}+...+7^{k-2}3^2+7^{k-1}3^1)+7^k3^0-7^k3^0\)
\(=\frac{7^{k+1}-3^{k+1}}{7^1-3^1}-7^k3^0\)
\(=\frac{7^{k+1}-3^{k+1}-4(7^k)}{4}\)
\(=\frac{3(7^k-3(3^k))}{4}\) Therefore, by using equations (1) and (2), we have
\(n_k=6\times7^k-(4(\frac{3}{4}(7^k-3^k))+(7^k-1))\) and \(\forall k\geq0, n_k=|V(Ca_k(C_6))|=2\times7^k+3^{k+1}+1. \)
By using a similar argument and (1), we can see that
\(e_k=7e_{k-1}-4ò=7^2e_{k-2}-7(4ò_{k-1})-4ò.\)
.
.
.
\(=7^ke_{k-k=0}-4\sum^{k-1}_{i=0}7^iò_{k-i}=7^ke_0-4\sum^{k-1}_{i=0}7^i3^{k-i}\).
It is easy to see that, the first member of recursive sequence \(e_k\) is \(e_0=6\), (Figure 2). Now, by using (2), we have \(e_k=6\times7^k-4(\frac{3}{4}(7^k-3^k))\) and the size of edge set \(E(Ca_k(C_6))\) is equal to: \(e_k=|E(Ca_k(C_6))|=3(^7k+3^k)\), \(\forall k\geq0\).
Also, according to Figures 2 and 3, we see that the number of vertices of degree two in the graph \(Ca_k(C_6)\) (we denote by \(v^{(k)}_2\) ) is equal to \(6\times3(\frac{(v_2^{(k-1))}}{6})-6\). The six removed vertices are the common ones between the six flowers \("Ca_{k-1}(C_6)"\) with degree three. By using a similar argument and simple induction, we have \(v_2^{k-1}\) the numbers of edges of graph \(Ca_k(C_6)\), which are in the set \(E_4\) or \(E_4^*\) (denoted by \(e_4^{(k)}\)).
Now, we solve the recursive sequence \(v_2^{(k)}=6(3(\frac{v_2^{(k-1)}}{6})-1)\) and we conclude \(v_2^{k}=3v_2^{(k-1)}-6=3(3v_2^{(k-2)}-6)-6=...=3^kv_2^{(0)}-6\sum^{k-1}_{i=0}3^i\).
It is obvious that, according to the structure of benzene, \(v_2^{(0)}=n_0=6\). Thus, \(v_2^{(k)}=6\times3^k-6(\frac{3^k-1}{3-1})=3^k+1+3.\)
Also, \(e_4^{(k)}=|E_4|=|E_4^*|=v_2^{(k-1)}=3^{k+1}+3\) and according to the above definition, it is obvious that, for Capra of planar benzenoid series \(G=Ca_k(C_6)\) we have two partitions: \(V_2=\{v\in V(Ca_k(C_6))|d_v=2\}\) and \(V_3=\{v\in V(Ca_k(C_6))|d_v=3\}\) with the size \(3^{k+1}+3\) and \(2(7^k-1)\) respectively.
On the other hand, according to the structure of Capra planar benzenoid series \(Ca_k(C_6)\), there are \(2v_2^{(k)}\) edges, such that the first point of them is a vertex with degree two. Among these edges, there exist \(v_2^{k-1}\) edges, of which the first and end point of them have degree 2 (the members of \(E_4\) or \(E_4^*\) ).
Thus, \(e_5^{(k)}=|E_5|=|E_6^*|=2v_2^{(k)}-2e_4^{(k)}=2v_2^{(k)}-2v_2^{(k-1)}\). So, the size of edge set \(E_5\) and \(E_6^*\) is equal to \(e_5^{k}=2(3^{k+1}+3-3^k-3)=4(3^k)\).
Now, it is obvious that:
\(e_6^{(k)}=|E_6|=|E_9^*|=3(7^k+3^k)-e_5^{(k)}-e_4^{(k)}\)
\(=3\times7^k+3^{k+1}-4\times3^k-3^k-3\)
\(=3\times7^k-2\times3^k-3\)
\(={3(7^k-2(3^{k-1})-1)}.\)
Now, we know the size of all sets \(V_2, V_3, E_4, E_4^*, E_5, E_6^*, E_6\) and \(E_9^*\). So, we can calculate the Generalized Zagreb Index Of Capra planar benzenoid series \(G=Ca_k(C_6)\), as follow:
Thus, we have following computations for the generalized Zagreb index of Capra planar benzenoid series \(G=Ca_k(C_6)\) $$M_{(r,s)}(G)=\sum_{uv \in E(G)} \big[ d_u^rd_v^s+d_u^sd_v^r \big] $$ $$=\sum_{uv \in E(G)} \big[ 3^r3^s+3^s3^r \big]+\sum_{uv \in E(G)} \big[ 2^r3^s+2^s3^r \big]+\sum_{uv \in E(G)} \big[ 2^r2^s+2^s2^r \big] $$ $$=\sum_{uv \in E(G)} \big[ 2(3^{r+s}) \big]+\sum_{uv \in E(G)} \big[ 2^r3^s+2^s3^r \big]+\sum_{uv \in E(G)} \big[2(2^{r+s})\big] $$ $$=3(7^k-2(3^{k-1})-1)\times \big[ 2(3^{r+s}) \big]+4(3^k) \times\big[ 2^r3^s+2^s3^r \big]+3(3^{k-1}+1)\times \big[2(2^{r+s}) \big].$$ Thus, we completed the proof of the theorem.

