1. Introduction

Let \( \mathbb{C} \) be complex plane and let \(\mathbb{ U } = \{z: z \in \mathbb{C} ~\text{and}~ |z| < 1\} = \mathbb{ U } \setminus \{0\}\) be an open unit disc in \( \mathbb{C} .\) Also let \(H(\mathbb{ U } )\) be a class of analytic functions in \(\mathbb{ U } .\) For \(n \in \mathbb{N}= \{1, 2, 3, \cdots , \} \) and \(a \in \mathbb{C} ,\) let \(H[a, n]\) be a subclass of \(H(\mathbb{ U } )\) formed by the functions of the form \[ f(z) = z+ a_n z^n + a_{n+1} z^{n+1} + \cdots \] with \(H_0 \equiv H[0, 1]\) and \(H \equiv H[1, 1].\) Suppose that \(A_n\) is a class of all analytic functions of the form

in the open unit disk \(\mathbb{ U } \) with \(A_1 = A.\) A function \(f \in H(\mathbb{ U } )\) is univalent if it is a one-to-one function in \(\mathbb{ U } .\) By \(S\), we denote a subclass of \(A\) formed by functions univalent in \(\mathbb{ U } .\) If a function \(f \in A\) maps \(\mathbb{ U } \) onto a convex domain and \(f\) is univalent, then \(f\) is called a convex function. By \[ K= \left\{ f\in A: \Re \left\{1+ \frac{zf''(z)}{f'(z)}\right\}> 0, \ \ z \in \mathbb{ U } \right\}, \] we denote a class of all convex functions defined in \(\mathbb{ U } \) and normalized by \(f(0) = 0\) and \(f'(0) = 1.\)

Let \(f\) and \(F\) be elements of \(H(\mathbb{ U } ).\) A function \(f\) is said to be subordinate to \(F\), if there exists a Schwartz function \(w\) analytic in \(\mathbb{ U } \) with \( w (0) = 0 ~~ \text{ and } ~~ |w(z) | < 1, \ \ z \in \mathbb{ U } , \) such that \(f(z) = F(w(z)).\) In this case, we write \( f(z) \prec F(z) ~~~ \text{ or } ~~ f \prec F.\) Furthermore, if the function \(F\) is univalent in \(\mathbb{ U } ,\) then we get the following equivalence [1,2]:

\[f(z) \prec F(z) \Leftrightarrow f(0) = F(0) ~~~ \text{ and}~~ f(\mathbb{ U } ) \prec F(\mathbb{ U } ). \] The method of differential subordinations (also known as the method of admissible functions) was first introduced by Miller and Mocanu in 1978 [3], and the development of the theory was originated in 1981 [4]. All details can be found in the book by Miller and Mocanu [2]. In recent years, numerous authors studied the properties of differential subordinations (see [5,6,7,8], etc.).

Let \( \Psi : \mathbb{C}^3 \times \mathbb{ U } \rightarrow \mathbb{C} \) and let \(h\) be univalent in \(\mathbb{ U } .\) If \(p\) is analytic in \(\mathbb{ U } \) and satisfies the second-order differential subordination:

then \(p\) is called the solution of differential subordination. The univalent function \(q\) is called a dominant of the solution of the differential subordination or, simply, a dominant if \( p \prec q\) for all \(p\) satisfying (2). The dominant \(q_1\) satisfying \(q_1 \prec q\) for all dominants \(q\) of (2) is called the best dominant of (2).

Let us recall lambda function [9] defined by:

\[ \lambda (z, s ) = \sum \limits _{k =2} ^{\infty} \frac{z^k }{ (2k +1) ^k }\] where \(z \in \mathbb{ U } , s \in \mathbb{C},\) when \(|z| < 1, \Re (s) > 1, \) when \(|z| = 1\) and let \(\lambda ^{(-1)} (z, s ) \) be defined such that \[ \lambda (z, s ) * \lambda ^{(-1)} (z, s ) = \frac{1}{(1- z) ^{ \mu +1}}, ~ \mu > -1 .\] We now define \(\left( z \lambda ^{(-1)} (z, s ) \right)\) as: \begin{align*} (z \lambda (z,s)) * \left( z \lambda ^{(-1)} (z, s ) \right) = \frac{z}{(1-z)^{\mu+1}} = z+ \sum \limits _{k =2} ^{\infty} \frac{ ( \mu +1)_{k -1}}{ (k -1) !} z^k , \mu > -1 \end{align*} and obtain the linear operator \( \mathcal{I}_{\mu}^{s} f(z)= \left( z \lambda ^{(-1)} (z, s ) \right) * f(z),\) where \(f \in A, z \in \mathbb{ U } \) and \( \left( z \lambda ^{(-1)} (z, s ) \right) = z+ \sum \limits _{k =2} ^{\infty} \frac{ ( \mu +1)_{k -1} (2k -1) ^s}{ (k -1) !} z^k .\) A simple computation gives us where where \(( \mu)_{k } \) is the Pochhammer symbol defined in terms of the Gamma function by: \[( \mu)_{k } = \frac{\Gamma ( \mu+k )}{\Gamma (\mu)}=\left\{ \begin{array}{ll} 1, & \hbox{ if \(k =0 \);} \\ \mu ( \mu +1) \cdots (\mu +k -1), & \hbox{ if \( k \in \mathbb{N} \)}. \end{array} \right . .\]

Definition 1. Let \(\mathfrak{ L} _{\mu , s } ( \varrho ) \) be a class of function \(f \in A\) satisfying the inequality \[ \mathfrak { \Re } \left( \mathcal{I}_ {\mu}^{s}f(z) \right) \geq \varrho ,\] where \( z\in \mathbb{ U }, \ \ 0 \leq \varrho < 1 \) and \(\mathcal{I}_ {\mu}^{s}f(z) \) is the Lambda operator.

Lemma 1. let \(h\) be a convex function with \(h(0)=a \) and let \(\gamma\in \mathbb { C^* }:= \mathbb { C } \setminus \{ 0 \} \) be a complex number with \( \mathfrak { \Re \{ \gamma \} }\geq 0. \) If \( p \in H[a, n] \) and

then \( p(z) \prec q(z) \prec h(z), \) where \( q(z) = \frac{\gamma}{nz^\frac{\gamma}{n}}\int \limits_{0}^{z}t^\frac{\gamma}{n-1}h(t)dt, ~~ z\in \mathbb{ U }.\) The function \(q\) is convex and is the best dominant for subordination (5).

Lemma 2. [10] Let \( \mathfrak { \Re \{ \mu \} }> 0, ~~n\in \mathbb { N } \) and \( w= \frac{n^2 + |\mu|^2-|n^2-\mu^2|}{4n \mathfrak { \Re \{ \mu \} }} . \) Also, let \(h\) be an analytic function in \(\mathbb { U } \) with \( h(0)=1.\) Suppose that \( \mathfrak { \Re }\left \{ 1+\frac{zh''(z)}{h'(z)} \right \} > -w. \) If \( p(z) = 1+p_nz^n+p_{n+1}z^{n+1}+ \cdots \) is analytic in \(\mathbb { U } \) and

then \( p(z) \prec q(z), \) where \(q\) is a solution of the differential equation \( q(z)+ \frac{n}{\mu}zq'(z) = h(z), ~~q(0)=1, \) given by \( q(z) = \frac{\mu}{nz^\frac{\mu}{n}}\int \limits_{0}^{z}t^\frac{\mu}{n-1}h(t) dt, ~~z\in \mathbb { U } . \) Moreover, \(q\) is the best dominant for the differential subordination (6).

Lemma 3. [11] Let \(r\) be a convex function in \(\mathbb { U } \) and let \( h(z)=r(z)+n\varrho zr'(z), ~~ z\in \mathbb { U } , \) where \(\varrho > 0 ~~ and ~~ n \in \mathbb { N }.\) If \( p(z)= r(0)+p_nz^n+p_{n+1}z^{n+1}+ \cdots , ~~z\in \mathbb { U } , \) is holomorphic in \( \mathbb { U } \) and \( p(z)+\varrho zp'(z)\prec h(z), ~~z\in \mathbb { U } , \) then \( p(z)\prec r(z) \) and this result is sharp.

In the present paper, we use the subordination results from [10] to prove our main results.

2. Main results

Theorem 1. The set \( \mathfrak{ L} _{\mu , s } ( \varrho ) \) is convex.

Proof. Let \( f_j(z)=z+\sum\limits_{k=2}^{\infty}a_{k,j}z^k, ~~ z\in \mathbb { U } , ~~ j= 1, \cdots , m \) be in the class \( \mathfrak{ L} _{\mu , s } ( \varrho ). \) Then, by Definition 1, we get

For any positive numbers \( \varsigma _1, \varsigma _2, \varsigma _3, \cdots ,\varsigma _m \) such that \( \sum\limits_{j=1}^{m}\varsigma _j=1 , \) it is necessary to show that the function \(h(z)=\sum\limits_{j=1}^{m}\varsigma _jf_j(z)\) is an element of \( \mathfrak{ L} _{\mu , s } ( \varrho ) \), i.e., Thus, we have If we differentiate (9) with respect to \(z\), then we obtain \[ (\mathcal{I}_ \mu ^s h(z))' = 1+\sum\limits_{k=2}^{\infty}kL(k,\mu,s) \left \{ \sum\limits_{j=1}^{m} \varsigma _ja_{k,j} \right \} z^{k-1}. \] Thus by using (8), we have \begin{align*} \mathfrak{ \Re } \left \{ (\mathcal{I}_ \mu ^s h(z))' \right \} = 1+\sum\limits_{j=1}^{m}\varsigma _j \mathfrak{ \Re } \left \{ \sum\limits_{k=2}^{\infty}kL(k,\mu,s)a_{k,j}z^{k-1} \right \} > 1+\sum\limits_{j=1}^{m}\varsigma _j(\varrho -1) = \varrho . \end{align*} Hence, inequality (7) is true and we arrive at the desired result.

Theorem 2. Let \(q\) be convex function in \(\mathbb {U} \) with \(q(0)=1\) and \( h(z)=q(z) + \frac{1}{\gamma+1}zq'(z), ~~ z\in \mathbb { U } , \) where \(\gamma\) is a complex number with \(\mathfrak{ \Re } \{ {\gamma} \} > -1\). If \(f\in \mathfrak{ L} _{\mu , s } ( \varrho ) \) and \( \aleph =\Upsilon_\gamma f,\) where

then implies that \( (\mathcal{I}_ \mu ^s \aleph (z))' \prec q(z) \) and this result is sharp.

Proof. In view of equality (10), we can write

Differentiating (12) with respect to \(z,\) we obtain \( (\gamma) \aleph (z)+z \aleph ~'(z)=(\gamma+1)f(z). \) Further, by applying the operator \(\mathcal{I}_ \mu ^s\) to the last equation, we get If we differentiate (13) with respect to \(z\), then we find By using the differential subordination given by (11) in equality (14), we obtain We define Hence, as a result of simple computations, we get \begin{align*} p(z) &= \left \{ z+ \sum\limits_{k=2}^{\infty}L(k,\mu,s)\frac{\gamma+1}{\gamma+k}a_kz^k \right \}' = 1+ p_1z+p_2z^2+ \cdots ,~~p\in H[1,1]. \end{align*} By using (16) in subordination (15), we obtain \begin{equation*} p(z)+\frac{1}{\gamma+1}zp'(z)\prec h(z)=q(z)+\frac{1}{\gamma+1}zq'(z),~~ z\in \mathbb{ U }. \end{equation*} If we use Lemma 2, then we write \( p(z)\prec q(z). \) Thus, we obtained the desired result and \(q\) is the best dominant.

Example 1. If we choose \( \gamma=i+1\) and \(q(z)= \frac{1+z}{1-z}, \) in Theorem 2, then we get \( h(z)=\frac{(i+2)-((i+2)z+2)z}{(i+2)(1-z)^2}. \) If \( f \in \mathfrak{ L} _{\mu , s } ( \varrho ) \) and \(\aleph\) is given as \( \aleph(z)= \Upsilon_if(z)=\frac{i+2}{z^{i+1}}\int\limits_0^zt^if(t)dt, \) then, by virtue of Theorem 2, we find \( (\mathcal{I}_ {\mu}^{s}f(z))'\prec h(z) = \frac{(i+2)-((i+2)z+2)z}{(i+2)(1-z)^2},\) implies \((\mathcal{I}_ {\mu}^{s}f(z))' \prec \frac{1+z}{1-z}.\)

Theorem 3. Let \( \mathfrak{ \Re }{ \left \{ \gamma \right \} } > -1 \) and \( w=\frac{1+|\gamma+1|^2-|\gamma^2+2\gamma|}{4\mathfrak{ \Re }{ \left \{ \gamma+1 \right \}}}. \) Suppose that \(h\) is an analytic function in \(\mathbb{ U } \) with \(h(0)=1\) and that \( \mathfrak { \Re }\left \{ 1+\frac{zh''(z)}{h'(z)} \right \} > -w. \) If \( f \in \mathfrak{ L} _{\mu , s } ( \varrho ) \) and \( \aleph = \Upsilon_\mu ^s f, \) where \(\aleph\) is defined by (10), then

implies that \( (\mathcal{I}_ \mu ^s \aleph(z))' \prec q(z), \) where \(q\) is the solution of the differential equation \( h(z) = q(z) + \frac{1}{\gamma+1}zq'(z), ~~q(0)=1, \) given by \( q(z) = \frac{\gamma+1}{z^{\gamma+1}}\int\limits_0^zt^\gamma f(t)dt. \) Moreover, \(q\) is the best dominant for subordination (17).

Proof. If we choose \( n=1 \) and \( \mu={\gamma+1} \) in Lemma 1, then the proof is obtained by means of the proof of Theorem 3.

Theorem 4. Let

be convex in \( \mathbb { U } \) with \( h(0)=1 .\) If \( f\in A \) and verifies the differential subordination \( (\mathcal{I}_ \mu ^s f(z))' \prec h(z), \) then \( (\mathcal{I}_ \mu ^s \aleph (z))' \prec q(z)=(2\varrho -1)+\frac{2(1-\varrho )(\gamma+1) \tau(\gamma)}{z^{\gamma+1}}, \) where \(\tau\) is given by the formula and \(\aleph\) is given by equation (10). The function \(q\) is convex and is the best dominant.

Proof. If \( h(z)=\frac{1+(2\varrho - 1)z}{1+z}, ~~ 0\leq \varrho < 1, \) then \( h\) is convex and, in view of Theorem 3, we can write \( (\mathcal{I}_ \mu ^s \aleph(z))' \prec q(z). \) Now, by using Lemma 1, we get \begin{align*} q(z) = \frac{\gamma+1}{z^{\gamma+1}}\int\limits_0^zt^\gamma h(t)dt = \frac{\gamma+1}{z^{\gamma+1}}\int\limits_0^zt^\gamma \left \{ \frac{1+(2\varrho -1)t}{1+t} \right \} dt = (2\varrho -1)+\frac{2(1-\varrho )(\gamma+1)}{z^{\gamma+1}}\tau(\gamma), \end{align*} where \(\tau\) is given by (19). Hence, we obtain \begin{equation*} (\mathcal{I}_ \mu ^s \aleph (z))' \prec q(z)=(2\varrho -1)+\frac{2(1-\varrho )(\gamma+1) \tau(\gamma)}{z^{\gamma+1}}. \end{equation*} The function \( q\) is convex. Moreover, it is the best dominant. Hence the theorem is proved.

Theorem 5. If \( 0 \leq \varrho < 1, 0 \leq \mu < 1, \delta \geq 0, \mathfrak{ \Re } \{ {\gamma} \} > -1, \) and \(\aleph= \Upsilon_\gamma f \) is defined by (10), then \( \Upsilon_\gamma( \mathfrak{ L} _{\mu , s } ( \varrho )) \subset \mathfrak{ L} _{\mu , s } ( \rho ), \) where

and \(\tau\) is given by (19).

Proof. Assume that \(h\) is given by equation (18), \(f \in \mathfrak{ L} _{\mu , s } ( \varrho ), \) and \( \aleph = \Upsilon_\gamma f \) is defined by (10). Then \(h\) is convex and, by Theorem 3, we deduce

where \(\tau\) is given by (19). Since \( q \) is convex, \( q( \mathbb{ U }) \) is symmetric about the real axis, and \( \mathfrak{ \Re } \{ {\gamma} \} > -1,\) we find \begin{align*} \mathfrak{ \Re } \left \{ (\mathcal{I}_ \mu ^s \aleph(z))' \right \} \geq \min \limits _{|z|=1} \mathfrak{ \Re } \{ {q(z)} \} = \mathfrak{ \Re } \{ q(1) \}= \rho(\gamma,\varrho ) = (2\varrho -1)+2(1-\varrho )(\gamma+1)(1-\varrho )\tau(\gamma). \end{align*} It follows from inequality (21) that \( \Upsilon_\gamma( \mathfrak{ L} _{\mu , s } ( \varrho )) \subset \mathfrak{ L} _{\mu , s } ( \rho ), \) where \(\rho\) is given by (20). Hence the theorem is proved.

Theorem 6. Let \(q\) be a convex function with \( q(0) = 1 \) and \(h\) be a function such that \( h(z)= q(z) + zq'(z), ~~ z\in \mathbb{ U }. \) If \( f \in A,\) then the subordination

implies that \( \frac{\mathcal{I}_ \mu ^s f(z)}{z} \prec q(z), \) and the result is sharp.

Proof. Let

Differentiating (23), we find \( (\mathcal{I}_ \mu ^s f(z))' = p(z)+zp'(z). \) We now compute \( p(z) \). This gives By using (24) in subordination (22), we find \( p(z)+zp'(z)\prec h(z)=q(z)+zq'(z). \) Hence, by applying Lemma 3, we conclude that \( p(z) \prec q(z) \) i.e., \( \frac{\mathcal{I}_ \mu ^s f(z)}{z} \prec q(z). \) This result is sharp and \(q\) is the best dominant. Hence the theorem is proved.

Example 2. If we take \( \mu = 0 \) and \( s = 1 \) in equality (4) and \( q(z) = \frac{1}{1-z} \) in Theorem 5, then \( h(z)=\frac{1}{(1-z)^2} \) and

Differentiating (25) with respect to \(z\), we get \begin{align*} (I_0^1f(z))' = 1+ \sum\limits_{k=2}^{\infty}\frac{(2k-1)}{(k-1)!}a_kz^{k-1} = 1+p_1z+p_2z^2+ \cdots , ~~ p \in H[1,1]. \end{align*} By using Theorem 5, we find \( (I_0^1f(z))'\prec h(z) = \frac{1}{(1-z)^2}. \) This yields \( \frac{I_0^1f(z)}{z} \prec q(z) = \frac{1}{1-z}. \)

Theorem 7. Let \( h(z)=\frac{1+(2\varrho -1)z}{1+z}, ~~ z\in \mathbb{ U } \) be convex in \( \mathbb { U } \) with \( h(0)=1\) and \( 0 \leq \varrho < 1.\) If \( f \in A \) satisfies the differential subordination

then \( \frac{\mathcal{I}_ \mu ^s f(z)}{z} \prec q(z)= (2\varrho -1)+\frac{2(1-\varrho )ln(1+z)}{z}. \) The function \(q\) is convex and, in addition, it is the best dominant.

Proof. Let

Differentiating (27), we find In view of (28), the differential subordination (26) becomes \( (\mathcal{I}_ \mu ^s f(z))' \prec h(z) = \frac{1+(2\varrho -1)z}{1+z},\) and by using Lemma 1, we deduce \( p(z)\prec q(z)=\frac{1}{z}\int h(t)dt = (2\varrho -1)+\frac{2(1-\varrho )ln(1+z)}{z}. \) Now, by virtue of relation (27) we obtained the desired result.

Corollary 1. If \( f \in \mathfrak{ L} _{\mu , s } ( \varrho ), \) then \( \mathfrak { \Re } \left( \frac{\mathcal{I}_ \mu ^s f(z)}{z} \right) > (2\varrho -1)+2(1-\varrho )ln(2). \)

Proof. If \( f \in \mathfrak { L} _{\mu , s } ( \varrho ), \) then it follows from Definition 1 that \( \mathfrak { \Re } \left \{ (\mathcal{I}_ \mu ^s f(z))' \right \} > \varrho , ~~ z \in \mathbb { U }, \) which is equivalent to \( (\mathcal{I}_ \mu ^s f(z))' \prec h(z) = \frac{1+(2\varrho -1)z}{1+z}. \) Now, by using Theorem 7, we obtain \begin{equation*} \frac{\mathcal{I}_ \mu ^s f(z)}{z} \prec q(z) = (2\varrho -1)+\frac{2(1-\varrho )ln(1+z)}{z}. \end{equation*} Since \(q\) is convex and \(q( \mathbb { U } ) \) is symmetric about the real axis, we conclude that \begin{equation*} \mathfrak { \Re } \left( \frac{\mathcal{I}_ \mu ^s f(z)}{z} \right) > \mathfrak { \Re } (q(1)) = (2\varrho -1)+2(1-\varrho )ln(2). \end{equation*}

Theorem 8. Let \(q\) be a convex function such that \(q(0)=1\) and \(h\) be the function given by the formula \( h(z)=q(z)+zq'(z), ~~z \in \mathbb { U }. \) If \( f \in A \) and verifies the differential subordination

then \( \frac{\mathcal{I}_ \mu ^s f(z)}{\mathcal{I}_ \mu ^s \aleph(z)} \prec q(z), ~~ z \in \mathbb { U }, \) and this result is sharp.