Competing Interests

The author(s) do not have any competing interests in the manuscript.

1. Introduction

The problem of accelerated expansion is a critical topic in cosmology since its discovery [1]. The main cause of accelerated expansion of the universe is a unknown force so-called dark energy (DE). To explain the nature of DE many cosmologists have proposed many models and theories. Many DE theories for dynamical DE scenario have been proposed to interpret the nature of accelerating universe. A number of DE models have been discussed in this context by many cosmologists. Cosmological constant \(\Lambda\) (\(\Lambda CDM\)) [2] is the simplest candidate for DE (has a constant energy and pressure with constant equation of state). But this model has faced two major problems, cosmic coincidence and fine tuning [2].

In order to describe accelerated expansion phenomenon, two different approaches has been adopted. One is the proposal of various dynamical DE models such as family of chaplygin gas [3], holographic [4, 5], new agegraphic [6], polytropic gas [7], pilgrim [8, 9, 10] \(DE\) models etc. A second approach for understanding this strange component of the universe is modifying the standard theories of gravity, namely, General Relativity (GR) or Teleparallel Theory Equivalent to GR (TEGR). Several modified theories of gravity are \(f(R)\), \(f(T)\) [11, 12, 13, 14, 15, 16], \(f(R,\mathcal{T})\) [17, 18], \(f(G)\) [19, 20, 21, 22, 23] (where \(R\) is the curvature scalar, \(T\) denotes the torsion scalar, \(\mathcal{T}\) is the trace of the energy momentum tensor and \(G\) is the invariant of Gauss-Bonnet defined as \(G=R^2-4R_{\mu\nu}R^{\mu\nu}+ R_{\mu\nu \lambda\sigma}R^{\mu\nu\lambda\sigma}\)). For clear review of DE models and modified theories of gravity, see the reference [24, ].

We arrange the paper as follow: Section 2 describes the basic equations of fractal cosmology. In section 3, we discuss the cosmological parameters (Hubble, Deceleration, EoS) and planes (\(\omega_{D}-\omega'_{D}\), state-finder). In the last section, we conclude our results.

2. Fractal Cosmology

According to Einstein gravity in a fractal space-time, the total action is [25, 26, ] where \(S_{G}\) is the gravitational part of the action and can be defined as and \(S_{m}\) is the matter part of the action is Where \(g\) is the determinant of the metric (dimensionless) \(g_{\mu\nu}\), \(\Lambda\) is the cosmological constant and \(R\) is the Ricci scalar, \(\nu\) and \(\omega\) are the fractional function and fractal parameter respectively, while the standard measure \(d^{4}x\) is replaced with a Lebesgue-Stieltjes measure \(d\varrho(x)\). The Friedmann equation in fractal universe can be obtained after variation of Eq.(1) with respect to the \(g_{\mu\nu}\) as Here \(H\) denotes the Hubble parameter (\(H=\frac{\dot{a}}{a}\)), \(\rho_{cdm}\) and \(\rho_{de}\) are the energy densities due to CDM and DE and \(p=p_{de}\) is the pressure of DE. \(k\) is the curvature constant with different values of \(k=0,+1,-1\) described as a flat closed and open universe respectively. \(\Lambda\) is the cosmological constant. The continuity equations in fractal universe are given by Where \(Q\) describes as the interaction term between DE and CDM with \(Q=3b^{2}H\rho_{m}\) and \(b^{2}\) is a coupling constant. By assuming a timelike fractal profile \(\nu=a^{-\gamma}\) (where \(\gamma\) is the constant), the Friedmann equation becomes and the continuity equations can be written as Where \(\rho_{m0}\) is the integrating constant.

3. QCD Ghost Dark Energy

Recent observations have been proved that Veneziane ghost of chromodynamics QCD is a good model and helps to solve the U(1) problem [27]. Veneziane ghost DE model contribute to the vacuum energy and proportional to \(\Lambda^{3}_{QCD}H\) (smallest \(QCD\) scale), where \(H\) is the Hubble parameter and \(\Lambda_{QCD}\) is the QCD mass scale. GDE is defined as [28, 29, 30, 31, 32] \(\rho_{de}=\alpha H\). This model is discussed for many cosmological parameter theories and observational schemes. Later on, it has been discussed in the form \(H+O(H^{2})\) [33] of Veneziane ghost in QCD has enough vacuum energy by which the early evolution of the universe is explained (with the help of \(H^{2}\)) [34]. This model is called generalized ghost DE model (GGDE). Garcia-Salcedo has proposed a new version of GGDE called QCD ghost DE model which depends on the radius of trapping horizon. For flat universe, it is defined as where \(\alpha\) is numerical constant, \(\epsilon=\frac{\dot{\tilde{r}}_T}{2H\tilde{r}_{T}}\) and \(\tilde{r}_{T}=\frac{1}{H}\). Using these values in Eq.(10), we get

3.1. Hubble Parameter

By using Eqs.(7) and (11), we get the Hubble parameter in the form

Figure 1 that Hubble parameter corresponds to future day observation of the universe.

3.2. Deceleration Parameter

The deceleration parameter is denoted by \(q\). This parameter tells us the transaction phase of the universe, either accelerating (\(-1 \leq q< 0\)) or decelerating (\(q\geq0\)). Its mathematical form is After solving the Eqs. (13) and (12), the deceleration parameter is

It is cleared from figure 2 that the deceleration parameter corresponds to acceleration expansion of the universe.

3.3. Equation of State Parameter

This parameter can be obtained by using Eqs.(9) and (12) as follows

The plot of EoS versus redshift parameter is shown in Figure 3. The EoS parameter behaves quintom-like nature for the interacting case \(d^2=0.2\). For \(d^2=0.3\), EoS parameter starts from phantom region and goes towards quintessence region of the universe. However, it remains in the phantom region for \(d^2=0.4\).

3.4. \(\omega_{D}-\omega'_{D}\) plane

The \(\omega_{D}-\omega'_{D}\) plane characterize thawing as well as the freezing region of universe, i.e., when \(\omega_{D}< 0\) and \({\omega}'_{D}> 0\) then the plane corresponding to thawing region. But if both \(\omega_{D}\) and \({\omega}'_{D}\) are negative, then this plane provides freezing region. Caldwell and Linder [35] discover this method. By taking derivative of Eq.(15), we obtain

The behavior of \(\omega_{D}-\omega'_{D}\) can be observed from Figure 4 which exhibits the freezing region.

3.5. \(r-s\) plane

With the help of this plane, we can identify different DE models. Trajectories of different DE models have different ranges in this plane. For example, \(\{r,s\}=\{1,0\}\) corresponds to \(\Lambda CDM\) model, \(\{r,s\}=\{1,1\}\) shows \(CDM\) limit, \(\{s>0,r< 1\}\) shows phantom and quintessence while \(\{s< 0,r >1\}\) denotes chaplygin gas region. Mathematical form of state-finder parameters are given by [36] To obtain the values of \(r-s\) plane we substitute Eq. (12) and (14) in (17),

The plane of this model is given in Figure 5. The \(r-s\) plane for this model shows the Chaplygin gas behavior as well as \(\Lambda CDM\) model.

4. Concluding Remarks

We have investigated the physical significance of QCD ghost DE model in fractal universe by developing various cosmological parameters as well as cosmological planes. These parameters as well as planes shamefully explain the current cosmic acceleration.