Proof. For function \( f \in A, \) given by Equation (1), we get \begin{equation*} \mathcal{I}_ \mu ^s \aleph (z) = z + \sum\limits_{k=2}^{\infty}L(k,\mu,s)\frac{\gamma+1}{k+\gamma}a_kb_kz^k, ~~z\in \mathbb { U }. \end{equation*} We now consider the function \begin{align*} p(z)=\frac{\mathcal{I}_ \mu ^s f(z)}{\mathcal{I}_ \mu ^s \aleph (z)} = \frac{z+\sum\limits_{k=2}^{\infty}L(k,\mu,s )a_kb_kz^k}{z+\sum\limits_{k=2}^{\infty}L(k,\mu,s)\frac{\gamma+1}{k+\gamma}a_kb_kz^k} = \frac{1+\sum\limits_{k=2}^{\infty}L(k,\mu,s)a_kb_kz^{k-1}}{1+\sum\limits_{k=2}^{\infty}L(k,\mu,s)\frac{\gamma+1}{k+\gamma}a_kb_kz^{k-1}}. \end{align*} In this case, we get \begin{equation*} (p(z))'=\frac{(\mathcal{I}_ \mu ^s f(z))'}{\mathcal{I}_ \mu ^s \aleph(z)} - p(z) \frac{(\mathcal{I}_ \mu ^s \aleph(z))'}{\mathcal{I}_ \mu ^s \aleph(z)}. \end{equation*} Then

By using relation (30) in inequality (29), we obtain \( p(z)+zp'(z)\prec h(z)=q(z)+zq'(z) \) and, by virtue of Lemma 3, \( p(z)\prec q(z), \) i.e., \( \frac{\mathcal{I}_ \mu ^s f(z)}{\mathcal{I}_ \mu ^s \aleph(z)} \prec q(z). \) Hence the theorem is proved.

Acknowledgments

The authors warmly thank the referees for the careful reading of the paper and their comments.

Author Contributions

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

Conflicts of Interest

The authors declare no conflict of interest.

1. Introduction

Let \(\mathcal{H}\) be an infinite dimensional complex Hilbert space and \(\mathcal{B(H)}\) be the algebra of all bounded linear operators on \(\mathcal{H}.\) In this paper, we discuss various types of norm inequalities for inner product type integral transformers in terms of Landau type inequality, Grüss type inequality and Cauchy-Schwarz type inequality. We shall also consider the applications in quantum theory. We begin by the following definition:

Definition 1. Grüss inequality states that if \(f\) and \(g\) are integrable real functions on \([a,b]\) such that \(C\leq f(x)\le D\) and \(E\leq g(x)\le F\) hold for some real constants \(C,D,E,F\) and for all \(x\in[a,b]\), then

Inequality (1) is very interesting to many researchers and it has been considered in many studies whereby conditions on functions are varied to give different estimates (see [1] and references therein). More on this inequality (and the classical one [2]) are discussed in the sequel.

Next, we discuss a very important definition of inner product type integral \((i.p.t.i)\) transformer which is key to our study.

Definition 2. Consider weakly \(\mu^*\)-measurable operator valued \((o.v)\) functions \(A, B:\Omega\rightarrow \mathcal{B(H)}\) and for all \(X\in \mathcal{B(H)}\). Let the function \(t\rightarrow A_t X B_t\) be also weakly \(\mu^*\)-measurable. If these functions are Gel'fand integrable for all \(X\in \mathcal{B(H)}\), then the inner product type linear transformation \(X\to\int_\Omega A_t X B_t dt\) is called an inner product type integral \((i.p.t.i)\) transformer on \(\mathcal{B(H)}\) and denoted by \(\int_\Omega A_t \otimes B_t dt\) or \({\mathcal I}_{A,B}\).

Remark 1. If \(\mu\) is the counting measure on \(\mathbb N\) then such transformers are known as elementary operators whose certain properties have been studied in details (see [3] and the references therein).

2. Preliminaries

In this section, we consider a special type of norms called the unitarily invariant norm. We give its description in details which will be useful in the sequel. Let \(\mathcal{C}_\infty(\mathcal{H})\) denote the space of all compact linear operators acting on a separable, complex Hilbert space \(\mathcal{H}\). Each symmetric gauge function \(\Phi,\) denoted by \((s.g.)\), on sequences gives rise to a unitarily invariant \((u.i)\) norm on operators defined by \(\left\|X\right\|_\Phi=\Phi(\{s_n(X)\}_{n=1}^\infty)\) with \(s_1(X)\ge s_2(X)\ge...\) being the singular values of \(X\), i.e., the eigenvalues of \(|X|=(X^*X)^\frac12.\) We denote any such norm by the symbol \(\left|\left|\left|\cdot\right|\right|\right|\), which is therefore defined on a naturally associated norm ideal \(\mathcal{C}_{\left|\left|\left|\cdot\right|\right|\right|}(\mathcal{H})\) of \(\mathcal{C}_\infty(\mathcal{H})\) and satisfies the invariance property \( |\|UXV|\|=|\|X|\|\) for all \(X\in\mathcal{C}_{\left|\left|\left|\cdot\right|\right|\right|}(\mathcal{H})\) and for all unitary operators \(U,V\in \mathcal{B(H)}\). One of the well known among \(u.i.\) norms are the Schatten \(p\)-norms defined for \(1\le p< \infty\) as \(\|X\|_p=\sqrt[p]{\,\sum_{n=1}^\infty s_n^p(X)}\), while \(\|X\|_\infty =\|X\|=s_1(X)\) coincides with the operator norm \(\|X\|\). Minimal and maximal \(u.i.\) norm are among Schatten norms, i.e., \(\|X\|_\infty\le|\|X\||\le\|X\|_1\) for all \(X\in\mathcal{C}_1(\mathcal{H})\) (see inequality (IV.38) [4]). For \(f,g\in\mathcal{H}\), we will denote by \(g^*\otimes f\) one dimensional operator \((g^*\otimes f)h=\langle h,g\rangle f\) for all \(h\in\mathcal{H}\) and it is known that the linear span of \(\{g^*\otimes f\,|\, f,g\in \mathcal{H}\}\) is dense in each of \(\mathcal{C}_p(\mathcal{H})\) for \(1\le p\le\infty\). Schatten \(p\)-norms are also classical examples of \(p\)-reconvexized norms. Namely, any \(u.i.\) norm \(\|.\|_\Phi\) could be \(p\)-reconvexized for any \(p\ge1\) by setting \(\|A\|_{\Phi^{(p)}} = \| |A|^p\|_{\Phi}^{\frac1p}\) for all \(A\in \mathcal{B(H)}\) such that \(|A|^p\in \Phi(\mathcal{H})\). For the proof of the triangle inequality and other properties of these norms, see [2] and for the characterization of the dual norm for \(p\)-reconvexized, see Theorem 2.1 [2].

The set \(\mathcal{C}_{|||\cdot|||}=\{A \in \mathcal{K}(\mathcal{H}) : \left\vert \left\vert \left\vert A \right\vert \right\vert \right\vert < \infty \}\) is a closed self-adjoint ideal \(\mathcal{J}\) of \(\mathcal{B}( \mathcal{H})\) containing finite rank operators. It enjoys the following properties. First, for all \(A,B\in \mathcal{B(H)}\) and \(X \in \mathcal{J}\), \( \left\vert \left\vert \left\vert AXB\right\vert \right\vert \right\vert \leq \left\vert \left\vert A\right\vert \right\vert \ \left\vert \left\vert \left\vert X\right\vert \right\vert \right\vert \ \left\vert \left\vert B\right\vert \right\vert\,. \) Secondly, if \(X\) is a rank one operator, then \( \left\vert \left\vert \left\vert X\right\vert \right\vert \right\vert =\|X\|\,. \) The Ky Fan norm as an example of unitarily invariant norms is defined by \(\| A\| _{(k)}=\sum_{j=1}^{k}s_{j}(A)\) for \(k=1,2,\ldots\). The Ky Fan dominance Theorem [5] states that \(\| A\| _{(k)}\leq \| B\| _{(k)}\,\,(k=1,2,\ldots )\) if and only if \(|||A||| \leq |||B|||\) for all unitarily invariant norms \(|||\cdot|||\), see [6] for more information on unitarily invariant norms. The inequalities involving unitarily invariant norms have been of special interest (see [5] and the references therein).

Lemma 1. Let \(\mathcal{T}\) and \(\mathcal{S}\) be linear mappings defined on \(\mathcal{C}_\infty(\mathcal{H}).\) If \(\|\mathcal{T}X\|\le\|\mathcal{S}X\|\mbox{ for all }X\in \mathcal{C}_\infty(\mathcal{H}), \;\|\mathcal{T}X\|_1\le\|\mathcal{S}X\|_1\mbox{ for all }X\in \mathcal{C}_\infty(\mathcal{H})\), then \( \mathcal{T}X\le\mathcal{S}X\) for all unitarily invariant norms.

Proof. The norms \(\|\cdot\|\) and \(\|\cdot\|_1\) are dual to each other in the sense that \(\|X\|=\sup_{\|Y\|_1=1}|tr(XY)|\) and \( \|X\|_1=\sup_{\|Y\|=1}|tr(XY)|.\) Hence \(\|\mathcal{T}^*X\|\le\|\mathcal{S}^*X\|\) and \(\|\mathcal{T}^*X\|_1\le\|\mathcal{S}^*X\|_1\). Consider the Ky Fan norm \(\|\cdot\|_{(k)}\). Its dual norm is \(\|\cdot\|_{(k)}^\sharp=\max\{\|\cdot\|,(1/k)\|\cdot\|_1\}\). Thus, by duality, \(\|\mathcal{T}X\|_{(k)}\le\|\mathcal{S}X\|_{(k)}\) and the result follows by Ky Fan dominance property [6].

An operator \(A\in \mathcal{B(H)}\) is called \(G_{1}\) operator if the growth condition

\[ \left\Vert (z-A)^{-1}\right\Vert =\frac{1}{{dist}(z,\sigma (A))} \] holds for all \(z\) not in the spectrum \(\sigma (A)\) of \(A\). Here \({dist}(z,\sigma (A))\) denotes the distance between \(z\) and \(\sigma (A)\). It is known that hyponormal (in particular, normal) operators are \( G_{1}\) operators [4].

Let \(A, B\in \mathcal{B(H)}\) and let \(f\) be a function which is analytic on an open neighborhood \( \Omega \) of \(\sigma (A)\) in the complex plane. Then \(f(A)\) denotes the operator defined on \(\mathcal{H}\) by \( f(A)=\frac{1}{2\pi i}\int\limits_{C}f(z)(z-A)^{-1}dz, \label{4} \) called the Riesz-Dunford integral, where \(C\) is a positively oriented simple closed rectifiable contour surrounding \(\sigma (A)\) in \(\Omega \) (see [2] and the references therein). The spectral mapping theorem asserts that \(\sigma (f(A))=f(\sigma (A))\). Throughout this paper, \(\mathbb{D}=\{z\in\mathbb{C}:\left\vert z\right\vert < 1\}\) denotes the unit disk, \(\partial\mathbb{D}\) stands for the boundary of \(\mathbb{D}\) and \(d_{A}={dist}(\partial\mathbb{D},\sigma (A))\). In addition, we adopt the notation \(\mathfrak{H}=\{f: \mathbb{D}\to \mathbb{C}: f \mbox{   is analytic}, \Re(f)>0 \mbox{   and  } f(0)=1\}.\)

In this work, we present some upper bounds for \(|||f(A)Xg(B)\pm X|||\), where \(A, B\) are \(G_{1}\) operators, \(|||\cdot|||\) is a unitarily invariant norm and \(f, g\in \mathfrak{H}\). Further, we find some new upper bounds for the the Schatten \(2\)-norm of \(f(A)X\pm Xg(B)\). Up-to this juncture, we find some upper estimates for \(|||f(A)Xg(B)+ X|||\) in terms of \(|||\,|AXB|+|X|\,|||\) and \(|||f(A)Xg(B)- X|||\) in terms of \(|||\,|AX|+|XB|\,|||\), where \(A, B\) are \(G_{1}\) operators and \(f, g\in \mathcal{H}\).

Proposition 1. If \(A,B\in \mathcal{B(H)}\) are \(G_{1}\) operators with \(\sigma (A)\cup \sigma (B)\subset\mathbb{D}\) and \(f, g \in \mathcal{H}\), then for every \(X\in \mathcal{B(H)}\) and for every unitarily invariant norm \(\left\vert \left\vert \left\vert\cdot \right\vert \right\vert \right\vert \), the inequality \( \left\vert \left\vert \left\vert f(A)Xg(B)+X\right\vert \right\vert \right\vert \leq \frac{2\sqrt{2}}{d_{A}d_{B}} \left\vert\left\vert \left\vert\,|AXB|+|X|\,\right\vert \right\vert \right\vert \label{5} \) holds.

Proof. From the Herglotz representation Theorem [1], it follows that \(f\in \mathcal{H}\) can be represented as

where \(\mu \) is a positive Borel measure on the interval \([0,2\pi ]\) with finite total mass \(\int\limits_{0}^{2\pi }d\mu(\alpha)=f(0)=1\). Similarly \(g(z)=\int\limits_{0}^{2\pi }\frac{e^{i\alpha}+z}{e^{i\alpha}-z}d\nu(\alpha)\) for some positive Borel measure \(\nu\) on the interval \([0,2\pi ]\) with finite total mass \(1\). We have \begin{eqnarray*} f(A)Xg(B)+X=\int\limits_{0}^{2\pi }\int\limits_{0}^{2\pi }\left[ \left( e^{i\alpha}-A\right)^{-1} \left( e^{i\alpha}+A\right)X\left( e^{i\beta}+B\right) \left( e^{i\beta}-B\right)^{-1}+X\right] d\mu(\alpha)d\nu(\beta). \end{eqnarray*} By some computation, we have \begin{eqnarray} \label{s1} \left\vert \left\vert \left\vert f(A)Xg(B)+X\right\vert \right\vert \right\vert \nonumber\leq \int\limits_{0}^{2\pi }\int\limits_{0}^{2\pi }2\left\Vert \left( e^{i\alpha }-A\right)^{-1}\right\Vert \left\vert \left\vert \left\vert AXB+ e^{i\alpha}Xe^{i\beta}\right\vert \right\vert \right\vert \left\Vert \left( e^{i\alpha }-B\right)^{-1}\right\Vert d\mu(\alpha)d\nu(\beta).\ \end{eqnarray} Since \(A\) and \(B\) are \(G_{1}\) operators, we deduce that and similarly \( \left\vert \left\vert \left( e^{i\beta}-B\right)^{-1}\right\vert \right\vert \leq \frac{1}{d_{B}}. \) Now, for every positive operators \(C, D\), every non-negative operator monotone function \(h(t)\) on \([0,\infty)\) and every unitarily invariant norm \(|||\cdot|||\), we have \(|||h(A+B)||| \leq |||h(A)+h(B)|||\). Now, from the Ky Fan dominance theorem, we infer that Therefore, it follows from inequality (3) and Equation (4) that \[ \left\vert \left\vert \left\vert f(A)Xg(B)+X\right\vert \right\vert \right\vert \leq \frac{2\sqrt{2}}{d_{A}d_{B}} \left\vert\left\vert \left\vert\,|AXB|+|X|\,\right\vert \right\vert \right\vert, \] which completes the proof.

Theorem 1. Let \(f, g\in \mathcal{H}\) and \(A\in\mathcal{B(H)}\) be a \(G_{1}\) operator with \(\sigma (A)\subset\mathbb{D}\). The inequality \( \left\vert \left\vert \left\vert f(A)Xg(A^*)+X\right\vert \right\vert \right\vert \leq \frac{2}{d_{A}^2} \left\vert\left\vert \left\vert\ A|X|A^*+|X|\ \right\vert \right\vert \right\vert \) holds for every normal operator \(X\in\mathcal{B(H)}\) commuting with \(A\) and for every unitarily invariant norm \(\left\vert \left\vert \left\vert\cdot \right\vert \right\vert \right\vert \).

Proof. Let \(X\) and \(AXB\) be normal. Since \(||| C+D |||\leq |||\,|C|+|D|\,|||\) for any normal operators \(C\) and \(D\), the constant \(\sqrt{2}\) can be reduced to \(1\) in Equation (4). Now from Fuglede-Putnam theorem, if \(A\in \mathcal{B(H)}\) is an operator, \(X\in {\mathcal(B)}({\mathcal(H)})\) is normal and \(AX=XA\), then \(AX^*=X^*A\). Thus if \(X\) is a normal operator commuting with a \(G_{1}\) operator \(A\), then \(AXA^*\) is normal, \(|AXA^*|=A|X|A^*\) and \(A^*\) is a \(G_1\) operator with \(d_{A^*}=d_A\). By Proposition 1 the proof is complete.

Next, letting \(A=B\) in Proposition 1, we obtain the following result.

Corollary 1. Let \(f, g\in \mathcal{H}\) and \(A\in \mathcal{B(H)}\) be a \(G_{1}\) operator with \(\sigma (A)\subset\mathbb{D}\). Then \( \left\vert \left\vert \left\vert f(A)Xg(A)-X\right\vert \right\vert \right\vert \leq \frac{2\sqrt{2}}{d_{A}^2} \left\vert\left\vert \left\vert \,|AX|+|XA|\,\right\vert \right\vert \right\vert \) for every \(X\in\mathcal{B}(\mathcal{H})\) and for every unitarily invariant norm \(\left\vert \left\vert \left\vert \cdot \right\vert \right\vert \right\vert \).

Setting \(X=I\) in Proposition 1 again, we obtain the following result.

Corollary 2. Let \(f, g\in \mathfrak{H}\) and \(A,B\in\mathbb{M}_n\) be \(G_{1}\) matrices such that \(\sigma (A)\cup \sigma (B)\subset\mathbb{D}\). Then \( \left\vert \left\vert \left\vert f(A)g(B)+I\right\vert \right\vert \right\vert \leq \frac{2\sqrt{2}}{d_{A}d_{B}} \left\vert\left\vert \left\vert\,|AB|+I\,\right\vert \right\vert \right\vert \) for every unitarily invariant norm \(\left\vert \left\vert \left\vert \cdot \right\vert \right\vert \right\vert. \)

Corollary 3. If \(A\in \mathcal{B}(\mathcal{H})\) is self-adjoint and \(f\) is a continuous complex function on \(\sigma(A)\), then \(f(UAU^*)=Uf(A)U^*\) for all unitaries \(U\).

Proof. By the Stone-Weierstrass theorem, there is a sequence \((p_n)\) of polynomials uniformly converging to \(f\) on \(\sigma(A)\). Hence, \[f(UAU^*)=\lim_np_n(UAU^*)=U(\lim_np_n(A))U^*=Uf(A)U^*.\] We note that \(\sigma(UAU^*)=\sigma(A)\).

We conclude this section by presenting some inequalities involving the Hilbert-Schmidt norm \(\|\cdot\|_2.\)

Theorem 2. Let \(A,B\in\mathbb{M}_n\) be Hermitian matrices satisfying \(\sigma(A)\cup \sigma(B)\subset \mathbb{D}\) and let \(f, g\in \mathfrak{H}\). Then \( \|f(A)X\pm Xg(B)\|_2\leq \left\|\frac{X+|A|X}{d_A}+\frac{X+X|B|}{d_B}\right\|_2. \)

Proof. Let \(A=UD(\nu_j)U^*\) and \(B=VD(\mu_k)V^*\) be the spectral decomposition of \(A\) and \(B\) and let \(Y=U^*XV:=[y_{jk}].\) Noting that \(|e^{i\alpha}-\lambda_j|\geq d_A\) and \(|e^{i\beta}-\mu_k|\geq d_B,\) we have from [7] that \begin{align*} \|f(A)X\pm Xg(B)\|_2^2&=\sum_{j,k}|f(\lambda_j)\pm g(\mu_k)|^2|y_{jk}|^2\\&\leq\sum_{j,k}\left(\frac{1+|\lambda_j|}{d_A}+\frac{1+|\mu_k|}{d_B}\right)^2|y_{jk}|^2\\ &=\left\|\frac{X+|A|X}{d_A}+\frac{X+X|B|}{d_B}\right\|_2^2, \end{align*} which completes the proof.

3. Operators in function spaces

In this section, we present some results on operator valued functions. From [5], if \((\Omega,\mathcal{M,}\mu)\) is a measure space, for a \(\sigma\)-finite measure \(\mu\) on \(\mathcal{M}\), the mapping \(\mathcal{A}:\Omega\rightarrow \mathcal{B(H)}\) will be called \([\mu]\) weakly\(^{*}\)-measurable if a scalar valued function \(t \rightarrow tr (A_{t} Y)\) is measurable for any \(Y\in\mathcal{C}_{1}(\mathcal{H})\). Moreover, if all these functions are in \(L^{1}(\Omega, \mu)\), then since \(\mathcal{B(H)}\) is the dual space of \(\mathcal{C}_{1}(\mathcal{H})\), for any \(E\in \mathcal{M}\), we have the unique operator \(I_{E}\in \mathcal{B(H)}\), called the Gel'fand or weak \(^*\)-integral of \(\mathcal{A}\) over \(E\), such that We denote it by \(\int_EA_{t}d\mu(t)\) or \(\int_E A d\mu.\) We consider the following important aspect.

Proposition 2. \(A:\Omega \rightarrow \mathcal{B(H)}\) is \([\mu]\) if and only if scalar valued functions \(t \rightarrow \langle A_{t} f,f\rangle\) are \([\mu]\) measurable (resp. integrable) for every \(f\in \mathcal{H}\).

Proof. Every one dimensional operator \(f^{*} \otimes f\) is in \(C_{1}(H)\) and \(tr(A_{t}( f^{*} \otimes f))=tr(f^{*} \otimes A_{t} f)=\left< A_{t} f,f\right>,\) so that \([\mu]\) weak \(^*\)-measurability (resp. \([\mu]\) weak \(^*\)-integrability) of \(A\) directly implies measurability (resp. integrability) of \(\left< A_{t} f,f\right>\) for any \(f\in \mathcal{H}\). The converse follows immediately from [4] and this completes the proof.