Competing Interests

The author(s) do not have any competing interests in the manuscript.

1. Introduction

To find the root of nonlinear equation of the form \(f(x)=0,\) is the oldest and basic problem in numerical analysis. Newton's method is one of oldest and most powerful formula to approximate the root of nonlinear equations. It has second order convergence. Many modifications in Newton's method have been made to increase its convergence order using various techniques. Recently, many iterative methods with higher order convergence have been established using different techniques like Taylor's series, Adomain decomposition, homotopy, modified homotopy, decomposition, modified decomposition, interpolation, quadrature rules and their many modifications. see [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12] and references therein. Chun [3] introduced some multi-step iterative methods using Adomain decomposition. These method requires higher order derivatives. Later on, Noor [4, 5] established some multi-step iterative methods that do not require higher order derivative using a different technique.

In this paper, some numerical methods based on decomposition technique are proposed for solving algebraic nonlinear equations. For this purpose, we write nonlinear equations as an equivalent system of equations and use technique introduced by Gejji and Jafari [9]. Recently, several iterative methods have been suggested by writing nonlinear equations as an equivalent system of equations and then using different techniques. In section 3, we give detailed proof regarding convergence of our newly established iterative methods. In the last section, numerical results are given to make comparison of these algorithm with some classical methods.

2. Iterative methods

Consider the nonlinear equation assume \(\alpha\) is a simple root of (1) and \(\gamma\) is an initial guess sufficiently close to \(\alpha\). We can write (1) as following coupled system; From Eq. (2), we have \[ x=\gamma -\frac{f(\gamma )}{f^{\prime }(\frac{x+\gamma }{2})}-\frac{g(x)}{% f^{\prime }(\frac{x+\gamma }{2})}. \] Let where and Now, we construct a sequence of higher-order iterative methods by applying modified decomposition technique introduced by Gejji and Jafari [8]. This technique consists a solution of equation (3) that can be written in the form of infinite series: \noindent and using Gejji and Jafari [8] technique, we decompose the nonlinear function \(N\) as; Thus from Eqs. (4), (7) and (8) we have \[ \sum\limits_{i=0}^{\infty }x_{i}=x_{0}+N(x_{0})+\sum_{i=1}^{\infty }\{N(\sum\limits_{j=0}^{i}x_{j})-N(\sum\limits_{j=0}^{i-1}x_{j})\}. \] So we have iteration scheme as \begin{eqnarray*} x_{0} &=&c \\ x_{1} &=&N(x_{0}) \\ x_{2} &=&N(x_{0}+x_{1})-N(x_{0}) \\ &&. \\ &&. \\ &&. \\ x_{n+1} &=&N(x_{0}+x_{1}+...+x_{n})-N(x_{0}+x_{1}+...+x_{n-1}),\text{ }% n=1,2,.. \end{eqnarray*} When \begin{eqnarray*} x &\approx &x_{0} \\ &=&c \\ &=&\gamma -\frac{f(\gamma )}{f^{\prime }(\frac{x+\gamma }{2})}. \end{eqnarray*} From above relation, we can write the algorithm as follows;
Algorithm 2.1: For any initial value \(x_{0},\) we compute the approximation solution \(x_{n+1}\), by the iteration scheme; \begin{eqnarray*} y_{n} &=&x_{n}-\frac{f(x_{n})}{f^{\prime }(x_{n})} \\ x_{n+1} &=&x_{n}-\frac{f(x_{n})}{f^{\prime }(\frac{x_{n}+y_{n}}{2})}. \end{eqnarray*} This algorithm has third order convergence and has been established by Frontini and Sormani [1]. When \begin{eqnarray*} x &\approx &x_{0}+x_{1} \\ &=&\gamma -\frac{f(\gamma )}{f^{\prime }(\frac{x+\gamma }{2})}-\frac{g(x_{0}) }{f^{\prime }(\frac{x+\gamma }{2})}. \end{eqnarray*} From Eq. (3), we see that \[ g(x_{0})=f(x_{0}), \] by substituting in above, we get \[ x=\gamma -\frac{f(\gamma )}{f^{\prime }(\frac{x+\gamma }{2})}-\frac{f(x_{0}) }{f^{\prime }(\frac{x_{0}+\gamma }{2})}, \] Using this, we suggest the following three step iterative methods for solving Eq. (1) as follows;
Algorithm 2.2: For any initial value \(x_{0},\) we compute the approximation solution \(x_{n+1}\), by the iteration scheme; \begin{eqnarray*} \text{ Predictor Steps;}\;\;\;\; y_{n} &=&x_{n}-\frac{f(x_{n})}{% f^{\prime }(x_{n})} \\ z_{n} &=&x_{n}-\frac{f(x_{n})}{f^{\prime }(\frac{x_{n}+y_{n}}{2})} \end{eqnarray*} \[ \text{Corrector Step; }\;\;\;\; x_{n+1}=z_{n}-\frac{f(z_{n})}{% f^{\prime }(\frac{x_{n}+z_{n}}{2})} \] When \begin{eqnarray*} x &\approx &x_{0}+x_{1}+x_{2} \\ &=&x_{0}+N(x_{0}+x_{1}) \\ &=&x_{0}-\frac{g(x_{0}+x_{1})}{f^{\prime }(\frac{x_{0}+x_{1}+\gamma }{2})} \\ &=&x_{0}-\frac{f(x_{0}+x_{1})+f(x_{0})}{f^{\prime }(\frac{x_{0}+x_{1}+\gamma }{2})} \end{eqnarray*} From this relation, we formulate four step iterative method for solving Eq. (1) as follows;
Algorithm 2.3: For any initial value \(x_{0},\) we compute the approximation solution \(x_{n+1}\), by the iteration scheme; \begin{eqnarray*} \text{ Predictor Steps;} \;\;\;\; y_{n} &=&x_{n}-\frac{f(x_{n})}{% f^{\prime }(x_{n})} \\ z_{n} &=&x_{n}-\frac{f(x_{n})}{f^{\prime }(\frac{x_{n}+y_{n}}{2})} \\ u_{n} &=&z_{n}-\frac{f(z_{n})}{f^{\prime }(\frac{x_{n}+z_{n}}{2})} \end{eqnarray*} \[ \text{Corrector Step;} \;\;\;\; x_{n+1}=z_{n}-\frac{f(u_{n})+f(z_{n})% }{f^{\prime }(\frac{x_{n}+u_{n}}{2})} \]