We note that in view of Proposition 2, the Equation (5) of Gel'fand integral for \(o.v.\) functions can be reformulated as follows [2]:

Proposition 3. If \(\left< A f,f\right>\in L^1(E,\mu)\) for all \(f\in \mathcal{H}\), for some \(E\in \mathcal{M}\) and a \(\mathcal{B(H)}\)-valued function \(A\) on \(E\), then the mapping \(f\rightarrow\int_E \left< A_{t} f,f\right>d\mu(t)\) represents a quadratic form of bounded operator \(\int_E A dm\) or \(\int_E A_t d\mu(t)\), satisfying the following \( \left< \left(\int_E A_t d\mu (t)\right) f,g\right>= \int_E \left< At f,g\right>\,d\mu (t), for\;\; all\;\; f,g\in \mathcal{H}.\)

Proof. It suffices to show that for all \(E\in \mathcal{M},\) \( \Phi_E(f,g)=\int_E \left< A_{t} f,g\right>\,d\mu (t),\) for all \(f, g\in \mathcal{H}\), defines a bounded sesquilinear functional \(\Phi\) on \(\mathcal{H}\). Indeed, by [1], we have \( | \Phi_E(f,g)| \le \int_E|\left< A_{t} f,g\right>|\,d\mu (t) \le \| A_{t} f,g\|_{L^1} \le M \|f\|\|g\| \) for all \(f,g\in \mathcal{H}\) since integration is a contractive functional on \(L^{1}(\Omega ,\mu)\). This completes the proof.

Remark 2. It is known from [1] that for a \([\mu]\) \(A:\Omega \rightarrow \mathcal{B(H)}\) we have that \(A^*A\) is Gel'fand integrable if and only if \( \int_\Omega \|A_t f\|^2d\mu (t)< \infty,\) for all \(f\in \mathcal{H}\). Moreover, for a \([\mu]\) function \(A:\Omega \rightarrow \mathcal{B(H)}\). Let us consider a linear transformation \(\vec{A}:D_{\vec{A}}\rightarrow L^{2}(\Omega,\mu , \mathcal{H})\), with the domain \(D_{\vec{A}}=\{ f\in \mathcal{H} \,| \, \int_\Omega \|A_t f\|^2 d\mu (t)< \infty\}\), defined by \( ({\vec{A}}f)(t)=A_t f .\) and all \(f\in D_{\vec{A}}.\)

In the next section, we devote our efforts to results on inner product type integral transformers in terms of Landau, Cauchy-Schwarz and Grüss type norm inequalities.

4. Norm inequalities

In this section, we consider various types of norm inequalities for inner product type integral transformers discussed in [1,2,4,7]. From [1], a sufficient condition is provided when \(A^*\) and \(B\) from Definition 2 are both in \(L^2_G(\Omega,d\mu, \mathcal{B(H)}).\) If each of families \((A_t)_{t\in\Omega}\) and \((B_t)_{t\in\Omega}\) consists of commuting normal operators, then by Theorem 3.2 [1], the \(i.p.t.i\) transformer \(\int_\Omega A_t \otimes B_t d\mu(t)\) leaves every \(u.i.\) norm ideal \(\mathcal{C}_{|\|\cdot|\|}(\mathcal{H})\) invariant and the following Cauchy-Schwarz inequality holds: for all \(X\in \mathcal{C}_{|\|\cdot|\|}(\mathcal{H})\). Normality and commutativity condition can be dropped for Schatten \(p\)-norms as shown in Theorem 3.3 [1]. In Theorem 3.1 [2], a formula for the exact norm of the \(i.p.t.i\) transformer \(\int_\Omega A_t \otimes B_td\mu(t)\) acting on \(\mathcal{C}_2(\mathcal{H})\) is found. In Theorem 2.1 [2], the exact norm of the \(i.p.t.i\) transformer \(\int_\Omega A_t^* \otimes A_t d\mu(t)\) is given for two specific cases: \begin{equation} \left\| \int_\Omega A_t^*\otimes A_t d\mu(t) \right\|_{\mathcal{C}_{\Phi}(\mathcal{H})\to\mathcal{C}_1(\mathcal{H})}= \left\| \int_\Omega A_t A_t^* d\mu(t) \right \|_{\mathcal{C}_{\Phi_*}(\mathcal{H})}, \nonumber%\label{biloshtaunuklearne} \end{equation} where \(\Phi_*\) stands for a \(s.g.\) function related to the dual space \((\mathcal{C}_{\Phi}(\mathcal{H}))^*\). The norm appearing in (7) and its associated space \(L_G^2(\Omega,d\mu,\mathcal{B(H)},\mathcal{C}_\Phi(\mathcal{H}))\) present only a special case of norming a field \(A=(A_t)_{t\in\Omega}\). A much wider class of norms \( \| \cdot\|_{\Phi,\Psi}\) and their associated spaces \(L_G^2(\Omega,d\mu,\mathcal{B(H)},\mathcal{C}_\Phi(\mathcal{H}))\) are given by [2]: for an arbitrary pair of \(s.g.\) functions \(\Phi\) and \(\Psi\). For the proof of completeness of the space \(L_G^2(\Omega,d\mu,\mathcal{C}_\Phi(\mathcal{H}), \mathcal{C}_\Psi(\mathcal{H}))\), see Theorem 2.2 [2]. Before going into the details of this section lets consider the following Proposition which will be useful in the sequel [7]. We give its proof for completion.

Proposition 4. Let \(\mu\) be a probability measure on \(\Omega\), then for every field \((\mathcal{A}_t)_{t\in\Omega}\) in \(L^2(\Omega,\mu,\mathcal{B}(\mathcal{H}))\), for all \(B\in\mathcal{B}(\mathcal{H})\), for all unitarily invariant norms \(|\|\cdot|\|\) and for all \(\theta>0\),

Thus, the considered minimum is always obtained for \(B=\int_\Omega A_t d\mu(t)\).

Proof. The expression (9) is trivial and the inequality (10) follows from (9), while identity (10) is just a a special case of Lemma 2.1 [1] applied for \(k=1\) and \(\delta_1=\Omega\).

As \(0\le A\le B\) for \(A,B\in \mathcal{C_{\infty}(H)}\) implies \( s_n^\theta(A)\le s_n^\theta(B)\) for all \(n\in \mathbb{N}\), as well as \(|\| A^\theta|\|\le |\| B^\theta|\|,\) then (12) follows.

Recall that, for a pair of random real variables \((Y,Z)\), its coefficient of correlation

\[\rho_{Y,Z}=\frac{| E(YZ)-E(Y)E(Z)|}{\sigma(Y)\sigma(Z)}= \frac{| E(YZ)-E(Y)E(Z)|}{ \sqrt{E(Y^2)-E^2(Y)} \sqrt{E(Z^2)-E^2(Z)}}\] always satisfies \(|\rho_{Y,Z}|\le 1.\) For square integrable functions \(f\) and \(g\) on \([0,1]\) and \(D(f,g)=\int_0^1f(t)g(t)\,d t- \int_0^1f(t)\,d t\int_0^1g(t)\,d t.\) Landau proved that \( | D(f,g)|\le \sqrt{D(f,f)D(g,g)}.\)

The following result represents a generalization of Landau inequality in \(u.i.\) norm ideals [2] for Gel'fand integrals of \(o.v.\) functions with relative simplicity of its formulation.

Theorem 3. If \(\mu\) is a probability measure on \(\Omega\). Let both fields \((A_t)_{t\in\Omega}\) and \((B_t)_{t\in\Omega}\) be in \(L^2(\Omega,\mu,\mathcal{B(H)})\) consisting of commuting normal operators and consider \[\sqrt{\,\int_\Omega|A_{t}|^2 -\left|\int_\Omega A_{t} d\mu(t)\right|^2}X \sqrt{\,\int_\Omega| B_{t}|^2 d\mu(t)-\left|\int_\Omega B_{t} d\mu(t)\right|^2},\] for some \(X\in B(H)\). Then \[\int_\Omega A_tX B_t d\mu(t)-\int_\Omega A_{t} dt X\!\!\int_\Omega B_{t} d\mu(t) \in C_{|\|.|\|}(H).\]

Proof. First, we have the following Korkine type identity for \(i.p.t.i\) transformers

In this representation, we have \((A_s-A_t)_{(s,t)\in\Omega^2}\) and \((B_s-B_t)_{(s,t)\in\Omega^2}\) to be in \(L^2(\Omega^2,\mu\times\mu, \mathcal{B(H)})\) because by an application of the identity (13), Both families \((A_s-A_t)_{(s,t)\in\Omega^2}\) and \((B_s-B_t)_{(s,t)\in\Omega^2}\) consist of commuting normal operators and by Theorem 3.2 [1] \[\dfrac12\int_{\Omega^2}(A_s-A_t)X(B_s-B_t)d(\mu\times\mu)(s,t)\in \mathcal{C}_{|\|\cdot| |\|}(\mathcal{H})\] due to identity (14), and so the conclusion (13) follows.

Lemma 2. Let \(\mu\) (resp. \(\nu\)) be a probability measure on \(\Omega\) (resp. \(\mho\)). Further, let both families \(\{A_s,C_t\}_{(s,t)\in\Omega\times\mho}\) and \(\{B_s, D_t\}_{(s,t)\in\Omega\times\mho}\) consist of commuting normal operators and let \begin{equation}\sqrt{\,\int_\Omega|A_s|^2d\mu(s) \int_\mho|C_t|^2d\nu(t)-\left|\int_\Omega A_sd\mu(s)\int_\mho C_td\nu(t)\right|^2} X \sqrt{\,\int_\Omega| B_s|^2d\mu(s) \int_\mho|D_t|^2d\nu(t)-\left|\int_\Omega B_sd\mu(s)\int_\mho D_td\nu(t)\right|^2} \end{equation} be in \(\mathcal{C}_{|\|\cdot| |\|}(\mathcal{H})\) for some \(X\in \mathcal{B(H)}\). Then \begin{eqnarray*}\int_\Omega \int_\mho A_s C_tX B_s D_t\,d\mu(s)\,d\nu(t) -\int_\Omega A_s \,d\mu(s)\int_\mho C_t\,d\nu(t) X\!\!\int_\Omega B_s \,d\mu(s) \int_\mho D_t\,d\nu(t) \in\mathcal{C}_{|\|\cdot| |\|}(\mathcal{H}).\end{eqnarray*}

Proof. Apply Theorem 3 to the probability measure \(\mu\times\nu\) on \(\Omega\times\mho\) and families \((A_s C_t)_{(s,t)\in\Omega\times\mho}\) and \(( B_s D_t)_{(s,t)\in\Omega\times\mho}\) of normal commuting operators in \(L_G^2(\Omega\times\mho,d\mu\times\nu,\mathcal{B(H)}).\) The rest follows trivially.

Next, we consider Landau type inequality for \(i.p.t.i\) transformers in Schatten ideals for the Schatten \(p\)-norms.

Proposition 5. Let \(\mu\) be a probability measure on \(\Omega\) and \((A_t)_{t\in\Omega}\) and \((B_t)_{t\in \Omega}\) be \(\mu\)-weak\({}^*\) measurable families of bounded Hilbert space operators such that \(\int_\Omega\left(\|A_t f\|^2+\|A_t^* f\|^2+\| B_t f\|^2+\|B_t^* f\|^2\right)d\mu(t)< \infty\) for all \(f\in \mathcal{H}\) and let \(p,q,r\ge1\) such that \(\dfrac1p=\dfrac1{2q}+\dfrac1{2r}\,\). Then for all \(X\in \mathcal{C}_p(\mathcal{H})\),

Proof. According to identity (14), applying Theorem 3.3 [1] to families \((\mathcal{A}_s-\mathcal{A}_t)_{(s,t)\in\Omega^2}\) and \((\mathcal{B}_s-\mathcal{B}_t)_{(s,t)\in\Omega^2}\) gives

By applying identity (14) once again, the last expression in (16) becomes \[\bigg\|\biggl(\dfrac12\int_{\Omega^2}(\mathcal{A}_s-\mathcal{A}_t)^* \left(\int_\Omega\left|\mathcal{A}_t^*-\int_\Omega \mathcal{A}^* _t d\mu(t)\right|^2d\mu(t)\right)^{q-1} (\mathcal{A}_s-\mathcal{A}_t)d(\mu\times\mu)(s,t)\biggr)^{\frac1{2q}}\] \[\biggl(\dfrac12\int_{\Omega^2}(\mathcal{B}_s-\mathcal{B}_t) \Bigl(\int_\Omega\left|\mathcal{B}_s-\int_\Omega \mathcal{B} _t d\mu(t)\right|^2d\mu(s)\Bigr)^{r-1}\kern-5pt(\mathcal{B}_s-\mathcal{B}_t)^*d(\mu\times\mu)(s,t)\biggr)^{\frac1{2r}}\bigg\|_p.\label{odvizraz} \] Denoting \(\Bigl(\int_\Omega\left|A_s^*-\int_\Omega\mathcal{A}^*d\mu\right|^2d\mu(s)\Bigr)^{\frac{p-1}2}\) (resp. \(\Bigl(\int_\Omega\left|B_s-\int_\Omega\mathcal{B}d\mu\right|^2d\mu(s)\Bigr)^{\frac{r-1}2}\)) by \(Y\) (resp. \(Z\)), then the expression (16) becomes Again applying identity (14) to families \((Y A_t)_{t\in\Omega}\) and \((Z B_t^*)_{t\in\Omega}\), (17) becomes \[\left\|\left(\int_\Omega\left|Y\mathcal{A}_t-\int_\Omega Y\mathcal{A}_t d\mu(t)\right|^2d\mu(t)\right)^{\frac1{2q}}\kern-4pt . \left(\int_\Omega\left|Z\mathcal{B}_t^*-\int_\Omega Z\mathcal{B}_t^* d\mu(t)\right|^2d\mu(t)\right)^{\frac1{2r}}\right\|_p,\] which obviously equals to the righthand side of (15).

The next result [1] is a special case of an abstract Hölder inequality presented in Theorem 3.1.(e) [1] for Cauchy-Schwarz inequality for \(o.v.\) functions in \(u.i.\) norm ideals.

Proposition 6. Let \(\mu\) be a measure on \(\Omega\). Further, let \((A_t)_{t\in\Omega}\) and \((B_t)_{t\in \Omega}\) be \(\mu\)-weak\({}^*\) measurable in \(\mathcal{B(H)}\) such that \(|\int_\Omega|A_t|^2 d\mu(t)|^\theta\) and \(|\int_\Omega|B_t|^2 d\mu(t)|^\theta\) are in \(\mathcal{C}_{\||.|\|}\mathcal{H}\) for some \(\theta>0\) and for \(u.i.\) norm. Then we have \[ \|||\int_\Omega A_t^* B_t d\mu(t)\|||^\theta \|||\le \|||\int_\Omega A_t^* A_t d\mu(t)\|||^\theta \|||^\frac12 \|||\int_\Omega B_t^* B_t d\mu(t)\|||^\theta \|||^\frac12. \]

Proof. Take \(\Phi\) to be a \(s.g.\) function that generates \(u.i.\) norm \(\||\cdot\||\), \(\Phi_1=\Phi\), \(\Phi_2=\Phi_3=\Phi^{(2)}\) (2-reconvexization of \(\Phi\)), \(\alpha=2\theta\) and \(X=I\), and then apply Theorem 3.1 [1], we get our desired result.

Now, we give another generalization of Landau inequality for Gel'fand integrals of \(o.v.\) functions in \(u.i.\) norm ideals.

Theorem 4. If \(\mu\) is a probability measure on \(\Omega\), \(\theta>0\) and \((A_t)_{t\in\Omega}\) and \((B_t)_{t\in \Omega}\) are as in Proposition 6, \(\mu\)-weak\({}^*\) measurable families of bounded Hilbert space operators such that \(\|||\int_\Omega|A_t|^2d\mu(t)\|||^\theta\) and \(\|||\int_\Omega|B_t|^2d\mu(t)\|||^\theta\) are in \(\mathcal{C}_{\||.|\|}\mathcal{H}\) for some \(\theta>0\) and for some \(u.i.\) norm \(\||\cdot\||\) we have \begin{eqnarray*} && \left\|\left|\int_\Omega A_t^* B_td\mu(t) -\int_\Omega A_t^*d\mu(t)\int_\Omega B_td\mu(t) \|||^\theta \right\|\right|^2\\ &&\le \||\int_\Omega \||| A_t \|||^2d\mu(t)- \|||\int_\Omega A_td\mu(t) \|||^2 \|||^\theta \||| \|||\int_\Omega \||| B_t \|||^2d\mu(t)- \|||\int_\Omega B_td\mu(t) \|||^2 \|||^\theta \||.\end{eqnarray*}

Proof. It suffices to invoke Proposition 6 to \(o.v.\) families \((A_s-A_t)_{(s,t)\in\Omega^2}\) and \((B_s-B_t)_{(s,t)\in\Omega^2}\) and use identity [7] to proof this result.

Now, we consider some interesting quantities that relate to norm inequalities. For bounded set of operators \(A=(\mathcal{A}_t)_{t\in\Omega}\), we see that the radius of the smallest disk that essentially contains its range is

\[r_\infty(A)=\inf_{A\in \mathcal{B(H)}}ess \sup_{t\in\Omega}\| A_t-A\|= \inf_{A\in \mathcal{B(H)}}\| A_t-A\|_\infty=\min_{A\in \mathcal{B(H)}}\| A_t-A\|_\infty.\] From the triangle inequality, we have \(\bigl|\|\mathcal{A}_t-A'\|-\|\mathcal{A}_t-A\|\bigr|\leq\|A'-A\|\), so the mapping \(A\to ess \sup_{t\in\Omega}\|A_t-A\|\) is nonnegative and continuous on \(\mathcal{B(H)}\). Since \((\mathcal{A}_t)_{t\in\Omega}\) is bounded field of operators, we also have \(\| A_{t}-A\|\to\infty\) when \(\|A\|\to\infty\), so this mapping attains minimum [5], and it actually attains at some \(A_0\in \mathcal{B(H)}\), which represents a center of the disk considered [6]. Any such field of operators is of finite diameter, therefore, we have that \(r_\infty(A)=ess \sup_{s,t\in\Omega}\| A_s-A_t\|,\) with the simple inequalities given as \(r_\infty(A)\le diam_\infty(A)\le 2r_\infty(A)\) relating those quantities. For such fields of operators we can now state the following stronger version of Grüss inequality [2].

Lemma 3. Let \(\mu\) be a \(\sigma\)-finite measure on \(\Omega\) and let \(A=(\mathcal{A}_t)_{t\in\Omega}\) and \(B=(\mathcal{B}_t)_{t\in\Omega}\) be \([\mu]\) a.e. bounded fields of operators. Then, for all \(X\in \mathcal{C_{|\|.|\|}(H)}\), \( \sup_{\mu(\delta)>0}\||\frac1{\mu(\delta)}\int_\delta\mathcal{A}_tX\mathcal{B}_t d\mu(t) - \frac1{\mu(\delta)}\int_\delta\mathcal{A}_t d\mu(t) \,X \frac1{\mu(\delta)}\int_\delta\mathcal{B}_t d\mu(t) |\|\le \min_{i} \mathcal{P}_{i}\cdot\|| X\||.\) (Here \(\sup\) is taken over all measurable sets \(\delta\subseteq\Omega\) such that \(0< \mu(\delta)< \infty\)).

Lemma 3 has the following immediate implication when \((\mathcal{A}_t)_{t\in\Omega}\) and \((\mathcal{B}_t)_{t\in\Omega}\) are bounded fields of self-adjoint operators.

Theorem 5. If \(\mu\) is a probability measure on \(\Omega\) and \(C,D,E,F\) be bounded self-adjoint operators. Also, let \((\mathcal{A}_t)_{t\in\Omega}\) and \((\mathcal{B}_t)_{t\in\Omega}\) be bounded self-adjoint fields satisfying \(C\le\mathcal{A}_t\le D\) and \(E\le\mathcal{B}_t\le F\) for all \(t\in\Omega\). Then for all \(X\in \mathcal{C_{|\|.|\|}(H)}\), we have

Proof. As \(\frac{C-D}2\le\mathcal{A}_t-\frac{C+D}2\le\frac{D-C}2\) for every \(t\in\Omega\), then \begin{eqnarray*} ess \sup_{t\in\Omega}\| \mathcal{A}_t-\frac{C+D}2\|= ess \sup_{t\in\Omega}\sup_{\| f\|=1}\||\langle\mathcal{A}_t-\frac{C+D}2 \| f,f\rangle|\le \sup_{\| f\|=1}|\langle\frac{D-C}2 f,f\rangle|= \frac{\| D-C\|}2, \end{eqnarray*} which implies \(r_\infty(A)\le \frac{\| D-C\|}2,\) and similarly \(r_\infty(B)\le\frac{\| F-E\|}2.\) Thus (18) follows directly.

In case of \(\mathcal{H}=\mathbb{C}\) and \(\mu\) being the normalized Lebesgue measure on \([a,b]\) (i.e. \(d\,\mu(t)=\frac{dt}{b-a}\)), then (1) follows from Theorem 5. This special case also confirms the sharpness of the constant \(\frac14\) in the inequality (18).

Lastly, we consider, the Grüss type inequality for elementary operators in the example below.

Example 1. Let \(A_1, \ldots, A_n\), \(B_1, \ldots, B_n\), \(C, D, E\) and \(F\) be bounded linear self-adjoint operators acting on a Hilbert space \(\mathcal{H}\) such that \(C\le A_i\le D\) and \(E\le B_i\le F\) for all \(i=1,2,\cdots,n\), then for arbitrary \(X\in\mathcal{C}_{\||.|\|}\mathcal{H}\), we have \begin{eqnarray} \nonumber%\label{mainc} \|| \frac1n\sum_{i=1}^n A_i XB_i-\frac1{n^2}\sum_{i=1}^nA_i\, X\sum_{i=1}^nB_i\|| \leq \frac{\|D-C\|\|F-E\|}4 \|| X\||. \end{eqnarray} Indeed, it is sufficient to prove that the elementary operator is normally represented and that Grüss type inequality holds for it [3].

In the next section, we dedicate our effort to the applications of this study in other fields. We consider quantum theory in particular, whereby, we describe the application in quantum chemistry and quantum mechanics.

5. Applications in quantum theory

Norm inequalities and other properties of \(i.p.t.i\) transformers have various applications in other fields. We discuss the applications in quantum theory involving two cases [3]. The first case is in quantum chemistry, whereby, we consider the Hamiltonian which is a bounded, self-adjoint operator on some infinite-dimensional Hilbert space which governs a quantum chemical system. The Hamiltonian helps in estimation of ground state energies of chemical systems via subsystems.

The quantum mechanics deals with commutator approximation. The discussions of approximation by commutators \(AX-XA\) or by generalized commutator \(AX-XB\) originates from quantum theory. For instance, the Heisenberg uncertainly principle may be mathematically deduced as saying that there exists a pair \(A,X\) of linear operators and a non-zero scalar \(\alpha\) for which \(AX - XA = \alpha I\). A natural question immediately arises: How close can \(AX - XA\) be to the identity? In [3], it is discussed that if \(A\) is normal, then, for all \(X \in B(H)\), \(||I - (AX - XA)|| \geq ||I||.\) In the inequality here, the zero commutator is a commutator approximate in \(B(H)\).

Acknowledgments

The author is grateful to the referees for the useful comments

Conflicts of Interest

The author declares no conflict of interest.

1. Introduction

Fractional derivatives and fractional order differential equations have received an increasingly high interest and attention due to its physical and modeling applications in science, engineering and mathematics[1,2,3,4]. On the other hand, researches have shown that two-point boundary value problems have found their applications in theoretical physics, applied mathematics, optimization and control theory and engineering [5]. The use of second-order Boundary Value Problem (BVP) arises in several areas of engineering and applied sciences such as celestial mechanics, circuit theory, astrophysics, chemical kinetics, and biology [6].

They are particularly encountered in the study of following natural phenomena: in the study of surface-tension-induced flows of a liquid metal or semiconductor [7,8], used in modeling biological materials (elastic and hyperelastic materials) [9]. In [10], authors studied the existence and uniqueness of a nontrivial solution of a two-point boundary value problems. Stochastic nonlinear fractional order differential equation and its associated BVPs, on the other hand, will undoubtedly give more realistic models for the above natural occurrences.

Motivated by the above applications and the results of the papers [2,3,4], we study the white noise pertubation of a nonlinear BVP and consider the following BVP for a stochastic nonlinear fractional differential equation

where \(\lambda>0\) is a noise level parameter, \(\sigma:[0,1]\times\mathbb{R}\rightarrow\mathbb{R}\) is continuous, \(\dot{w}(t)\) is a Gaussian white noise, \(D^\alpha=D^\alpha_{0_+}\) is the Riemann-Liouville (R-L) fractional derivative of order \(\alpha\in (3, 4)\) and \(I^\beta\) is the fractional integral of order \(\beta>0\). Existence and uniqueness results are very crucial in the study of BVP since they are not well behaved like the initial value problems, therefore, we aim to establish the existence and uniqueness of solution to Equation (1). To the best of our knowledge, the above model has not been studied before and we therefore, try to make sense of the solution to the above equation as follows:

Definition 1. We say that \(\{u(t)\}_{0\leq t\leq 1}\) is a mild solution to Equation (1) if \(a. s\), the following is satisfied \begin{eqnarray*} u(t)=\lambda\int_0^1 G(t,s)\sqrt{I^\beta[\sigma^2(s,u(s))]}\dot{w}(s)ds=\lambda\int_0^1 G(t,s)\sqrt{I^\beta[\sigma^2(s,u(s))]}dw(s), \end{eqnarray*} where \(G(t,s)\) is as defined in (3).

If \(\{u(t)\}_{0< t< 1}\) satisfies the additional condition \(\sup_{0\leq t\leq 1}{\mathbb E}|u(t)|^2< \infty,\) then we say that \(\{u(t)\}_{0\leq t\leq 1}\) is a random field solution to Equation (1).

Definition 2. The R-L fractional integral of order \(\beta>0\) of a given continuous function \(f:(0,\infty)\rightarrow\mathbb{R}\) is defined by \[I^\beta f(t)=\frac{1}{\Gamma(\beta)}\int_0^t(t-s)^{\beta-1}f(s)ds,\] provided the right integral exists. Denote \(I^0 f(t)=f(t).\)

Definition 3. The R-L fractional derivative of order \(\alpha>0\) of a given continuous function \(f:(0,\infty)\rightarrow\mathbb{R}\) is defined by \[D^\alpha f(t)=\frac{1}{\Gamma(n-\alpha)}\bigg(\frac{d}{dt}\bigg)^n\int_0^t\frac{f(s)}{(t-s)^{\alpha-n+1}}ds,\] where \(n=[\alpha]+1\), provided the right integral converges.

Many authors studied different boundary value problems of nonlinear fractional order differential equations, see [1,11,12,13,14] and their references. Xu in [3] considered the following boundary value problem of a nonlinear fractional differential equation:

Lemma 1 ([3]). Given \(h\in C[0,1]\) and \(3< \alpha\leq 4\), the unique solution of \begin{equation*} \begin{cases} D^\alpha u(t)=h(t),\,\,0< t< 1\\ u(0)=u(1)=u'(0)=u'(1)=0, \end{cases}\end{equation*} is \(u(t)=\displaystyle\int_0^1 G(t,s)h(s)ds\) where

Lemma 2 ([4]). The Green function \(G(t,s)\) in (2) has the following properties

• (1) \(G(t,s)=G(1-s,1-t)\) for \(t,s\in[0,1];\)
• (2) \(t^{\alpha-2}(1-t)^2q(s)\leq G(t,s)\leq (\alpha-1)q(s)\) and \(G(t,s)\leq \frac{(\alpha-1)(\alpha-2)}{\Gamma(\alpha)}t^{\alpha-2}(1-t)^2\) for \(t,s\in[0,1]\) where \(q(s)=\frac{(\alpha-2)}{\Gamma(\alpha)}s^2(1-s)^{\alpha-2}.\)

Xu in [3] gave more properties of the function \(G(t,s)\) in (2) as follows:

Lemma 3([3]). The Green function \(G(t,s)\) defined by (2) satisfies the following conditions:

• (1) \(G(t,s)=G(1-s,1-t)\) for \(t,s\in(0,1)\);
• (2) \((\alpha-2)t^{\alpha-2}(1-t)^2s^2\leq\Gamma(\alpha)G(t,s)\leq M_0 s^2(1-s)^{\alpha-2}\), for \(t,s\in(0,1)\);
• (3) \(G(t,s)>0\), for \(t,s\in(0,1);\)
• (4) \((\alpha-2)s^2(1-s)^{\alpha-2}t^{\alpha-2}(1-t)^2\leq\Gamma(\alpha)G(t,s)\leq M_0 t^{\alpha-2}(1-t)^2\), for \(t,s\in(0,1)\), where \(M_0=\max\{\alpha-1,(\alpha-2)^2\}\).

Stanek in [2] studied the following fractional boundary value problems:

Lemma 4([2]). Suppose that \(\rho\in L^1[0,1]\). Then \(u(t)=\int_0^1 G(t,s)\rho(s)ds\) for \(t\in [0,1]\) is the unique solution of the following equation in \(C^1[0,1]\): \begin{equation*} \begin{cases} D^\alpha u(t)+\rho(t)=0,&\alpha\in(2,3)\\ u(0)=0,\,\, u'(0)=u'(1)=0,& \end{cases} \end{equation*} where

Lemma 5([2]). The function \(G(t,s)\) given in (3) has the following properties:

• (1) \(G\in C([0,1]\times [0,1])\) and \(G^\alpha>0\) on \((0,1)\times (0,1)\);
• (2) \(G(t,s)\leq\frac{1}{\Gamma(\alpha)}\) for \((t,s)\in [0,1]\times [0,1]\);
• (3) \(\displaystyle\int_0^1 G(t,s)ds\geq\frac{t^{\alpha-1}}{(\alpha-1)\Gamma(\alpha+1)}\) for \(t\in[0,1]\);
• (4) \(\frac{\partial}{\partial t}G(t,s)\in C([0,1]\times [0,1])\) and \(\frac{\partial}{\partial t}G(t,s)>0\) on \((0,1)\times (0,1)\);
• (5) \(\frac{\partial}{\partial t}G(t,s)\leq\frac{1}{\Gamma(\alpha-1)}\) for \((t,s)\in [0,1]\times [0,1]\);
• (6) \(\displaystyle\int_0^1 \frac{\partial}{\partial t}G(t,s)ds\geq\frac{t(1-t)}{\Gamma(\alpha)}\) for \(t\in[0,1]\).

The paper is outlined as follows. A short preliminary on second order boundary problems is given in Section 2. In Section 3, we gave the main results and their proofs while Section 4 contains a concise summary of the paper.

2. Preliminaries

Let \((a,b)\in\mathbb{R}\) be an interval and \(p,q,f:(a,b)\rightarrow\mathbb{R}\) be continuous functions. Now, consider the linear second order equation given by \(y''+p(x)y'+q(x)=f(x)\) subject to the following boundary conditions:

• (a.) Dirichlet or first kind: \(y(a)=\eta_1,\,\,y(b)=\eta_2\),
• (b.) Neuman or second kind: \(y'(a)=\eta_1,\,\,y'(b)=\eta_2\),
• (c.) Robin or third or mixed kind: \(\alpha_1 y(a)+\alpha_2 y'(a)=\eta_1,\,\,\alpha_1 y(b)+\alpha_2 y'(b)=\eta_2\),
• (d.) Periodic: \(y(a)=y(b),\,\,y'(a)=y'(b)\).

Remark 1.

• 1. Unlike the initial value problems, the BVPs do not behave nicely because there are BVPs for which their solutions fail to exist and even when a solution does exist, there might be many of them. This of course makes existence and uniqueness for BVPs to fail generally.
• 2. For example, \(y''+y=0\) has a unique solution \(y(x)=\sin x+\cos x\) with the boundary conditions \(y(0)=y(\frac{\pi}{2})=1\). It has no solution with the boundary conditions \(y(0)=y(\pi)=1\) and has infinitely many solutions with the boundary conditions \(y(0)=y(2\pi)=1\).

Thus, for the linear boundary value problem with the Dirichlet boundary conditions: where \(L\) is a second-order differential operator given by \(L=\frac{d}{dx}\bigg[p(x)\frac{d}{dx}\bigg]+q(x).\) The Equation (4) has the following integro-differential equation as its solution: \[u(x)=\int_a^b G(x,\xi)f(\xi,u(\xi))d\xi,\] with \(G(x,\xi)\) the Green's function.

Lemma 6([10]). Let \(y(t)\in X\), then the Boundary Value Problem \begin{eqnarray*} \left \{ \begin{array}{lll} u''(t)-y(t)=0,& 0< t< 1\\ u(0)=u(1)=0,& \end{array}\right. \end{eqnarray*} has a unique solution \(\displaystyle\int_0^1G(t,s)y(s)ds,\) where

is the Green's function of BVP \begin{eqnarray*} \left \{ \begin{array}{lll} u''(t)=0,& 0< t< 1\\ u(0)=u(1)=0.& \end{array}\right. \end{eqnarray*}

Remark 2. The Green function has the following bound: \(\displaystyle\sup_{0\leq t,s\leq 1}G(t,s)\leq\frac{1}{2}.\)

3. Main results

Now, consider the following global Lipschitz continuity condition on \(\sigma\):

Condition 1. Let there exist a nonnegative function \(p\in L^2[0,1],\) such that \(|\sigma(t,x)-\sigma(t,y)|\leq p(t)|x-y|,\,\,\forall\,t\in [0,1],\,x,y\in\mathbb{R},\) with \(\sigma(t,0)=0\) for convenience, and there exists \(t_0\in [0,1]\) such that \(p(t_0)\neq 0.\)

In particular, for \(p(t)=Lip_{\sigma}\) and for all \(t\in[0,1]\), we have the following:

Condition 2. There exist a finite positive constant \(Lip_{\sigma},\) such that for all \(x,\,y\in\mathbb{R}\), we have \( |\sigma(t,x)-\sigma(t,y)|\leq Lip_{\sigma}|x-y|, \) with \(\sigma(t,0)=0\) for convenience.

Now, define \(L^2({\mathbb P})\) norm of the solution \(u\) by \[\|u\|_2:=\bigg\{\sup_{0\leq t\leq 1}{\mathbb E}|u(t)|^2\bigg\}^{1/2}.\]

3.1. Proof of results for Equation (1)

Using Condition 2, we obtain the following results for Equation (1):

Theorem 1. Suppose \(\lambda< \frac{\sqrt{\beta(\beta+1)\Gamma(\beta)\Gamma^2(\alpha)}}{Lip_{\sigma}}\), where \(\alpha\in(3,4)\) and \(\beta>0\). For a positive constant \(Lip_{\sigma}\) together with Condition 2, there exists a solution \(u\) for Equation (1) that is unique up to modification.

To proof the Theorem 2, let \(u(t)=\mathcal{A} u(t)\), where the operator \(\mathcal{A}\) is given by

\[\mathcal{A} u(t)=\lambda\int_0^1 G(t,s)\sqrt{I^\beta[\sigma^2(s,u(s))]}dw(s),\] and we will use the fixed point of \(\mathcal{A}\). The proof follows using the Lemma(s) below:

Lemma 7. Given a random solution \(u\) such that \(\|u\|_2< \infty\) and Condition 2 holds. Then \[\|\mathcal{A} u\|^2_2\leq c_{\alpha,\beta}\lambda^2Lip_{\sigma}^2\|u\|_2^2,\] where \(c_{\alpha,\beta}=\frac{1}{\beta(\beta+1)\Gamma(\beta)\Gamma^2(\alpha)}.\)

Proof. Take second moment of both sides and use Itó isometry together with Lemma 5(2) to obtain \begin{eqnarray*} \mathbb{E}|\mathcal{A} u(t)|^2&=& \lambda^2\int_0^1G^2(t,s){\mathbb E}|\sqrt{I^\beta[\sigma^2(s,u(s))]}|^2ds\\ &\leq&\lambda^2\int_0^1G^2(t,s)\bigg[\frac{1}{\Gamma(\beta)}\int_0^s(s-r)^{\beta-1}{\mathbb E}|\sigma^2(r,u(r))|dr\bigg]ds\\ &\leq&\lambda^2Lip_{\sigma}^2\int_0^1G^2(t,s)\bigg[\frac{1}{\Gamma(\beta)}\int_0^s(s-r)^{\beta-1}{\mathbb E}|u(r)|^2dr\bigg]ds\\ &\leq&\lambda^2Lip_{\sigma}^2\sup_{0\leq s\leq 1}G^2(t,s)\|u\|_2^2\int_0^1\bigg[\frac{1}{\Gamma(\beta)}\int_0^s(s-r)^{\beta-1}dr\bigg]ds\\ &\leq&\lambda^2Lip_{\sigma}^2\bigg[\sup_{0\leq s\leq 1}G(t,s)\bigg]^2\|u\|_2^2\int_0^1\bigg[\frac{1}{\Gamma(\beta)}\int_0^s(s-r)^{\beta-1}dr\bigg]ds\\ &\leq&\frac{\lambda^2Lip_{\sigma}^2}{\Gamma^2(\alpha)\Gamma(\beta)}\|u\|_2^2\int_0^1\bigg[\int_0^s(s-r)^{\beta-1}dr\bigg]ds=\frac{\lambda^2Lip_{\sigma}^2}{\beta(\beta+1)\Gamma(\beta)\Gamma^2(\alpha)}\|u\|_2^2. \end{eqnarray*} Now, by taking supremum over \(t\in [0,1]\), the result follows.

Remark 3. The operator \(\mathcal{A}\) is a contraction for \(\frac{\lambda^2Lip_{\sigma}^2}{\beta(\beta+1)\Gamma(\beta)\Gamma^2(\alpha)}< 1\).

Lemma 8. Suppose \(u\) and \(v\) are random solutions such that \(\| u\|_{2}+\|v\|_{2}< \infty\) and Condition 2 holds. Then \[\|\mathcal{A} u-\mathcal{A} v\|^2_2\leq c_{\alpha,\beta}\lambda^2Lip_{\sigma}^2\|u-v\|_2^2.\]

Proof. The proof follows similarly to the proof of Lemma 7.

Now, we present the proof of Theorem 1.

Proof of Theorem 1. From Lemma 7, we have \[\|u\|^2_2=\|\mathcal{A} u\|^2_2\leq c_{\alpha,\beta}\lambda^2Lip_{\sigma}^2\|u\|_2^2,\] which follows that \[\|u\|^2_2\bigg[1-c_{\alpha,\beta}\lambda^2Lip_{\sigma}^2\bigg]\leq 0,\] and this shows that \(\|u\|_2< \infty\) whenever \(c_{\alpha,\beta}\lambda^2Lip_{\sigma}^2< 1\) or \( \lambda< \frac{1}{\sqrt{c_{\alpha,\beta}}Lip_{\sigma}}\).

Similarly from Lemma 8, we have

\[\|u-v\|^2_2=\|\mathcal{A} u-\mathcal{A} v\|^2_2\leq c_{\alpha,\beta}\lambda^2Lip_{\sigma}^2\|u-v\|_2^2,\] and therefore \[\|u-v\|^2_2\bigg[1-c_{\alpha,\beta}\lambda^2Lip_{\sigma}^2\bigg]\leq 0.\] This implies that \(\|u-v\|_2< 0\) if and only if \(c_{\alpha,\beta}\lambda^2Lip_{\sigma}^2< 1\) and thus \(\|u-v\|_2=0\Rightarrow u=v\). This shows the existence and uniqueness result by Banach's contraction principle.

3.2. Existence and uniqueness for stochastic second order BVP

Here, we consider the following stochastic second order boundary value problem: where \(\lambda>0\) is a noise level parameter, \(\sigma:[0,1]\times\mathbb{R}\rightarrow\mathbb{R}\) is continuous and \(\dot{w}(t)\) is a Gaussian white noise.

Definition 4. We say that \(\{u(t)\}_{0\leq t\leq 1}\) is a mild solution of Equation (6) if \(a. s\), \begin{equation*} u(t)=\lambda\int_0^1 G(t,s) \sigma(s,u(s))\dot{w}(s)ds=\lambda\int_0^1 G(t,s) \sigma(s,u(s))dw(s), \end{equation*} is satisfied, where \(G(t,s)\) is as given in Equation (5).

If \(\{u(t)\}_{0< t< 1}\) satisfies the additional condition \(\displaystyle\sup_{0\leq t\leq 1}{\mathbb E}|u(t)|^2< \infty,\) then we say that \(\{u(t)\}_{0\leq t\leq 1}\) is a random field solution to Equation (6).

Theorem 2. Suppose Condition 1 holds and there exists a positive constant \(\lambda^*\) such that for any \(0< \lambda\leq\lambda^*\), then Equation (6) has a unique solution.

To proof the Theorem 2, we define the operator

\[\mathcal{B} u(x,t)=\lambda\int_0^1 G(t,s) \sigma(s,u(s))dw(s),\] and use the fixed point of the operator \(\mathcal{B}\). The proof follows using the Lemma(s) below:

Lemma 9. Given a random solution \(u\) such that \(\|u\|_2< \infty\) and Condition 1 holds. Then there exists a positive constant \(\lambda^*\) such that for \(0< \lambda\leq\lambda^*\), \[\|\mathcal{B} u\|_2\leq \frac{1}{2}\|u\|_2.\]

Proof. By Itó Isometry, we obtain \begin{eqnarray*} {\mathbb E}|\mathcal{B} u(t)|^2&\leq&\lambda^2\int_0^1 G^2(t,s) {\mathbb E}|\sigma(s,u(s))|^2ds\\ &\leq&\lambda^2\sup_{0\leq t,s\leq 1}G^2(t,s)\int_0^1 p^2(s) {\mathbb E}|u(s)|^2ds\\ &\leq&\lambda^2\bigg[\sup_{0\leq t,s\leq 1}G(t,s)\bigg]^2\int_0^1 p^2(s){\mathbb E}|u(s)|^2ds\\ &\leq&\frac{\lambda^2}{4}\|u\|_2^2\int_0^1 p^2(s) ds. \end{eqnarray*} Taking suprimum of both sides over \(t\in[0,1]\) and letting \(\lambda^*:=\displaystyle\bigg(\int_0^1 p^2(s) ds\bigg)^{-2}\) such that \(0< \lambda\leq\lambda^*\), we have \(\|\mathcal{B} u\|^2_2\leq \frac{1}{4}\|u\|_2^2.\)

Following similar steps of Lemma 9, we obtain the following result.