3. Convergence Analysis

In this section, we discuss in detail the convergence analysis of Algorithm 2.2 and 2.3 established above.

Theorem 3.1. For an open interval \(I \subset \mathbb{R}\) let \(f:I \rightarrow\mathbb{R}\) and \(\alpha \in I\) be simple zero of \(f(x)=0.\) If \(f\) is differentiable and \(x_{0}\) is sufficiently close to \(\alpha\) then three-step iterative method defined by Algorithm 2.2 has fourth order convergence.

Proof. Expanding \(f(x)\) by Taylor's series about \(\alpha ,\) we have

where \(c_{k}=\frac{1}{k!}\frac{f^{(k)}(\alpha )}{f^{^{\prime }}(\alpha )}\), and \(e_{n}=x_{n}-\alpha .\) \[ y_{n}=x_{n}-\frac{f(x_{n})}{f^{\prime }(x_{n})} \] From above equations,we get Now, expanding \(f^{\prime }(\frac{x_{n}+y_{n}}{2})\) by Taylor's series about \(\alpha\), Dividing Eq. (9) by Eq. (12) Now \[ z_{n}=x_{n}-\frac{f(x_{n})}{f^{\prime }(\frac{x_{n}+y_{n}}{2})} \] Putting values in above, we get Expanding \(f(z_{n})\) and \(f^{\prime }(\frac{x_{n}+z_{n}}{2})\) by Taylor's series about \(\alpha,\) Dividing (15) by Eq. (16), we have \begin{align*} \frac{f(z_{n})}{f^{\prime }(\frac{x_{n}+z_{n}}{2})} &=(c_{2}^{2}-\frac{1}{4}% c_{3})e_{n}^{3}+(4c_{2}c_{3}-4c_{2}^{3}-\frac{1}{2}c_{4})e_{n}^{4}+(\frac{27% }{8}c_{3}^{2}-\frac{73}{4}c_{3}c_{2}^{2}\\&+\frac{5}{2}c_{2}c_{4}+ 10c_{2}^{4}-\frac{11}{16}c_{5})e_{n}^{5}+... \end{align*} Corrector step of Algorithm 2.2 is \[ x_{n+1}=z_{n}-\frac{f(z_{n})}{f^{\prime }(\frac{x_{n}+z_{n}}{2})} \] By substitution and simplification, we have Hence Algorithm 2.2 has fourth order convergence.