Lemma 10. Suppose \(u\) and \(v\) are random solutions such that \(\| u\|_{2}+\|v\|_{2}< \infty\) and Condition 1 holds. Then there exists a positive constant \(\lambda^*\) such that for \(0< \lambda\leq\lambda^*\), \[\|\mathcal{B} u-\mathcal{B} v\|_2\leq \frac{1}{2}\|u-v\|_2.\]

Proof of Theorem 2. Following Lemma 9 and Lemma 12, it is clear that \(\mathcal{B}\) is a contraction. Thus by Banach fixed point theorem, the existence of a unique solution for Equation (6) follows: Let \(u=\mathcal{B}\) and \[\|u\|^2_2=\|\mathcal{B} u\|^2_2\leq \frac{1}{4}\|u\|_2^2\Rightarrow \|u\|^2_2\left[1-\frac{1}{4}\right]\leq 0\Rightarrow \|u\|_2=0\Rightarrow u=0.\] Assume a nontrivial solution \(u\) of Equation (6) and we show that it is unique. Suppose for contradiction that there exists another solution \(v\) of Equation (6) such that \[\|u-v\|^2_2=\|\mathcal{B} u-\mathcal{B} v\|^2_2\leq \frac{1}{4}\|u-v\|_2^2,\] then \(\|u-v\|^2_2[1-\frac{1}{4}]\leq 0\), which follows that \(\|u-v\|=0\). Thus \(u=v\), a unique solution.

Next, we seek to establish the existence and uniqueness of solution for Equation (6) using Condition 2 as follows:

Theorem 3. Suppose \(\lambda< \frac{2}{Lip_{\sigma}}\), for positive constant \(Lip_{\sigma}\) together with Condition 2. Then there exists solution \(u\) that is unique up to modification.

Lemma 11. Given a random solution \(u\) such that \(\|u\|_2< \infty\) and Condition 2 holds. Then \(\|\mathcal{B} u\|_2\leq\frac{\lambda Lip_{\sigma}}{2}\|u\|_2.\)

Lemma 12. Suppose \(u\) and \(v\) are random solutions such that \(\| u\|_{2}+\|v\|_{2}< \infty\) and Condition 2 holds. Then \(\|\mathcal{B} u-\mathcal{B} v\|_2\leq\frac{\lambda Lip_{\sigma}}{2}\|u-v\|_2.\)

Proof of Theorem 3. By fixed point theorem we have \(u(t)=\mathcal{A} u(t)\) and \( \|u\|^2_{2}=\|\mathcal{B} u\|^2_{2}\leq \frac{\lambda^2Lip_{\sigma}^2}{4}\|u\|^2_{2}, \) which follows that \(\|u\|^2_{2}\left[1-\frac{\lambda^2Lip_{\sigma}^2}{4}\right]\leq 0\Rightarrow \|u\|_{2}< \infty \Leftrightarrow \lambda< \frac{2}{Lip_{\sigma}}.\)

Similarly, \(\|u-v\|^2_{2}=\|\mathcal{B} u-\mathcal{B} v\|^2_{2}\leq \frac{\lambda^2Lip_{\sigma}^2}{4}\|u-v\|^2_{2},\) thus \(\|u-v\|^2_{2}\left[1-\frac{\lambda^2Lip_{\sigma}^2}{4}\right]\leq 0\) and therefore \(\|u-v\|_{2}< 0\) if and only if \(\lambda< \frac{2}{Lip_{\sigma}}.\) Hence, the existence and uniqueness result follows by Banach's contraction principle.

4. Conclusion

We studied the boundary value problems for both stochastic nonlinear fractional order differential equation and stochastic nonlinear second order equation. The existence and uniqueness result for both boundary value problems were given under different linearity conditions on \(\sigma\) using contraction fixed point theorem.

Acknowledgments

The first author wishes to acknowledge the continuous support of the University of Hafr Al Batin, Saudi Arabia.

Author Contributions

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

Conflicts of Interest

The authors declare no conflict of interest.

1. Introduction

Fixed point theory is an important tool in the study of nonlinear analysis as it is considered to be the key connection between pure and applied mathematics with wide applications in all branches of Mathematics, Economics, Biology, Chemistry, Physics and almost all engineering fields.

The famous Banach contraction principle is one of the powerful tools in metric fixed point theory and It has been extended and generalized in different directions by different researchers. The notion of metric space has been extended, improved and generalized in many different ways. Bakhtin [2] introduced a \( b \) - metric space as a generalization of metric space and investigated some fixed point theorem in such spaces. Hitzler [3] introduced the notion of dislocated metric spaces. Zeyada et al., [4] generalized the results of Hitzler [3] and introduced the concept of complete dislocated quasi metric space. Aage et al., [5] proved common fixed point theorem in dislocated quasi \( b \) - metric space. Zoto and Kumari [1] constructed theorems on common fixed point results on \( b \) - dislocated metric spaces and proved the existence and uniqueness.

In this research work, we concentrate in establishing and proving common fixed point results for a pair of maps satisfying \( s-\alpha \) contraction condition in the setting \( b \) - dislocated metric spaces.

2. Preliminaries

Throughout this manuscript \(\Re^{+}\) represents the set of non-negative real numbers and \(\mathbf{N}\) represents the set of natural numbers.

Definition 1. [6] Let \( X \) be nonempty set and a mapping \( d_{l}:X\times X\rightarrow\Re^{+} \) is called a dislocated or \( d_{l} \) - metric if the following conditions hold:

• (a) \( d_{l}(x,y)=0\Rightarrow x=y \) ;
• (b) \( d_{l}(x,y)=d_{l}(y,x) \) ;
• (c) \( d_{l}(x,y)\leq d_{l}(x,z)+d_{l}(z,y),\) for all \( x,y\in X \) .

Then the pair \( (X,d_{l}) \) is called a \(d_{l}\) - metric space.

Definition 2. [7] Let \( X \) be nonempty set and \( s\geq 1 \) be a real number, then a mapping \( b_{d}:X\times X\rightarrow \Re^{+}\) is called \( b \) - dislocated metric if the following conditions hold:

• (a) \( b_{d}(x,y)=0 \Rightarrow x=y \) ;
• (b) \( b_{d}(x,y)=b_{d}(y,x) \) ;
• (c) \( b_{d}(x,y)\leq s[b_{d}(x,z)+b_{d}(z,y)],\) for all \( x,y,z\in X \) .

Then the pair \( (X,b_{d}) \) is called a \( b \) - dislocated metric space.

Remark 1. The class of \( b \) - dislocated metric space is larger than that of dislocated metric space.

Definition 3. Let \((X,d)\) be a metric space and \(T : X \rightarrow X\) be a self-map, then \(T\) is said to be a contraction map if there exists a constant \(k \in [0, 1)\) such that \(d(Tx,Ty) \leq kd(x,y)\) for all \(x,y \in X.\)

Definition 4. [1] Let \( (X, b_{d}) \) be a complete \( b \) - dislocated metric space with parameter \( s\geq 1 .\) If \( T:X\rightarrow X \) is self-mapping that satisfy

for all \( x,y\in X \) and \( \alpha \in [ 0,\dfrac{1}{2}) \), then \( T \) is called a \( s-\alpha \) quasi-contraction.

Lemma 1. Let \( (X,b_{d}) \) be a \( b \) - dislocated metric space with parameter \( s\geq 1 .\) Suppose that \( \{x_n\}\) and \( \{y_n\} \) are \( b \) - dislocated convergent to \( x,y\in X \) respectively. Then we have \[\frac{1}{s^{2}}b_{d}(x,y)\leq\lim\limits_{n\to\infty}infb_{d}(x_{n},y_{n})\leq\lim\limits_{n\to\infty} \text{Sup}\,b_{d}(x_{n},y_{n})\leq s^{2}b_{d}(x,y).\] In particular, if \( b_{d}(x_{n},y_{n})=0 \), then we have \( \lim\limits_{n\to\infty} b_{d}(x_{n},y_{n})=0=b_{d}(x,y) .\) Moreover, if each \( z\in X \), we have \[\frac{1}{s}b_{d}(x,z)\leq\lim\limits_{n\to\infty}infb_{d}(x_{n},z)\leq\lim\limits_{n\to\infty} \text{Sup}\,b_{d}(x_{n},z)\leq sb_{d}(x,z).\] In particular, if \( b_{d}(x,z)=0 \) , then we have \( \lim\limits_{n\to\infty} b_{d}(x_{n},z)=0=b_{d}(x,z) \).

Theorem 1. [1] Let \( (X,b_{d}) \) be complete \( b \) - dislocated metric space with parameter \( s\geq 1 .\) If \( T:X\rightarrow X\ \) is a self-map that is a \( s-\alpha \) quasi contraction, then \( T \) has a unique fixed point in \( X .\)

Example 1. [1] Let \(X=[0,\infty)\) and \( b_{d}(x,y)=(x+y)^{2} \) for all \( x,y\in X .\) Then \( b_{d} \) is a \( b \) - dislocated metric on \( X \) with parameter \( s=2 \) and is complete.

Definition 5. [8] Let \( (X,b_{d}) \) be a \( b \) - dislocated metric space and \( \{x_n\} \) be a sequence of points in \( X .\) A point \( x \in X \) is said to be the limit of the sequence \( \{x_n\} \) if \(\lim\limits_{n \to\infty} d(x_n,\ x)=0 \) and we say that the sequence \( \{x_n\} \) is \( b \) - dislocated convergent to \( x \) and denote it by \( x_{n}\rightarrow x \) as \( n\rightarrow\infty .\)

Lemma 2. [7] The limit of a convergent sequence in a \( b \) - dislocated metric space is unique.

Definition 6. [7] A sequence \( \{x_n\} \) in a \( b \) - dislocated metric space \( (X,b_{d}) \) is called a \( b \) - dislocated Cauchy sequence if and only if given \( \epsilon> 0 ,\) there exists \( n_{0}\in\mathbb{N} \) such that for all \(n,m>n_{0},\) we have \( b_{d}(x_{n},x_{m})< \epsilon \) or \(\lim\limits_{n,m\to\infty} d(x_n\,\ x_m)=0 .\)

Lemma 3. [7] Every \( b \) - dislocated convergent sequence in \( b \) - dislocated metric space is a \( b \) - dislocated Cauchy.

Definition 7. [7] A \( b \) - dislocated metric space \( (X, b_{d}) \) is called complete if every \( b \) - dislocated Cauchy sequence in \( X \) is a \( b \) - dislocated convergent.

Definition 8. [6] Let \( T:X\rightarrow X \) and \( S:X\rightarrow X \) be self-maps in \((X,b_{d}).\) An element \(x \in X \) is said to be a coincidence point of \( T \) and \( S \) if and only if \( Tx=Sx=u .\) A point \( u \in X \) is point of coincidence of \( T \) and \( S .\)

Definition 9. [9] Let \( T \) and \( S \) be two self-maps on a metric space \( (X,d) .\) Then \( T \) and \( S \) are said to be weakly compatible if they commute at their coincident point; that is \( Tu=Su \) for some \( u \in X \) implies \( STu=TSu.\)

Definition 10. [10] Two self-maps \( T:X\rightarrow X \) and \( S:X\rightarrow X \) are said to be occasionally weakly compatible (OWC) if there exists some point \( u \in X \) such that \( Tu=Su \) and \( STu=TSu.\)

Remark 2. Clearly weakly compatible maps are occasionally weakly compatible. However the converse is not true in general.

Definition 11. [11] Let \( T:X\rightarrow X \) and \( S:X\rightarrow X \) be two self-maps on a metric space \( (X,d) .\) Then \( T \) and \( S \) are said to satisfy the common limit in the range of \( S \) property, denoted by (CLRs) if there exists a sequence \(\{x_n\} \in X \), such that \[\lim\limits_{n\to\infty} Tx_{n}=\lim\limits_{n\to\infty} Sx_{n}=Sx,\] for some \( x \in X .\)

Inspired and motivated by the result of Zoto and Kumari [1], the purpose of this research was to extend and generalize their main theorem to common fixed point theorem involving pairs of \( s-\alpha \) contraction condition in the setting of \( b \) - dislocated metric space.

3. Main results

In this section, we shall state and prove our main results.

Definition 12. Let \( (X,b_{d}) \) be a \( b \) - dislocated metric space with parameter \( s\geq 1 \). If \( T,S :X\rightarrow X \) are self-mapping that satisfy

for all \( x,y\in X \) and \( \alpha \in [ 0,\dfrac{1}{2}) \), then \( T \) and \( S \) are called an \( s-\alpha \) contraction maps.

Theorem 2. Let \( (X,b_{d}) \) be a complete \( b \) - dislocated metric space with parameter \( s\geqslant 1 .\) If \( T,S \) be self-maps of \( X \) such that: The pair \( (T,S) \) satisfy common limit in the range of S property (CLRs) in \( X \) and also \( T \) and \( S \) are \( s-\alpha \) contraction maps, then

• 1. The pair \( (T,S) \) has a coincidence point in \( X \).
• 2. The pair \( (T,S) \) has a unique common fixed point provided that \( T \) and \( S \) are weakly compatible mapping.

Proof. Since \( T \) and \( S \) satisfy (CLRs) property, there exists a sequence \( \{x_{n}\} \in X \) such that: \[ \lim\limits_{n\to \infty}Tx_{n}=\lim\limits_{n\to \infty}Sx_{n}=Su \] for some \( u\in X .\) By Equation (2), we have

By replacing \( x=u \) and \( y=x_{n} \) in the above condition, we obtain \[s^{2}b_{d}(Tu,Tx_{n})\leq\alpha\max\{b_{d}(Su,Sx_{n}),b_{d}(Su,Tu),b_{d}(Sx_{n},Tx_{n}),b_{d}(Su,Tu),b_{d}(Sx_{n},Tu)\}.\] By applying Lemma 1 and taking the upper limit as \( n\rightarrow\infty \) on Equation (3), we get \begin{eqnarray*} s^{2}b_{d}(Tu,Su)&\leq&\alpha\max\{b_{d}(Su,Su),b_{d}(Su,Tu),b_{d}(Su,Su),b_{d}(Su,Su),b_{d}(Su,Tu)\}\\ &=& \alpha\max\{b_{d}(Su,Tu),b_{d}(Su,Su)\}. \end{eqnarray*} We consider the following cases as follows;

Case 1: If \( max\{b_{d}(Su,Tu),b_{d}(Su,Su)\} =b_{d}(Su,Tu)\), then we have \(b_{d}(Tu,Su)\leq\frac{\alpha}{s^2} b_{d}(Su,Tu).\) It follows that \(b_{d}(Su,Tu)=0\) since \( 0\leq\alpha< \frac{1}{2}.\) Hence \( Su=Tu=z \) (say).

Case 2: If \( max\{b_{d}(Su,Tu),b_{d}(Su,Su)\} =b_{d}(Su,Su)\), then we have \begin{eqnarray*} s^{2}b_{d}(Tu,Su)&\leq&\alpha\ b_{d}(Su,Su)\\ &\leq&\ s \alpha [b_{d}(Su,Tu)+b_{d}(Tu,Su)]\\ &=&\ 2s\alpha b_{d}(Su,Tu). \end{eqnarray*}

It follows that \(b_{d}(Tu,Su)\leq\frac{2\alpha}{s} b_{d}(Su,Tu),\) which in turn implies that \(b_{d}(Su,Tu)=0\) since \( 0\leq\alpha< \frac{1}{2}. \) Hence \( Su=Tu=z \) (say). Therefore, \( T \) and \( S \) have a coincidence point. Now, by weakly compatibility property of the pair \( (T,S) \), we have \[ Tz=T(Su)=STu=Sz\] which implies that \( Tz=Sz .\) Now, we show existence of a common fixed point. First, we show that \( z \) is a fixed point of \( T \). By Equation (2), we have \[s^{2}b_{d}(Tz,Tx_{n})\leq\alpha\max\{b_{d}(Sz,Sx_{n}),b_{d}(Sz,Tz),b_{d}(Sx_{n},Tx_{n}),b_{d}(Sz,Tz),b_{d}(Sx_{n},Tz)\}.\] Taking the upper limit as \( n\rightarrow\infty \), we get \begin{eqnarray*} s^{2}b_{d}(Tz,z)&\leq&\alpha\max\{\lim\limits_{n\to\infty}sup[b_{d}(Sz,Sx_{n}),b_{d}(Sz,Tz),b_{d}(Sx_{n},Tx_{n}),b_{d}(Sz,Tx_{n}),b_{d}(Sx_{n},Tz)]\}\\ &=& \alpha\max\{b_{d}(Sz,Su),b_{d}(Sz,Tz),b_{d}(Su,Su),b_{d}(Sz,Su),b_{d}(Su,Tz)\}. \end{eqnarray*} Since \( Su=Tu=z \) and \( Tz=Sz \), we have \begin{eqnarray*} s^{2}b_{d}(Tz,z)&\leq&\alpha\max\{b_{d}(Tz,z),b_{d}(Sz,Sz),b_{d}(z,z),b_{d}(Tz,z),b_{d}(z,Tz)\}\\&=&\alpha\max\{b_{d}(Tz,z),b_{d}(Tz,Tz),b_{d}(z,z),b_{d}(Tz,z),b_{d}(z,Tz)\}. \end{eqnarray*} Now, we consider the following cases as follows;

Case 1: If \( max\{b_{d}(Tz,z),b_{d}(Tz,Tz),b_{d}(z,z)\} =b_{d}(Tz,z)\), then we have \(b_{d}(Tz,z)\leq\frac{\alpha}{s^2} b_{d}(Tz,z),\) which implies \(b_{d}(Tz,z)=0\) since \( 0\leq\alpha< \frac{1}{2}. \) Hence it follows that \( Tz=z. \)

Case 2: If \( max\{b_{d}(Tz,z),b_{d}(Tz,Tz),b_{d}(z,z)\} =b_{d}(Tz,Tz)\), then we have \begin{eqnarray*} s^{2}b_{d}(Tz,z)&\leq&\alpha\ b_{d}(Tz,Tz)\\ &\leq&\ s\alpha [b_{d}(z,Tz)+b_{d}(Tz,z)]\\ &=&\ 2s\alpha b_{d}(Tz,z). \end{eqnarray*} It follows that \(b_{d}(Tz,z)\leq\frac{2\alpha}{s}b_{d}(Tz,z),\) which implies \(b_{d}(Tz,z)=0\) Since \( 0\leq\alpha< \frac{1}{2}.\) Hence \( Tz=z \).

Case 3: If \( max\{b_{d}(Tz,z),b_{d}(Tz,Tz),b_{d}(z,z)\} =b_{d}(z,z)\), then we have \begin{eqnarray*} s^{2}b_{d}(Tz,z)&\leq&\alpha\ b_{d}(z,z)\\ &\leq&\ s\alpha [b_{d}(z,Tz)+b_{d}(Tz,z)]\\ &=&\ 2s\alpha b_{d}(Tz,z). \end{eqnarray*} It follows that \(b_{d}(Tz,z)\leq\frac{2\alpha}{s^2}b_{d}(Tz,z),\) which implies \(b_{d}(Tz,z)=0\) since \( 0\leq\alpha< \frac{1}{2}. \) Hence \( Tz=z \). But we know that \( Tz=Sz\) which gives us \( Tz=Sz=z\). Therefore, \(z\) is a common fixed point of \(T\) and \(S\).

Uniqueness

Let \(z\) and \(z\prime\) be fixed points of \(T\) and \(S\) with \(z\neq z\prime\) . Then by Equation (2), we have \begin{eqnarray*} s^{2}b_{d}(Tz,Tz\prime) &\leq&\alpha\max \{b_{d}(Sz,Sz\prime),b_{d}(Sz,Tz),b_{d}(Sz\prime,Tz\prime),b_{d}(Sz,Tz\prime),b_{d}(Sz\prime,Tz)\}\\ &=&\alpha\max\{b_{d}(z,z\prime),b_{d}(z,z),b_{d}(z\prime,z\prime),b_{d}(z,z\prime),b_{d}(z\prime,z)\}. \end{eqnarray*} We consider the following cases as follows.

Case 1: If \( max\{b_{d}(z\prime,z),b_{d}(z,z),b_{d}(z\prime,z\prime)\} =b_{d}(z,z\prime)\), then we have \(b_{d}(z\prime,z)\leq\frac{\alpha}{s^2} b_{d}(z,z\prime)\) which implies \(b_{d}(z,z\prime)=0\) since \( 0\leq\alpha< \frac{1}{2}.\) Hence it follows that \( z=z\prime \).

Case 2: If \( max\{b_{d}(z\prime,z),b_{d}(z,z),b_{d}(z\prime,z\prime)\} =b_{d}(z,z)\), then we have \begin{eqnarray*} s^{2}b_{d}(z,z\prime)&\leq&\alpha\ b_{d}(z,z)\\ &\leq&\ s\alpha [b_{d}(z,z\prime)+b_{d}(z\prime,z)]\\ &=&\ 2s\alpha b_{d}(z,z\prime). \end{eqnarray*} It follows that \(b_{d}(z,z\prime)\leq\frac{2\alpha}{s} b_{d}(z,z\prime)\) which implies \(b_{d}(z\prime,z)=0\) since \( 0\leq\alpha< \frac{1}{2}. \) Hence \( z=z\prime \).

Case 3: If \( max\{b_{d}(z,z\prime),b_{d}(z,z),b_{d}(z\prime,z\prime)\} =b_{d}(z\prime,z\prime)\), then we have \begin{eqnarray*} s^{2}b_{d}(z,z\prime)&\leq&\alpha\ b_{d}(z\prime,z\prime)\\ &\leq&\ s\alpha [b_{d}(z,z\prime) b_{d}(z\prime,z)]\\ &=&\ 2s\alpha b_{d}(z\prime,z). \end{eqnarray*} It follows that \(b_{d}(z\prime,z)\leq\frac{2\alpha}{s} b_{d}(z\prime,z\prime)\) which implies \(b_{d}(z\prime,z)=0\) since \( 0\leq\alpha< \frac{1}{2}. \) Hence \( z\prime=z \) which contradicts to our assumption \( z\neq z\prime \).

Hence \( z \) is a unique common fixed point of \( T \) and \( S .\)

Theorem 3. Let \( (X,b_{d}) \) be a complete \( b \) - dislocated metric space with parameter \( s\geqslant 1 .\) If \( T,S \) be self-maps of \( X \) such that the pair \( (T,S) \) satisfy occasionally weakly compatible property (OWC) in \( X \) and \(T\) and \(S\) are an \( s-\alpha \) contraction maps. Then the pair \( (T,S) \) has a unique common fixed point.

Proof. Since \( T \) and \( S \) satisfy (OWC) property, there exists a point \( u \in X \) such that \(Tu=Su\) and \( TSu=STu.\) This implies that \( TSu=TTu=STu=SSu.\) It follows that \( TTu=SSu. \) We claim that \( Tu \) is the unique common fixed point of \( T \) and \( S .\) First, we assert that \( Tu \) is a fixed point of \( T .\) For, if \( TTu\neq Tu ,\) then by Equation (2), we get \begin{equation*} s^{2}b_{d}(Tu,TTu)\leq\alpha\max\Bigl\{b_{d}(Su,STu),b_{d}(Su,Tu),b_{d}(STu,TTu),b_{d}(Su,TTu),b_{d}(STu,Tu)\Bigr\}. \end{equation*} Since \( Su=Tu, \) then we have \begin{eqnarray*} s^{2}b_{d}(Tu,TTu)&\leq&\alpha\max\Bigl\{b_{d}(Tu,TTu),b_{d}(Tu,Tu),b_{d}(TTu,TTu),b_{d}(Tu,TTu),b_{d}(TTu,Tu)\Bigr\}\\ &=&\alpha\max\Bigl\{b_{d}(Tu,TTu),b_{d}(Tu,Tu),b_{d}(TTu,TTu)\Bigr\}. \end{eqnarray*} We consider the following three cases as follows.

Case 1: If \( max\{b_{d}(Tu,TTu),b_{d}(Tu,Tu),b_{d}(TTu,TTu)\} =b_{d}(Tu,TTu)\), then we have \(b_{d}(Tu,TTu)\leq\frac{\alpha}{s^2} b_{d}(Tu,TTu),\) which implies \(b_{d}(Tu,TTu)=0\) since \( 0\leq\alpha< \frac{1}{2}. \) Hence \( Tu=TTu \).

Case 2: If \( max\{b_{d}(Tu,TTu),b_{d}(Tu,Tu),b_{d}(TTu,TTu)\} =b_{d}(Tu,Tu)\), then we have \begin{eqnarray*} s^{2}b_{d}(Tu,TTu)&\leq&\alpha\ b_{d}(Tu,Tu)\\ &\leq&\ s \alpha [b_{d}(TTu,Tu)+b_{d}(Tu,TTu)]\\ &=&\ 2s\alpha b_{d}(TTu,Tu). \end{eqnarray*} It follows that \(b_{d}(TTu,Tu)\leq\frac{2\alpha}{s} b_{d}(TTu,Tu)\), which implies that \(b_{d}(TTu,Tu)=0\) since \( 0\leq\alpha< \frac{1}{2}. \) Hence \( TTu=Tu \).

Case 3: If \( max\{b_{d}(Tu,TTu),b_{d}(Tu,Tu),b_{d}(TTu,TTu)\} =b_{d}(TTu,TTu)\), then we have \begin{eqnarray*} s^{2}b_{d}(Tu,TTu)&\leq&\alpha\ b_{d}(TTu,TTu)\\ &\leq&\ s \alpha [b_{d}(TTu,Tu)+b_{d}(Tu,TTu)]\\ &=&\ 2s\alpha b_{d}(TTu,Tu). \end{eqnarray*} It follows that \(b_{d}(TTu,Tu)\leq\frac{2\alpha}{s} b_{d}(TTu,Tu),\) which implies that \(b_{d}(TTu,Tu)=0\) since \( 0\leq\alpha< \frac{1}{2} .\) Hence \( Tu=TTu \) which is a contradiction with \( Tu \neq TTu .\) There fore, \( TTu=Tu \) and \( Tu \) is the fixed point of \( T. \) Since \( TTu=TSu=STu=Tu=SSu, \) it implies \(STu=Tu.\) Thus \( Tu \) is fixed point of \( S. \) Therefore, \(Tu\) is a common fixed point of \( T \) and \( S .\)

Uniqueness

Suppose that \( u, v \in X \) such that \( Tu=Su=u \) and \( Tv=Sv=v \) and \( u\neq v.\) Then by Equation (2), we get \begin{eqnarray*} s^{2}b_{d}(u,v)&=& s^{2}b_{d}(Tu,Tv)\\ &\leq&\alpha\max\Bigl\{b_{d}(Su,Sv),b_{d}(Su,Tu),b_{d}(Sv,Tv),b_{d}(Su,Tv),b_{d}(Sv,Tu)\Bigr\}\\ &=&\alpha\max\Bigl\{b_{d}(u,v),b_{d}(u,u),b_{d}(v,v),b_{d}(u,v),b_{d}(v,u)\Bigr\}\\ &=&\alpha\max\Bigl\{b_{d}(u,v),b_{d}(u,u),b_{d}(v,v)\Bigr\}. \end{eqnarray*} We consider the following cases as follows.

Case 1: If \( max\{b_{d}(u,v),b_{d}(u,u),b_{d}(v,v)\} =b_{d}(v,u)\), then we have \(b_{d}(u,v)\leq\frac{\alpha}{s^2} b_{d}(v,u),\) which implies that \(b_{d}(v,u)=0\), since \( 0\leq\alpha< \frac{1}{2}. \) Hence it follows that \( u=v .\)

Case 2: If \( max\{b_{d}(v,u),b_{d}(u,u),b_{d}(v,v)\} =b_{d}(u,u)\), then we have \begin{eqnarray*} s^{2}b_{d}(v,u)&\leq&\alpha\ b_{d}(u,u)\\ &\leq&\ s\alpha [b_{d}(u,v)+b_{d}(u,v)]\\ &=&\ 2s\alpha b_{d}(u,v). \end{eqnarray*} It follows that \(b_{d}(v,u)\leq\frac{2\alpha}{s} b_{d}(v,u),\) which implies that \(b_{d}(v,u)=0\) since \( 0\leq\alpha< \frac{1}{2}. \) Hence \( v=u \).

Case 3: If \( max\{b_{d}(u,v),b_{d}(v,v),b_{d}(u,u)\} =b_{d}(v,v)\), then we have \begin{eqnarray*} s^{2}b_{d}(u,v)&\leq&\alpha\ b_{d}(v,v)\\ &\leq&\ s\alpha [b_{d}(u,v) b_{d}(v,u)]\\ &=&\ 2s\alpha b_{d}(u,v). \end{eqnarray*} It follows that \(b_{d}(u,v)\leq\frac{2\alpha}{s} b_{d}(v,v),\) which implies that \(b_{d}(v,u)=0\), since \( 0\leq\alpha< \frac{1}{2}. \) Hence \( u=v \). Therefore, it contradicts with our assumption \( u\neq v .\)

Hence \( u \) is a unique common fixed point of \( T \) and \( S .\)

Theorem 4. Let \( (X, b_{d}) \) be a complete \( b \) - dislocated metric space with parameter \( s\geqslant 1.\) If \( T,S \) be self-maps of \( X \) such that \[s^{2}b_{d}(Tx,Ty)\leq\alpha\max\Bigl\{b_{d}(Sx,Sy),b_{d}(Sx,Tx),b_{d}(Sy,Ty),b_{d}(Sx,Ty),b_{d}(Sy,Tx)\Bigr\}\] for all \( x,y\in X \) and \( 0\leq\alpha< \frac{1}{2}.\) Then the pair \( (T,S) \) has a unique common fixed point.

Proof. Let \( x_{0} \) be arbitrary given point in \( X \). Define the sequence \( {y_{n} }\in X \) such that; \( y_{2n}=Tx_{2n}=Sx_{2n+1} ,\) for all \( n\geq 0. \) We show that \( {y_{n} }\in X \) for all \( n\in\mathbb{N}.\) Since \(y_{2n}=Tx_{2n}=Sx_{2n+1},\) we have from (1) that \begin{eqnarray*} s^{2}b_{d}(y_{2n}, y_{2n+1})&=& s^{2}b_{d}(Tx_{2n},Tx_{2n+1})\\ &\leq&\alpha\max\Bigl\{b_{d}(Sx_{2n},Sx_{2n+1}),b_{d}(Sx_{2n},Tx_{2n}),b_{d}(Sx_{2n+1},Tx_{2n+1}),b_{d}(Sx_{2n},Tx_{2n+1}),{}b_{d}(Sx_{2n+1},Tx_{2n}x)\Bigr\}\\ &=&\alpha\max\Bigl\{b_{d}(y_{2n-1},y_{2n}),b_{d}(y_{2n-1},y_{2n}),b_{d}(y_{2n},y_{2n+1}),b_{d}(y_{2n-1},y_{2n+1}),b_{d}(y_{2n},y_{2n})\Bigr\}\\ &\leq&\alpha\max\Bigl\{b_{d}(y_{2n-1},y_{2n}),b_{d}(y_{2n-1},y_{2n}),b_{d}(y_{2n},y_{2n+1}),s[b_{d}(y_{2n-1},y_{2n})+b_{d}(y_{2n},{}y_{2n+1})],s[b_{d}(y_{2n},y_{2n+1}+b_{d}(y_{2n-1},y_{2n})]\Bigr\} \end{eqnarray*} \begin{eqnarray*} &=&\alpha\max\Bigl\{b_{d}(y_{2n-1},y_{2n}),b_{d}(y_{2n-1},y_{2n}),b_{d}(y_{2n},y_{2n+1}),s[b_{d}(y_{2n-1},y_{2n})+b_{d}(y_{2n},{}y_{2n+1})],2s[b_{d}(y_{2n-1},y_{2n})]\Bigr\}. \end{eqnarray*} If \(b_{d}(y_{2n-1},y_{2n})\leq b_{d}(y_{2n},y_{2n+1}),\) for some \( n\in\mathbb{N},\) then by (1), we have \(s^{2}b_{d}(y_{2n},y_{2n+1})\leq2\alpha b_{d}(y_{2n},y_{2n+1}),\) which implies that \(b_{d}(y_{2n},y_{2n+1})\leq\frac{2\alpha}{s^2} b_{d}(y_{2n},y_{2n+1}).\) This is not true because \(\frac{2\alpha}{s^2}< 1.\) Thus \( b_{d}(y_{2n},y_{2n+1})\leq b_{d}(y_{2n-1},y_{2n})\) for all \( n\in\mathbb{N}.\) Also, by the above inequality, we get \begin{eqnarray*} s^2b_{d}(y_{2n},y_{2n+1})&\leq& 2\alpha s b_{d}(y_{2n-1},y_{2n}),\\ b_{d}(y_{2n-1},y_{2n})&\leq&\frac{2\alpha}{s} b_{d}(y_{2n-2},y_{2n-1}),\\ b_{d}(y_{2n-2},y_{2n-1})&\leq&\frac{2\alpha}{s} b_{d}(y_{2n-3},y_{2n-4}). \end{eqnarray*} Continuing like this, we have \[b_{d}(y_{2n},y_{2n+1})\leq cb_{d}(y_{2n-1},y_{2n})\leq c^{2}b_{d}(y_{2n-2},y_{2n-1}), \cdots \leq c^{n}b_{d}(y_{0},y_{1}),\] where \( c=\frac{2\alpha}{s} \) and \( 0\leq c< 1. \) Taking the upper limit as \( n\rightarrow\infty \) in the inequality above, we have \(b_{d}(y_{2n},y_{2n+1})\rightarrow 0.\) Now, we prove that \(\{y_{m}\}\) is a \( b \) - dislocated Cauchy sequence where \( m=2n. \) To do this let \(m,n\geq 0 \) with \( m>n.\) Now, using the triangle inequality, we have

Therefore \( b_{d}(y_{n},y_{m})\leq\frac{sc^{n}}{(1-sc)}b_{d}(y_{0},y_{1}).\) Taking the upper limit as \( m,n\rightarrow\infty, \) we have \( b_{d}(y_{n},y_{m})\rightarrow 0 \) as \( sc< 1. \) Therefore, \(\{\ y_{m}\} \) is a \( b \) -dislocated Cauchy sequence in \( b \) - dislocated metric space \( (X, b_{d}).\) So there is some \( u \in X\) such that \(\{\ y_{m}\} \) is a \( b \) -dislocated converges to \( u \). Since a subsequence of a Cauchy sequence in \( b \) -dislocated metric space is a Cauchy sequence, then \(\{Tx_{2n}\}\) and \(\{Sx_{2n+1}\}\) are also Cauchy sequences.

If \(T\) and \( S \) are continuous mappings, we get

\[ T(u)=T(\lim\limits_{n\to \infty}x_{n})=\lim\limits_{n\to \infty}Tx_{n}= S(u)= \lim\limits_{n\to \infty}Sx_{n+1}=u .\] Thus, \( u \) is a common fixed point of \( T \) and \( S \). If the self-map \( T \) is not continuous then, we consider \begin{eqnarray*} s^{2}b_{d}(y_{2n}, Tu)&=& s^{2}b_{d}(Tx_{2n},Tu)\\ &\leq&\alpha\max\Bigl\{b_{d}(Sx_{2n},Su),b_{d}(Sx_{2n},Tx_{2n}),b_{d}(Su,Tu),b_{d}(Sx_{2n},Tu),b_{d}(Su,Tx_{2n})\Bigr\}\\ &=&\alpha\max\Bigl\{b_{d}(y_{2n-1},Su),b_{d}(y_{2n-1},y_{2n}),b_{d}(Su,Tu),b_{d}(y_{2n-1},Tu),b_{d}(Su,y_{2n})\Bigr\}. \end{eqnarray*} On taking upper limit as \( n\rightarrow \infty ,\) we get \( s^{2}b_{d}(u,Tu)\leq\alpha\max \Bigl\{b_{d}(Su,Su),b_{d}(Su,Tu),b_{d}(Su,Tu)\Bigr\}\). We consider the following cases:

Case 1: If \( max\{b_{d}(u,u),b_{d}(Tu,Tu),b_{d}(u,Tu)\} =b_{d}(Tu,u)\), then we have \(b_{d}(Tu,u)\leq\frac{\alpha}{s^2} b_{d}(Tu,u),\) which implies \(b_{d}(Tu,u)=0\), since \( 0\leq\alpha< \frac{1}{2}. \) Hence \( Tu=u \).

Case 2: If \( max\{b_{d}(u,u),b_{d}(Tu,Tu),b_{d}(Tu,u)\} =b_{d}(Tu,Tu)\), then we have \begin{eqnarray*} s^{2}b_{d}(Tu,u)&\leq&\alpha\ b_{d}(Tu,Tu)\\ &\leq&\ s\alpha [b_{d}(u,Tu)+b_{d}(Tu,u)]\\ &=&\ 2s\alpha b_{d}(Tu,u). \end{eqnarray*} It follows that \(b_{d}(Tu,u)\leq\frac{2\alpha}{s} b_{d}(Tu,u),\) which implies \(b_{d}(Tu,u)=0\) since \( 0\leq\alpha< \frac{1}{2}. \) Hence \( Tu=u \).

Case 3: If \( max\{b_{d}(u,u),b_{d}(Tu,Tu),b_{d}(Tu,u)\} =b_{d}(u,u)\), then we have \begin{eqnarray*} s^{2}b_{d}(Tu,u)&\leq&\alpha\ b_{d}(u,u)\\ &\leq&\ s\alpha [b_{d}(u,Tu) b_{d}(Tu,u)]\\ &=&\ 2s\alpha b_{d}(Tu,u). \end{eqnarray*} It follows that \(b_{d}(Tu,u)\leq\frac{2\alpha}{s} b_{d}(Tu,u),\) which implies \(b_{d}(Tu,u)=0\) since \( 0\leq\alpha< \frac{1}{2}. \) Hence \( Tu=u \).

In all cases \( b_{d}(u,Tu)=0 \) which implies that \( u=Tu. \) Thus, \( u \) is fixed point of \( T.\)

In similar cases, we have \(b_{d}(Su,u)\leq\frac{2\alpha}{s} b_{d}(Su,u),\) which implies \(b_{d}(Su,u)=0\) since \( 0\leq\alpha< \frac{1}{2}. \) Hence \( Su=u \). Thus, \( u \) is fixed point of \( S.\) Since, \( Su=u=Tu, \) then \( u \) is a common fixed point of \( T \) and \( S .\)

Uniqueness

Let \( u \) and \( v \) are fixed points of \( T \) and \( S \) with \( u\neq v.\) Then by using Equation (2), we have \begin{eqnarray*} s^{2}b_{d}(u,v)=s^{2}b_{d}(Tu,Tv)&\leq&\alpha\max \Bigl\{b_{d}(Su,Sv),b_{d}(Su,Tu),b_{d}(Sv,Tv),b_{d}(Su,Tv),b_{d}(Sv,Tu)\Bigr\}\\ &=&\alpha\max \Bigl\{b_{d}(u,v),b_{d}(u,u),b_{d}(v,v),b_{d}(u,v),b_{d}(v,u)\Bigr\}\\ &=&\alpha\max \Bigl\{b_{d}(u,v),b_{d}(v,v),b_{d}(v,u)\Bigr\}. \end{eqnarray*} We consider the following three cases;

Case 1: If \( max\{b_{d}(u,v),b_{d}(u,u),b_{d}(v,v)\} =b_{d}(v,u)\), then we have \(b_{d}(u,v)\leq\frac{\alpha}{s^2} b_{d}(v,u),\) which implies \(b_{d}(v,u)=0\) since \( 0\leq\alpha< \frac{1}{2}. \) Hence it follows that \( u=v \).

Case 2: If \( max\{b_{d}(v,u),b_{d}(u,u),b_{d}(v,v)\} =b_{d}(u,u)\), then we have \begin{eqnarray*} s^{2}b_{d}(v,u)&\leq&\alpha\ b_{d}(u,u)\\ &\leq&\ s\alpha [b_{d}(u,v)+b_{d}(u,v)]\\ &=&\ 2s\alpha b_{d}(u,v). \end{eqnarray*} It follows that \(b_{d}(v,u)\leq\frac{2\alpha}{s} b_{d}(v,u),\) which implies \(b_{d}(v,u)=0\) since \( 0\leq\alpha< \frac{1}{2}. \) Hence it follows that \( v=u \).

Case 3: If \( max\{b_{d}(u,v),b_{d}(v,v),b_{d}(u,u)\} =b_{d}(v,v)\), then we have \begin{eqnarray*} s^{2}b_{d}(v,u)&\leq&\alpha\ b_{d}(v,v)\\ &\leq&\ s\alpha [b_{d}(u,v)+b_{d}(v,u)]\\ &=&\ 2s\alpha b_{d}(u,v). \end{eqnarray*} It follows that \(b_{d}(u,v)\leq\frac{2\alpha}{s} b_{d}(v,u),\) which implies \(b_{d}(v,u)=0\) since \( 0\leq\alpha< \frac{1}{2}. \) Hence in all cases \( u=v \). Therefore, it contradicts with our assumption \( u\neq v \). Hence \( u \) is a unique common fixed point of \( T \) and \( S .\)

Remark 3. If we take \(S=I\) (\(I\) is the identity map on \(X\)) in Theorem 4, we get Theorem 1.

Now, we give an example in support of Theorem 4.

Example 2. Let \(X=[0,1]\) and \(b_{d}(x,y)=(x+y)^2\) for all \(x, y \in X\) when \(s=2\) is a b-dislocated metric on \(X\). Then \((X,b_{d})\) is a b-dislocated metric space.

Solution. We take the \(s-\alpha\) contraction map and define the following \[Tx=\begin{cases} \frac{1}{25}x, \text{ if } x \in [0,1)\\ \frac{1}{30}, \text{ if } x=1 \end{cases} \text{ and }\;\;\;\; Sx=\frac{1}{5}x.\] Consider the sequence \(\{x_n\}\) given by \(X_n= \frac{1}{n}\) for all \(n \in \mathbf{N}.\) \(T\) and \(S\) satisfies the common limit in the range of \(S\) property. By using (CLRs) properties, we have \(\lim\limits_{n \to \infty}Tx_n = \lim\limits_{n \to \infty}Sx_n=0,\) for \( 0 \in Tx\) or \( 0 \in Sx.\) We have the following cases with \(\alpha = \frac{4}{25}. \)

Case 1: For \(x, y \in [0,1)\), we have \begin{equation*} s^2 b_{d}(Tx,Ty) = 4 b_{d}\biggl(\frac{1}{25}x,\frac{1}{25}y\biggr) = 4 \biggl(\frac{1}{25}x+\frac{1}{25}y\biggr)^2 = \frac{4}{25} \biggl(\frac{1}{5}x+\frac{1}{5}y\biggr)^2 \leq \frac{4}{25} b_{d}(Sx,Sy). \end{equation*} Therefore, \(s^2b_{d}(Tx,Ty)\leq \alpha b_{d}(Sx,Sy).\)

Case 2: For \(y< x\) and \(x=1\), we have \begin{equation*} s^2 b_{d}(T1,Ty) = 4 b_{d}\biggl(\frac{1}{30},\frac{1}{25}y\biggr) = 4 \biggl(\frac{1}{30}+\frac{1}{25}y\biggr)^2 \leq \frac{4}{25} \biggl(\frac{1}{5}+\frac{1}{5}y\biggr)^2 = \frac{4}{25} b_{d}\biggl(\frac{1}{5},\frac{1}{5}y\biggr) \leq \frac{4}{25} b_{d}(Sx,Sy). \end{equation*} Therefore, \(s^2b_{d}(T1,Ty) \leq \alpha b_{d}(S1,Sy)= \alpha b_{d}(Sx,Sy).\)

Case 3: For \(x< y\) and \(y=1\), we have \begin{equation*} s^2b_{d}(Tx,T1) = 4 b_{d}\biggl(\frac{1}{25}x,\frac{1}{30}\biggr) = 4 \biggl(\frac{1}{25}x+\frac{1}{30}\biggr)^2 \leq \frac{4}{25} \biggl(\frac{1}{5}x+\frac{1}{5}\biggr)^2 = \frac{4}{25} b_{d}\biggl(\frac{1}{5}x,\frac{1}{5}\biggr) \leq \frac{4}{25} b_{d}(Sx,Sy). \end{equation*} Therefore, \(s^2b_{d}(Tx,T1) \leq \alpha b_{d}(Sx,S1)= \alpha b_{d}(Sx,Sy).\)

Case 4: For \(x=y=1\), we have \begin{equation*} s^2 b_{d}(T1,T1) = 4 b_{d}\biggl(\frac{1}{30},\frac{1}{30}\biggr) = \frac{4}{25} \biggl(\frac{1}{6}+\frac{1}{6}\biggr)^2 \leq \frac{4}{25} \biggl(\frac{1}{5}+\frac{1}{5}\biggr)^2 = \frac{4}{25} b_{d}\biggl(\frac{1}{5},\frac{1}{5}\biggr) = \frac{4}{25} b_{d}\biggl(S1,S1\biggr) \leq \frac{4}{25} b_{d}(Sx,Sy). \end{equation*}

Therefore, \(s^2b_{d}(T1,T1) \leq \alpha b_{d}(S1,S1)=\alpha b_{d}(Sx,Sy).\)

From cases 1 up to 4, we see that all the conditions of Theorem 4 are satisfied and \(0\) is the unique common fixed point of \(T\) and \(S\).