Theorem 3.2. Let \(f:I\subset \mathbb{R}\rightarrow\mathbb{R}\) for an open interval \(I\) and \(\alpha \in I\) be simple zero of \(f(x)=0.\) If \(f\) is differentiable and \(x_{0}\) is sufficiently close to \(\alpha\) then three-step iterative method defined by Algorithm 2.3 has fifth order convergence and satisfies the error equation \[ x_{n+1}=\alpha +(-\frac{1}{4}c_{3}c_{2}^{2}+c_{2}^{4})e_{n}^{5}+O(e_{n}^{6}),% \text{ and }e_{n}=x_{n}-\alpha. \]

Proof. From Eq. (17), we have \[ u_{n}=\alpha +(4c_{2}c_{3}-4c_{2}^{3}-\frac{1}{2}c_{4})e_{n}^{4}+(\frac{27}{8% }c_{3}^{2}-\frac{73}{4}c_{3}c_{2}^{2}+\frac{5}{2}c_{2}c_{4}+10c_{2}^{4}-% \frac{11}{16}c_{5})e_{n}^{5}+..., \] Expanding \(f(u_{n})\) and \(f^{\prime }(\frac{x_{n}+u_{n}}{2})\) by Taylor's series about \(\alpha ,\)

Now \[ x_{n+1}=z_{n}-\frac{f(u_{n})+f(z_{n})}{f^{\prime }(\frac{x_{n}+u_{n}}{2})}, \] by substituting values and simplifying, we have \[ x_{n+1}=\alpha +(-\frac{1}{4}c_{3}c_{2}^{2}+c_{2}^{4})e_{n}^{5}+O(e_{n}^{6}). \] Hence Algorithm 2.3 has fifth order convergence.