4. Conclusion

Zoto and Kumari [1] established the existence and uniqueness of fixed point for a mapping satisfying \( s-\alpha \) type contraction condition in a complete dislocated metric space. In this thesis, we have explored the properties of \( s-\alpha \) type contraction mapping in \( b \) - dislocated metric spaces. We established the theorem on common fixed points of two mapping satisfying \( s-\alpha \) contraction condition in the setting of \( b \) - dislocated metric spaces and proved the existence and uniqueness of common fixed point for a pair of maps T and S in the setting of \( b \) - dislocated metric space. Also we provided an example in support of our main results. Our work extended fixed point result in single map to common fixed point result in a pair of maps. The presented theorem extends and generalizes several well-known comparable results in literature.

Acknowledgments

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

Author Contributions

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

Conflicts of interest

The authors declare no conflict of interest.

1. Introduction

The differential and integral equations with deviating arguments usually involve the deviation of the argument only the time itself, however, another case, in which the deviating arguments depend on both the state variable \(x\) and the time \(t\) is important in theory and practice. This kind of equations is called self-reference or state dependent equations. Differential equations with state-dependent delays attract interests of specialists since they widely arise from application models, such as two-body problem of classical electrodynamics also have may applications in the class of problems that have past memories, for example in hereditary phenomena see [1,2,3,4]. For papers studying such kind of equations, see for example [5,6,7,8,9,10,11,12,13,14,15,16] and references therein.

One of the first papers studying this class of equations was introduced by Buicá [7], the author proved the existence and the uniqueness of the solution of the initial value problem

\begin{eqnarray*} \frac{d x(t)}{dt} &=& f(t,x(x(t))), ~~~t\in[a,b],\\ x(0) &=& x_{0}, \end{eqnarray*} where \( f\in C([a,b]\times[a,b])\) and satisfied Lipshitz condition.

EL-Sayed and Ebead [13] relaxed the assumptions of Buicá and generalized their results, they studied the functional integral equation of the more general form

\begin{eqnarray*}\label{equ8555} x(t)~=~f\bigg(t,~\int_{0}^{t}g\big(s,x(x(s))\big)ds\bigg),~~~~t\in[0,T], \end{eqnarray*} where \( g\) satisfies Carathéodory condition.

El-Sayed and Ahmed [9] studied the existence of solutions and its continuous dependence of the initial value problem

\begin{eqnarray*} \frac{d}{dt} x(t)&=& f(t,x(\int_0^{\phi(t)} g(s,x(s)) ds)),~~a.e.~~t\in(0,T],\nonumber\\ x(0)&=& x_{0}, \end{eqnarray*} where \(g: [0,T]\times R^+\rightarrow R^+\) is continuous and \(g(t,x(t))\leq1\) and \(\phi(t) \leq t.\)

El-Sayed and Ebead [11,12] studied the existence of solution and its continuous dependence of the initial value problem of the delay-refereed differential equation

\begin{eqnarray*} \frac{d}{dt} x(t)&=& f(t,x(g(t,x(t)))),~~~a.e.~~t\in(0,T],\nonumber\\ x(0)&=& x_{0},\nonumber \end{eqnarray*} with the two cases of state-delay functions

• (i) \(g: [0,T]\times R^+\rightarrow [0,T]\) is continuous and \(g(t,x(t))\leq t\),
• (ii) \(g: [0,T]\times [0,T] \rightarrow [0,T]\) is continuous and \(g(t,x(\phi(t)))\leq x(\phi(t)).\)

Here we shall study the initial value problem of state-dependent neutral functional differential equation with two state-delay functions with the initial data where \(g_{i},~i=1,~2\) are continuous and \(g_{1}(t,x)\leq \phi (t)\) and \(g_{1}(t,x)\leq x(t) .\)

Our aim in this work is to study the existence of at least one and exactly one positive solution of the Problem (1)-(2). The continuous dependence of the unique solution on the two functions \(g_{1}~\) and \(~g_{2} \) will be proved. To illustrate our results some examples will be given.

In order to achieve our goal, we study the existence of positive solutions \(x \in C[0; T] \) for the state-dependent functional integral equation

and we will show later that this integral equation is equivalent to the initial value Problem (1)-(2).

2. Existence of solutions

Here we study the existence of solutions \(x\in C[0,T]\) for the integral Equation (3) under the following assumptions

• (1) \(f_{1}:[0,T]\times[0,T] \rightarrow R^+\) is continuous and there exist two positive constants \(k_{1}~and~k_{2}\) such that \[ |f_{1}(t_{2},x)-f_{1}(t_{1},y)|\leq k_{1}|t_{2}-t_{1}|+k_{2}|x-y|. \]
• (2) \(g_{1}:[0,T]\times [0,T] \rightarrow [0,T]\) is continuous and there exist the two positive constants \(k_{3},~k_{4}~\) such that \[ |g_{1}(t_{2},x)-g_{1}(t_{1},y)|\leq k_{3}|t_{2}-t_{1}|+k_{4}|x-y| \] and \(g_{1}(t,x)\leq \phi(t)\).
• (3) \(f_{2}:[0,T]\times [0,T] \rightarrow R^+~\) satisfies Carathéodory condition i.e. \(f(t,x)\) is measurable in \(t\) for all \(x\in C [0,T]\) and continuous in \(x\) for almost all \(t\in[0,T]\).
• (4) There exists a bounded measurable function\(~m : [0,T] \rightarrow R^{+},~m(t)\leq A,\) and a constant \(~k_{5}\geq 0\) such that \[ f_{2}(t,x)\leq m(t) +k_{5}~x. \]
• (5) \( g_{2}:[0,T]\times [0,T]\rightarrow [0,T]\) is continuous and \(g_{2}(t,x)\leq x\)
• (6) \( \phi:[0,T]\rightarrow [0,T],~\phi(0)=0\) and \[ |\phi(t)-\phi(s)|\leq |t-s|, \] which implies that \(\phi(t)\leq t\).
• (7) There exists a real positive solution \(L\in (0,1)\) of the equation \[k_{2}~ k_{4} L^{2}+(k_{2}~k_{3}-1)L+ M+k_{1}=0,\] where \(M=A+k_{5}T\).
• (8) \(LT+x(0)\leq T.\)

Now we are in a position to prove the following existence theorem.

Theorem 1. Let the assumptions \((1)-(8)\) be satisfied, then the state-dependent integral Equation (3) has at least one positive solution \(x\in C[0,T]\).

Proof. Let \(X=C[0,T]\) be the class of real valued continuous functions defined on \([0,T]\). Define the subset \(S_{L}\) of \(X \) by \[ S_{L}=\big\{x\in X:|x(t_{2})-x(t_{1})|\leq L|t_{2}-t_{1}|\big\}. \] It is clear that \(S_{L}\) is nonempty, closed, bounded and convex subset of \(C[0,T]\).

Now define the operator \(F\) associated with Equation (3) by

\begin{eqnarray*} Fx(t)=f_{1}\big(t,x(g_{1}(t,x(\phi(t))))\big)+\int_{0}^{t}~f_{2}\big(s,x(g_{2}(s,x(\phi(s))))\big)ds,~~ t\in[0,T]. \end{eqnarray*} It is clear that F makes sense and well-defined.

Now, first we prove that the class of functions \(\{Fx\}\) is uniformly bounded on the set \(S_L.\) Let \(x \in X\), then for \(t \in [0, T],\) we obtain

\begin{eqnarray*} |Fx(t)|&=&|f_{1}\big(t,x(g_{1}(t,x(\phi(t))))\big)+\int_{0}^{t}~f_{2}\big(s,x(g_{2}(s,x(\phi(s))))\big)ds|\\ &\leq &|f_{1}\big(t,x(g_{1}(t,x(\phi(t))))\big)|+\int_{0}^{t}~|f_{2}\big(s,x(g_{2}(s,x(\phi(s))))\big)|ds. \end{eqnarray*} Using assumptions (1),(2) and (6) we can get Using assumptions (3) and (4) we can get Now from (4) and (5) and by assumption (8), we get \begin{eqnarray*} |Fx(t)| &\leq & (k_{1}+ k_{2}~k_{3}~L +k_{2}k_{4}~L^{2})~T+x(0)+\int_{0}^{t} (k_{5}(L~x(\phi(s))+x(0))+A)ds\\ &\leq & (k_{1}+ k_{2}~k_{3}~L +k_{2}k_{4}~L^{2})~T+x(0)+\int_{0}^{t} (k_{5}(L~T+x(0))+A)ds\\ &\leq &(k_{1}+ k_{2}~k_{3}~L +k_{2}k_{4}~L^{2})~T+x(0)+(k_{5}~T+ A)\int_{0}^{t}ds \end{eqnarray*} \begin{eqnarray*} &\leq & (k_{1}+ k_{2}~k_{3}~L +k_{2}k_{4}~L^{2}+M)~T+x(0)\\ &= & L~T+x(0)\leq T. \end{eqnarray*} This proves that the class of functions \(\{~Fx~\}\) is uniformly bounded on the set \(~S_L.~\)

Next, we prove that \(~F:S_{L}~\rightarrow S_{L} \) and the class of functions \(\{Fx\}\) is equi-continuous on the set \(S_L.\) Let \(x\in S_{L}\) and \(t_{1}, ~t_{2}\in [0,T]\) with \(t_{1}< t_{2} \) such that \( |t_{2}, -t_{1}|< \delta\), then

\begin{eqnarray*}\label{2} |Fx(t_2)-Fx(t_1)|&=&\bigg| f_{1}\big(t_{2},x(g_{1}(t_{2},x(\phi(t_{2}))))\big)~+\int_{0}^{t_{2}}~f_{2}\big(s,x(g_{2}(s,x(\phi(s))))\big)ds \nonumber\\ &&-f_{1}\big(t_{1},x(g_{1}(t_{1},x(\phi(t_{1}))))\big)-\int_{0}^{t_{1}}~f_{2}\big(s,x(g_{2}(s,x(\phi(s))))\big)ds \bigg|\nonumber\\ &\leq& |f_{1}\big(t_{2},x(g_{1}(t_{2},x(\phi(t_{2}))))\big)- f_{1}\big(t_{1},x(g_{1}(t_{1},x(\phi(t_{1}))))\big)|+\int_{t_{1}}^{ t_{2}}|f_{2}\big(s,x(g_{2}(s,x(\phi(s))))\big)|ds \nonumber\\ &\leq& k_{1}|t_{2}-t_{1}|+k_{2}|x(g_{1}(t_{2},x(\phi(t_{2}))))-x(g_{1}(t_{1},x(\phi(t_{1}))))|+ \int_{t_{1}}^{t_{2}} (k_{5}~T+ A)ds\\ &\leq& k_{1}|t_{2}-t_{1}|+k_{2}|x(g_{1}(t_{2},x(\phi(t_{2}))))-x(g_{1}(t_{1},x(\phi(t_{1}))))| +M|t_{2}-t_{1}|. \end{eqnarray*} Using assumptions (2) and \(x\in S_{L}\), we can get \begin{eqnarray}\label{20} |x( g_{1}\big(t_{2},x(\phi(t_{2})))\big)- x(g_{1}\big(t_{1},x(\phi(t_{1}))))|& \leq & L | g_{1}\big(t_{2},x(\phi(t_{2})))\big)- g_{1}\big(t_{1},x(\phi(t_{1}))))|\nonumber\\& \leq & L~k_{3}|t_{2}-t_{1}|+ L~k_{4}|x(\phi(t_{2})))-x(\phi(t_{1})))|\nonumber\\ &\leq&L~ k_{3}|t_{2}-t_{1}|+L^{2} k_{4}|\phi(t_{2})-\phi(t_{1})|\nonumber \\ &\leq&L~ k_{3}|t_{2}-t_{1}|+ L^{2}k_{4}~|t_{2}-t_{1}|.\nonumber \end{eqnarray} Then, we have \begin{eqnarray*} |Fx(t_2)-Fx(t_1)|&\leq& k_{1}|t_{2}-t_{1}|+k_{2}(L~ k_{3}|t_{2}-t_{1}|+ L^{2}k_{4}~|t_{2}-t_{1}|)+M|t_{2}-t_{1}|\\ &=& \big(k_{2}~k_{4}~L^{2}+ k_{2}~k_{3}~L+k_{1}+M\big)|t_{2}-t_{1}|\\ &=& L|t_{2}-t_{1}|. \end{eqnarray*} Hence, we proved that \(~F:S_{L}~\rightarrow S_{L}~ \) and the class of functions \(\{Fx\}\) is equi-continuous on the set \(S_L.\) Applying Arzela-Ascoli Theorem [17], we deduce that \(F\) is compact operator. Now we show that \(F\) is continuous. Let \(\{x_{n}\} \subset S_{L},~x_{n}\rightarrow x \) on \([0,T]\), i.e. \(|x_{n}(\phi(t))-x(\phi(t))|\leq \epsilon_{1}\) this implies that \( | x_{n}(g_{i}(t,x(\phi(t))))-x(g_{i}(t,x(\phi(t)))) |\leq \epsilon_{2}\) for arbitrary \(\epsilon_{1},\epsilon_{2}\geq0\), \(i=1,~2,\) then \begin{eqnarray*} &~&|x_{n}(g_{i}(t,x_n(\phi(t))))-x(g_{i}(t,x(\phi(t)))) |\\ &&\leq |x_{n}(g_{i}(t,x_n(\phi(t))))-x_{n}(g_{i}(t,x(\phi(t))))|+ | x_{n}(g_{i}(t,x(\phi(t))))-x(g_{i}(t,x(\phi(t)))) |\\ &&\leq L|g_{i}(t,x_n(\phi(t)))-g_{i}(t,x(\phi(t)))|+| x_{n}(g_{i}(t,x(\phi(t))))-x(g_{i}(t,x(\phi(t)))) |\\ &&\leq \epsilon,~~i=1,~2. \end{eqnarray*} Then \begin{eqnarray*} x_{n}(g_{i}(t,x_n(\phi(t)))) \rightarrow x(g_{i}(t,x(\phi(t))))~in~ S_{L}, ~ ~i=1,~2 \end{eqnarray*} and by using the continuity of the functions \(f_{1} ,\) we obtain \begin{eqnarray*} f_{1}\big(t,x_{n}(g_{1}(t,x_{n}(\phi(t))))\big) &\rightarrow & f_{1}\big(t,x(g_{1}(t,x(\phi(t))))\big). \end{eqnarray*} Now by using the continuity of the functions \(f_{2},\) assumption \((5)\) and Lebesgues dominated convergence theorem [17], we obtain \begin{eqnarray*} \int_{0}^{t} f_{2}\big(t,x_{n}(g_{2}(t,x_{n}(\phi(t))))\big)ds &\rightarrow & \int_{0}^{t} f_{2}\big(t,x(g_{2}(t,x(\phi(t))))\big)ds, \end{eqnarray*} and \begin{eqnarray*} \lim_{n\rightarrow \infty}\big(F x_{n}\big)(t) &=& \lim_{n\rightarrow \infty} f_{1}\big(t,x_{n}(g_{1}(t,x_{n}(\phi(t))))\big)+\lim_{n\rightarrow \infty}\int_{0}^{t} f_{2}\big(t,x_{n}(g_{2}(t,x_{n}(\phi(t))))\big)ds\\& =& f_{1}\big(t,x(g_{1}(t,x(\phi(t))))\big)+\int_{0}^{t} f_{2}\big(t,x(g_{2}(t,x(\phi(t))))\big)ds\\ &=&\big(F x \big)(t). \end{eqnarray*} This proves that the operator \(F \) is continuous.

Now all conditions of Schauder fixed point theorem [17] are satisfied, then the operator \(F\) has at least one fixed point \(x\in S_{L}\). Consequently there exists at leat one solution \(x\in C[0,T]\) of Equation (3). This completes the proof.

Now, we introduce the following equivalence theorem.

Theorem 2. Let the assumptions (1)-(8) be satisfied, then the initial value Problem (1)-(2) has at least one positive solution \(x\in C[0,T]\).

Proof. Let \(x\) be a solution of the Problem (1)-(2). Integrate (1) and substitute by (2), we obtain the integral Equation (3). Let \(x\) be a solution of (3) differentiate (3) we obtain (1) and the initial value (2). This proves the equivalence between the Problem (1)-(2) and the integral Equation (3). Then the Problem (1)-(2) has at least one positive solution \( x \in C[0,T]\).

3. Applications

As application of our results, we introduce the following corollaries;

Corollary 1. Let the assumptions \((1)-(8)\) of Theorem 2 be satisfied, if

• (i) \(g_1(t,x(\phi(t)))=\int_0^{\phi(t)} g_{3}(s,x(s))ds,~~~g_{3}: [0,1]\times [0,1]\rightarrow R^+\) is continuous and \(g_{3}(t,x(t))\leq 1 ,\)
• (ii) \(g_{2}(t,x(t)) = x(t).\)

Then the initial value problem \begin{eqnarray*} &~&\frac{d}{dt}~\big[x(t)-f_{1}\big(t,x\big(\int_0^{\phi(t)}~g_3(s,x(s))ds)\big)\big] =f_{2}\big(t,x(x(\phi(t)))\big), ~~a.e.~~t\in(0,1], \\&~&x(0)= f_{1}(0,x(0)) \end{eqnarray*} has at least one positive solution \(x\in C[0,T]\).

Corollary 2. Let the assumptions of Theorem 1 be satisfied with \(g_{1}(t,x(\phi(t)))~= \phi(t)\) and \(g_{2}(t,x(\phi(t))) = x(\phi(t))\). Then the integral equation \begin{eqnarray*} x(t)=f_{1}(t,x(\phi(t)))+\int_{0}^{t}f_{2}(s,x(x(\phi(s))))ds,~~~t\in[0,T] \end{eqnarray*} has at least one solution \(x\in C[0,T].\) Consequently the initial value problem \begin{eqnarray*} \frac{d}{dt}~\big[x(t) -f_{1}(t,x(\phi(t)))\big]= f_{2}(t,x(x(\phi(t)))),~~~a.e.~~t\in[0,T] \end{eqnarray*} with the initial data \( x(0) = f_{1}(0,x(0))\) has at least one positive solution \( x \in C[0,T]\).

Corollary 3. Let the assumptions of Corollary 2 be satisfied with \(f_{1}(t,x)=t,\) then the initial value problem \begin{eqnarray*} \frac{d x(t)}{dt} &=& f(t,(x(x(\phi(t)))))~~a.e,~~t\in(0,T], \label{i1} \\ x(0)&=&0\label{i2} \end{eqnarray*} has at least one positive solution \( x \in C[0,T]\) where \(f(t,x)=f_{2}(t,x)+1\).

4. Uniqueness of the solution

In this section, we prove the uniqueness of the solution for the Problem (1)-(2). Therefore, we have to assume the following assumptions

• (\(1^{\prime}\)) \(|f_{2}(t,x)-f_{2}(t,y)|\leq k_{5}~|x-y|\),
• (\(2^{\prime}\)) \(~\sup_{t} |f_{2}(t,0)|\leq A,~~t\in [0,T]\),
• (\(3^{\prime}\)) \(|g_{2}(t,x)-g_{2}(t,y)|\leq k_{6}|x-y|\).