4. Applications

In this section, we present some numerical examples to examine the validity and efficieny of our newly devolped algorithms namely, Algorithm 2.2 and Algorithm 2.3. We also provide comparison of these algorithms with Newton's method(NM), Abbasbandy's method (AM), Householder's method(HHM), Halley's method (HM), Chun's method (CM) [3] and Noor's method(NR)[4]. We use \(\epsilon =10^{-25}\). We use the following stopping criteria; \vskip-1cm \begin{eqnarray*} (i)\text{}\;\;\;\; |x_{n}-x_{n-1}| &<&\in \\ (ii)\text{}\;\;\;\;|f(x_{n})| &<&\in \end{eqnarray*} We consider the following nonlinear scalar equations for comparison: \begin{eqnarray*} f_{1}(x) &=&\sin ^{2}x-x^{2}+1=0 \\ f_{2}(x) &=&x^{2}-e^{x}-3x+2=0 \\ f_{3}(x) &=&\cos x-x=0 \\ f_{4}(x) &=&(x-1)^{3}-1=0 \\ f_{5}(x) &=&x^{3}-10=0 \\ f_{6}(x) &=&e^{x^{2}+7x-30}-1=0. \end{eqnarray*} Comparison Table \begin{array}{ccccc} \text{Examples} & & \text{Iterations} & x_{n} & f(x_{n}) \\ f_{1},\text{ }x_{0}=1 & & & & \\ NM & & 7 & 1.40449164831534122635086 & -1.05e^{-50} \\ AM & & 5 & 1.40449164831534122635086 & -5.82e^{-54} \\ HM & & 4 & 1.40449164831534122635086 & -5.45e^{-63} \\ CM & & 5 & 1.40449164831534122635086 & -2.01e^{-64} \\ HHM & & 6 & 1.40449164821534122603508 & 1.82e^{-25} \\ NR & & 5 & 1.40449164821534122603508 & 9.23e^{-26} \\ \text{Alg. }2.2 & & 3 & 1.40449164821534122603508 & 3.40e^{-25} \\ \text{Alg. }2.3 & & 3 & 1.40449164821534122603509 & 2.32e^{-44} \\ f_{2},\text{ }x_{0}=2 & & & & \\ NM & & 6 & 0.257530285439860760455367 & 2.94e^{-55} \\ AM & & 5 & 0.257530285439860760455367 & 1.03e^{-63} \\ HM & & 5 & 0.257530285439860760455367 & 0 \\ CM & & 4 & 0.257530285439860760455367 & 1.05e^{-62} \\ HHM & & 5 & 0.257530285439860760455367 & -6.01e^{-25} \\ NR & & 5 & 0.257530285439860760455367 & 1.08e^{-23} \\ \text{Alg. }2.2 & & 3 & 0.257530285439860934975933 & 1.72e^{-69} \\ \text{Alg}.2.3 & & 2 & 0.257530285439860934975933 & 1.01e^{-29} \\ \end{array} \begin{array}{ccccc} \text{Examples} & & \text{Iterations} & x_{n} & f(x_{n}) \\ f_{3},\text{ }x_{0}=1.7 & & & & \\ NM & & 5 & 0.739085133215160641655372 & -2.04e^{-31} \\ AM & & 4 & 0.739085133215160641655372 & -7.13e^{-47} \\ HM & & 4 & 0.739085133215160641655372 & -5.04e^{-58} \\ CM & & 4 & 0.739085133215160641655372 & 0 \\ HHM & & 4 & 0.739085133215160641655372 & 3.91e^{-17} \\ NR & & 4 & 0.739085133215160645628855 & 6.66e^{-18} \\ \text{Alg. }2.2 & & 3 & 0.739085133215160641655312 & 3.21e^{-50} \\ \text{Alg.}2.3 & & 3 & 0.739085133215160641655312 & 1.21e^{-100} \\ f_{4},\text{ }x_{0}=1.5 & & & & \\ NM & & 8 & 2.0000000000000000000001234 & 2.05e^{-42} \\ AM & & 5 & 2 & 0 \\ HM & & 5 & 2 & 0 \\ CM & & 5 & 2 & 0 \\ HHM & & 7 & 2.000000000000000000000828 & 2.47e^{-22} \\ NR & & 5 & 2.000000000000000000000302 & 1.12e^{-24} \\ \text{Alg. }2.2 & & 4 & 2.000000000000000000000007 & 1.83e^{-44} \\ \text{Alg. }2.3 & & 4 & 2.000000000000000000000000 & 5.07e^{-27}\\ f_{5},\text{ }x_{0}=1.5 & & & & \\ N.M. & & 7 & 2.154434690031883721759235 & 2.05e^{-54} \\ A.M. & & 5 & 2.154434690031883721759235 & -5.01e^{-63} \\ H.M. & & 5 & 2.154434690031883721759235 & -5.03e^{-62} \\ C.M. & & 5 & 2.154434690031883721759235 & -5.01e^{-64} \\ H.H.M. & & 6 & 2.154434690031883721759293 & 7.86e^{-27} \\ NR.M. & & 4 & 2.154434690031883721759292 & 8.98e^{-24} \\ \text{Alg. }2.2 & & 3 & 2.154434690031883721759294 & 2.00e^{-27} \\ \text{Alg. }2.3 & & 3 & 2.154434690031883721759293 & 1.33e^{-50} \\ f_{6,}\text{ }x_{0}=3.5 & & & & \\ N.M. & & 13 & 3.000000000000000000000000 & 1.52e^{-48} \\ A.M. & & 7 & 3.000000000000000000000000 & -4.32e^{-48} \\ H.M. & & 8 & 3.000000000000000000000000 & 2.01e^{-62} \\ C.M. & & 8 & 3.000000000000000000000000 & 2.05e^{-62} \\ H.H.M. & & 12 & 3.000000000000000000000000 & 5.47e^{-25} \\ NR.M. & & 6 & 3.00000010285594873990664 & 1.34e^{-06} \\ \text{Alg. }2.2 & & 4 & 3.00000000000000000000000 & 4.76e^{-30} \\ \text{Alg. }2.3 & & 4 & 3.00000000000000000000000 & 1.99e^{-102}% \end{array}

5. Conclusion

We have established two new algorithms of fourth and fifth order convergence by using modified decomposition technique to coupled system. We have discussed convergence analysis of these newly established algorithms. Computational comparison of these algorithms has been made with some well known methods for solving nonlinear equations. From numerical table, we see that our new methods converge fast to the true solution of equations and are comparable with some classical methods.

Competing Interests

The author(s) do not have any competing interests in the manuscript.