Theorem 3. Let the assumptions (1)-(3), (5)-(8) and (1\(^{\prime}\))-(3\(^{\prime}\)) be satisfied, if \(k_{2}~(L~k_{4}+1)+k_{5}~T~(L~k_{6}+1) < 1,\) then the initial value Problem (1)-(2) has a unique positive solution \(x \in C[0,T].\)

Proof. Assumption (4) of Theorem 1 can be deduced from assumptions (1\(^{\prime}\)) and (2\(^{\prime}\)) as follows \begin{eqnarray}\label{w1} |f_{2}(t,x)|&\leq & k_{5}~|x|+|f_{2}(t,0)|\nonumber\\ &\leq & k_{5}~x+A,\nonumber \end{eqnarray} then we deduce that all assumptions of Theorem 1 are satisfied and the solution of Equation (3) exists. Now let \(x,~y\) be two solutions of (3), then \begin{eqnarray*} |x(t)-y(t)| &=&\big|f_{1}\big(t,x(g_{1}(t,x(\phi(t))))\big)+\int_{0}^{t}~f_{2}\big(s,x(g_{2}(s,x(\phi(s))))\big)ds \\ &&- f_{1}\big(t,y(g_{1}(t,y(\phi(t))))\big)-\int_{0}^{t}~f_{2}\big(s,y(g_{2}(s,y(\phi(s))))\big)ds\big|\\ &\leq & |f_{1}\big(t,x(g_{1}(t,x(\phi(t))))\big)- f_{1}\big(t,y(g_{1}(t,y(\phi(t))))\big)|\\ &&+\int_{0}^{t}|f_{2}\big(s,x(g_{2}(s,x(\phi(s))))\big)- f_{2}\big(s,y(g_{2}(s,y(\phi(s))))\big)|ds\\ &\leq & k_{2}~|x(g_{1}(t,x(\phi(t))))-y(g_{1}(t,y(\phi(t))))|\\ &&+ k_{5}\int_{0}^{t}|x(g_{2}(s,x(\phi(s))))-y(g_{2}(s,y(\phi(s))))|ds. \end{eqnarray*} But \begin{eqnarray}\label{q6} &~&|x(g_{1}(t,x(\phi(t))))-y(g_{1}(t,y(\phi(t)))) |\nonumber\\&&= |x(g_{1}(t,x(\phi(t))))-x(g_{1}(t,y(\phi(t))))\nonumber+x(g_{1}(t,y(\phi(t))))-y(g_{i}(t,y(\phi(t)))) |\nonumber \\ &&\leq |x(g_{1}(t,x(\phi(t))))-x(g_{1}(t,y(\phi(t))))|\nonumber+| x(g_{1}(t,y(\phi(t))))-y(g_{1}(t,y(\phi(t)))) |\nonumber\\ &&\leq L|g_{1}(t,x(\phi(t)))-g_{1}(t,y(\phi(t)))|\nonumber+| x(g_{1}(t,y(\phi(t))))-y(g_{1}(t,y(\phi(t)))) |\nonumber\\ &&\leq L ~k_{4}~\|x-y\|+\|x-y\|\nonumber\\ &&=(L ~k_{4}+1)\|x-y\|.\nonumber \end{eqnarray} Similarly, we can obtain \begin{eqnarray*}\label{q06} |x(g_{2}(t,x(\phi(t))))-y(g_{2}(t,y(\phi(t)))) |\leq (L ~k_{6}+1)\|x-y\|. \end{eqnarray*} Now \begin{eqnarray*} |x(t)-y(t)|&\leq & k_{2}~(L ~k_{4}+1)\|x-y\| +k_{5}~\int_{0}^{t}(L ~k_{6}+1)\|x-y\|ds, \end{eqnarray*} and \begin{eqnarray*} \|x-y\| &\leq&\big(k_{2}~(L~k_{4}+1)+k_{5}~T~(L~k_{6}+1)\big)\|x-y\|. \end{eqnarray*} Then we deduce that \[ \bigg(1- \big(k_{2}~(L~k_{4}+1)+k_{5}~T~(L~k_{6}+1)\big)\bigg)~|| x - y || \leq 0, \] and from the assumptions \(\big(k_{2}~(L~k_{4}+1)+k_{5}~T~(L~k_{6}+1)\big)< 1~~\) we can obtain \(x=y\) and the solution of (3) is unique. Consequently, the initial value Problem (1)-(2) has a unique positive solution \(x \in C[0,T].\)

5. Continuous dependence

5.1. Continuous dependence on the function \( g_{1}\)

Here we prove that the solution of the Problem (1)-(2) depends continuously on the function \(g_{1} \).

Definition 1. The solution of the Problem (1)-(2) depends continuously on the function \(g_{1},\) if \(\forall ~\epsilon > 0, ~\exists ~\delta(\epsilon)>0~\) such that \begin{eqnarray*}\label{cont} ||g_{1}-g_{1}^{*}||\leq \delta \Rightarrow \|x-x^{*}\|\leq \epsilon, \end{eqnarray*} where \(x^{*}\) is the unique solution of the equation

with the initial data

Theorem 4. Let the assumptions of Theorem 3 be satisfied, then the solution of initial value Problem (1)-(2) depends continuously on the function \(g_{1}. \)

Proof. Let \(x\) and \(x^*\) be the solution of the initial value Problems (1)-(2) and (6)-(7) respectively, then we have \begin{eqnarray*} |x(t)-x^{*}(t)| & =& \big|f_{1}\big(t,x(g_{1}(t,x(\phi(t))))\big)+\int_{0}^{t}~f_{2}\big(s,x(g_{2}(s,x(\phi(s))))\big)ds \\ &&- f_{1}\big(t,x^{*}(g^{*}_{1}(t,x^{*}(\phi(t))))\big)+ \int_{0}^{t}~f_{2}\big(s,x^{*}(g_{2}(s,x^{*}(\phi(s))))\big)ds\big|\\ &\leq& |f_{1}\big(t,x(g_{1}(t,x(\phi(t))))\big) -f_{1}\big(t,x^{*}(g^{*}_{1}(t,x^{*}(\phi(t))))\big) | \\ &&+ \int_{0}^{t} |f_{2}\big(s,x(g_{2}(s,x(\phi(s))))\big)- f_{2}\big(s,x^{*}(g_{2}(s,x^{*}(\phi(s))))\big)|ds\\ &\leq& k_{2}~ |x(g_{1}(t,x(\phi(t))))\big) -x^{*}(g^{*}_{1}(t,x^{*}(\phi(t)))\big) | \\ &&+ k_{5}\int_{0}^{t} |x(g_{2}(s,x(\phi(s))))\big)- x^{*}(g_{2}(s,x^{*}(\phi(s)))\big)|ds\\ &\leq & k_{2}~ |x(g_{1}(t,x(\phi(t))))\big) -x^{*}(g_{1}(t,x^{*}(\phi(t)))\big) | \\ &&+ k_{2}~ |x^{*}(g_{1}(t,x^{*}(\phi(t))))\big) -x^{*}(g^{*}_{1}(t,x^{*}(\phi(t)))\big) | + k_{5}~T (L~k_{6}+1)\|x-x^{*}\|\\ &\leq& k_{2} (L~k_{4}+1)\|x-x^{*}\|+ k_{2}~ L~|g_{1}(t,x^{*}(\phi(t)))) -g_{1}^{*}(t,x^{*}(\phi(t))) | \\ &&+ k_{5}~T (L~k_{6}+1)\|x-x^{*}\|\\ &\leq& \big( k_{2}~ (L~k_{4}+1)+k_{5}~T (L~k_{6}+1)\big)\|x-x^{*}\|+k_{2}~L ~\delta. \end{eqnarray*} Then \[ \|x-x^{*}\|\leq \frac{k_{2}~L ~\delta}{\big(1-\big(k_{2}~ (L~k_{4}+1)+k_{5}~T (L~k_{6}+1)\big)}=\epsilon, \] and by the assumption \(\big(k_{2}~ (L~k_{4}+1)+k_{5}~T (L~k_{6}+1)\big)< 1,\) then the solution of (3) depends continuously on the functions \(g_{1}.\) Consequently the solution of the Problem (1)-(2) depends continuously on the functions \(g_{1} \) which complete the proof.

5.2. Continuous dependence on the function \( g_{2}\)

Here we prove that the solution of the Problem (1)-(2) depends continuously on the function \(g_{2} \).

Definition 2. The solution of the Problem (1)-(2) depends continuously on the function \(g_{2},\) if \(\forall ~\epsilon > 0, ~\exists ~\delta(\epsilon)>0~such~that,\) \begin{eqnarray*} ||g_{2}-g_{2}^{*}||\leq \delta \Rightarrow \|x-x^{*}\|\leq \epsilon, \end{eqnarray*} where \(x^{*}\) is the unique solution of the equation

with the initial data

Theorem 5. Let the assumptions of Theorem 3 be satisfied, then the solution of initial value Problem (1)-(2) depends continuously on the function \(g_{2}. \)

Proof. The proof follow similarly as the proof of Theorem 4.

6. Examples

Example 1. Consider the following problem

with the initial data where \(t\in (0,1]\) and \(\beta \in (0,1]\). Here we have

\(\phi(t)=\beta~t\),

\(g_{1}\big(t,x(\phi(t))\big)=\frac{\beta~t}{1+x^{2}(\beta~t)},\)

\(g_{1}\big(t,x)\leq \beta~t,\)

\( f_{1}\big(t,x(g_{1}\big(t,x(\phi(t)))\big)\big) =\frac{1}{18}(1+t^{2})+\frac{1}{8}x\bigg(\frac{\beta~t}{1+x^{2}(\beta~t)}\bigg),\)

\( f_{1}\big(t,x)=\frac{1}{18}(1+t^{2})+\frac{1}{8}x,\)

\(g_{2}\big(t,x(\phi(t))\big)=\frac{x(\beta~t)~e^{-x^{2}(\beta~t)}}{1+\sin^{2}x(\beta~t)},\)

\(g_{2}\big(t,x)\leq x,\)

\( f_{2}\big(t,x(g_{2}\big(t,x(\phi(t)))\big)\big) =\frac{1}{7-t}+\frac{e^{-t}}{16}~x\bigg(\frac{x(\beta~t)~e^{-x^{2}(\beta~t)}}{1+\sin^{2}x(\beta~t)}\bigg), \)

\( f_{2}\big(t,x)=\frac{1}{7-t}+\frac{e^{-t}}{16}x,.\)

Thus we have \(k_{1}=\frac{1}{9},~k_{2}=\frac{1}{8},~k_{3}=2, ~k_{4}=2, ~k_{5}=\frac{1}{16},~A=\frac{1}{6},~ M=\frac{11}{48} ,~L\simeq 0.557< 1~ and~ L~T+x(0)\approx0.62< T=1.\)

Now all the assumptions of Theorem 1 are satisfied, then the Problem (10)-(11) has at least one solution \(x\in C[0,T]\).

Example 2. Consider the following problem

with the initial data where \(t\in (0,\frac{1}{2}]\). Here we have

\(\phi(t)=t^{2}\),

\(g_{1}\big(t,x(\phi(t))\big)=\frac{ \frac{1}{4}~t^{2}}{1+32~x(t^{2})} ,\)

\(g_{1}\big(t,x)\leq t^{2},\)

\( f_{1}\big(t,x(g_{1}\big(t,x(\phi(t)))\big)\big) =\frac{1}{48}(1+t)+ \frac{1}{4}~t^{2}~x\bigg(\frac{\frac{1}{4}~t^{2}}{1+ 32~x(t^{2})}\bigg), \)

\( f_{1}\big(t,x)=\frac{1}{48}(1+t)+ \frac{1}{4}~t^{2}~x,\)

\(g_{2}\big(t,x(\phi(t))\big)=\frac{x(t^{2})~\sin^{2}(x(t^{2}))}{1+x^{2}(t^{2})} ,\)

\(g_{2}\big(t,x)\leq x,\)

\( f_{2}\big(t,x(g_{2}\big(t,x(\phi(t)))\big)\big) =\frac{1}{5+2t} \sin^{2}(3(t+1)) +\frac{1}{12}~x\bigg(\frac{x(t^{2})~\sin^{2}(x(t^{2}))}{1+x^{2}(t^{2})}\bigg), \)

\( f_{2}\big(t,x)=\frac{1}{5+2t} \sin^{2}(3(t+1)) +\frac{1}{12}~x.\)

Thus, we have \(k_{1}=\frac{7}{48},~k_{2}=\frac{1}{16},~k_{3}=\frac{17}{4}, ~k_{4}=2, ~k_{5}=\frac{1}{12},~A=\frac{1}{5},~ M=\frac{29}{120} ,~L\simeq 0.586< 1~ and~ L~T+x(0)\approx 0.3139< T=\frac{1}{2}.\)

Now all the assumptions of Theorem 1 are satisfied, then the Problem (12)-(13) has at least one solution \(x\in C[0,T]\).

7. Conclusion

Here we relaxed the assumptions and generalized the results in [8,11,14,18] and [1]. We proved the existence of at lease one solution of the Problem (1)-(2). The sufficient condition for the uniqueness of the solution have been given and the continuous dependence of the unique solution have been proved. Also some examples and applications have been given.

Author Contributions

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

Conflicts of interest

The authors declare no conflict of interest.

1. Introduction

Consider the following problem:

where \(\Omega \) is a bounded regular domain in \(\mathbb{R}^{n},\) \(n\geq 1\) with a smooth boundary \(\partial \Omega \). \(\omega \), \(\mu \) and \(m\), \(p\) are real numbers.

The nonlinear wave equations

has been investigated by many authors [1,2,3,4,5,6,7,8,9,10]. In the absence of the nonlinear source term, it is well know that the presence of one damping term ensures global existence and decay of solutions for arbitrary initial condition [5,6]. For \(\omega =\mu =0\) the nolinear term \(u\mid u\mid ^{p-2}\) causes finite time blow up of solutions with negative energy [2]. The interaction between the damping and the source terms was first considered by Levine [11]. He showed that solutions with negative initial energy blows up in finite time. When \(\omega =0\) and the linear term \(u_{t}\) is replaced by \(\mid u_{t}\mid ^{r-2}u_{t}\), Georgiev and Todorowa [12] extended Levin's result to the case where \(r>2\). In their work, the authors introduced a method different from the one know as the concavity method. The termined suitable relations between \(r\) and \(p\), for whith there is global existence or alternatively fnite time blow-up.

For the initial boundary value problem of a quasilinear equation

\begin{equation*} u_{tt}-div\left( {\mid \nabla u\mid ^{m-2}\nabla u}\right) +au_{t}\mid u_{t}\mid ^{p-2}-\Delta u_{t}=bu\mid u\mid ^{r-2}, \end{equation*} Yang and Chen [13,14] studied the problem (7)-(9) and obtained global existence results under the growth assumptions on the nonlinear terms and the initial value. This global existence results have been improved by Liu and Zhao [15] by using a new method. In [13], the author considered a similar problem to (7)-(9) and proved a blow-up result under the condition \(p>max(r,m)\) and the energy is sufficiently negative. Messaoudi and Said-Houari [16] improved the results in [15] and showed that blow-up takes place for negative initial data only regardless of the size of \(\Omega\) Messaoudi in [17] showed that for \(m=2\), the decay is exponential. In absence of strong damping \(-\Delta u_{t}\) equation (7) becames For \(b=0\), it is well known that the damping term assures global existence and decay of the solution energy for arbitrary initial value [18]. For \(a=0\), the source term causes finite time blow-up of solutions with negative initial energy if \(r>m\) (see [2]). When the quasilinear operator \(-div\left( {\mid \nabla u\mid ^{m-2}\nabla u}\right) \) is replaced by \(\Delta ^{2}u\) , Wu and Tsai [19] showed that the solution is global in time under some conditions without the relation between \(p\) and \(r\). They also proved that the local solution blows up infinite time if \(r>p\) and the initial energy is nonnegative, and gave the decay estimates of the energy function and the lifespan of solutions. In this paper, we show that the local solutions of the problem (1)-(3) can be extented in infinite time to global solutions with the some conditions on initial data in the stable set for which the solutions decay expontially with \(L_{p}\) norm. The key tool in the proof is an idea of Haraux and Zuazua [6] and [9] with is based on the construction of a suitable Lyapunov function.

2. Assumptions and preliminaries

In this section, we present some material needed in the proof in our result.

Lemma 1. \(\left( Young^{\prime }s\ inequality\right) \) \(Let\ a,\ b\geq 0\) and \(\frac{1}{p}+\frac{1}{q}\)=1 for \( 1 < p, q < + \infty \), then one has the inequality \(ab \leq \delta a^{p} +C\left( \delta \right) b^{q}, \) where \(\delta >0 \) is an arbitrary constant, and \(C\left( \delta \right) \) is a positive constant depending on \(\delta \).

Lemma 2. Let s be a number with \(2\leq s < +\infty \) if \(\ n\leq r\) and \(2\leq s\leq \frac{nr}{n-r}\) if \(n>r\). Then there is a constant \(C\) depending on \(\Omega \) and s such that \(\left\Vert u\right\Vert _{s}\) \(\leq C\left\Vert \nabla u\right\Vert _{r}\), \(u\in W_{0}^{1,r}\left( \Omega \right) .\)

We denote the total energy related to the problem (1)-(3) by

We also introduce the following functionals: As in [20], we can now define the so called " Nehari manifold" as follows: \begin{equation*} \mathcal{N}\text{=}\left\{ u\in W_{0}^{1,m}\left( \Omega \right) \backslash \left\{ 0\right\} ;\text{ }I\left( t\right) =0\right\} . \end{equation*} \(\mathcal{N}\) separates the two unbounded sets: \begin{equation*} \mathcal{N}^{+}\text{=}\left\{ u\in W_{0}^{1,m}\left( \Omega \right) ;\text{ }I\left( t\right) >0\right\} \cup \left\{ 0\right\} \end{equation*} and \begin{equation*} \mathcal{N}^{-}\text{=}\left\{ u\in W_{0}^{1,m}\left( \Omega \right) ;\text{ }I\left( t\right) < 0\right\} . \end{equation*} Assumptions:

• \(\left( A1\right) :\) Assume that \(I\left( 0\right) >0,\) and \(0< E\left( 0\right) \) such that
  \begin{equation} B=c^{p}\left( \frac{mp}{p-m}E\left( 0\right) \right) ^{\frac{p-m}{m}}< 1.% \text{ } \label{14} \end{equation}

  (14)
  where \(c\) is the Poincaré constant.
• \(\left( A2\right) :\) \(p\) satisfies

\[(2< m< p\leq\dfrac{nm}{n-m}, n\geq m; 2< m < p\leq+\infty, n < m.\] For simplicity, we define the weak solutions of (1)-(3) over the interval \(\left[ 0,T\right) \), but it is to be understood throughout that \(T\) is either infinity or the limit of the existence interval.

Definition 1. We say that \(u\left( x,t\right) \) is a weak solution of the problem (1)-(3) on the interval \(\Omega \times \left[ 0,T\right) ,\) if \(u\in L^{\infty }\left( \left[ 0,T\right) ;W_{0}^{1,m}\left( \Omega \right) \right) ,u_{t}\ \in L^{\infty }\left( % \left[ 0,\text{ }T\right) ;\text{ }L^{2}\left( \Omega \right) \right) \cap L^{2}\left( \left[ 0,\text{ }T\right) ;\ H_{0}^{1}\left( \Omega \right) \right) \) satisfy the following conditions:

• \(\left( i\right) \)
  \begin{equation} \left( u^{\prime \prime }\left( t\right) ,\phi \right) +\left( \frac{\mid \nabla u\mid ^{2m-2}\nabla u}{\sqrt{1+\mid \nabla u\mid ^{2m}}},\nabla \phi \right) +\omega \left( \nabla u^{\prime },\nabla \phi \right) +\mu \left( u^{\prime },\phi \right) =\left( u\lvert u\rvert ^{p-2},\phi \right) , \label{15} \end{equation}

  (15)
  for any function \(\phi \in W_{0}^{1,m}\left( \Omega \right) \) and a.e. \(t\in \left[ 0,T\right)\).
• \(\left( ii\right) \)
  \begin{equation} u\left( x,0\right) =u_{0}\left( x\right) \in L^{2}\left( \Omega \right) ,\ u_{t}\left( x,0\right) =u_{1}\left( x\right) \in L^{1}\left( \Omega \right) . \label{16} \end{equation}

  (16)

Theorem 1. (Local existence) Suppose that \(u_{0}\in L^{2}\left( \Omega \right) ,\ u_{1}\in L^{1}\left( \Omega \right) \) and \(E\left( 0\right) >0,\) then there exists \(T>0\) such that problem (1)-(3) has a unique solution u satisfying \(u\ \in L^{\infty}\left( [0, T ];\ W_{0}^{1,m}\left( \Omega\right)\right), \) \(u_{t}\ \in L^{\infty }\left( \left[ 0,\text{ }T\right) ;\text{ }L^{2}\left( \Omega \right) \right) \cap L^{2}\left( \left[ 0,\text{ }T\right) ;\ H_{0}^{1}\left( \Omega \right) \right) .\)

3. Global existence and exponential decay of solutions

In this section we are going to obtain the existence of local solutions to the problem (1)-(3) and exponential decay of solution. We will use the Faedo- Galerkin's method approximation.

Let \(\{w_{l}\}_{l=1}^{\infty }\) be a basis of \(W_{0}^{1,m}\left( \Omega \right) \) wich constructs a complete orthonormal system in \(L^{2}\left( \Omega \right) .\) Denote by \(V_{k}=span\{w_{1},w_{2},...,w_{k}\}\) the subspace generated by the first \(k\) vectors of the basis \( \{w_{l}\}_{l=1}^{\infty }.\) By the normalization, we have \(\lVert w_{l}\rVert =1.\) for any given integer k, we consider the approximation solution

\begin{equation*} u_{k}\left( t\right) =\sum_{l=1}^{k}u_{lk}\left( t\right) v_{l}, \end{equation*} where \(u_{k}\) is the solutions to the following Cauchy problem where \(l=1,...,k,\) with initial conditions \(u_{k}\left( 0\right) =u_{0k}\) and \(u_{k}^{\prime }\left( 0\right) =u_{1k},\) \(u_{k}\left( 0\right) \) and \( u_{k}^{\prime }\left( 0\right) \) are chosen in \(V_{k}\) such that Well known results on the solvability of nonlinear ODE provide the existence of a solution to problem (17)-(18) on interval \(\left[ 0,\tau\right) \) for some \(\tau >0\) and we can extend this solution to the whole interval \(\left[ 0,T\right] \) for any given \(T>0\) by making use of the a priori estimates below. Multiplying equation (17) by \(u_{lk}^{\prime }\left( t\right) \) and sum for \(l=1,...,k,\) we obtain Integrating (19) over \(\left( 0,t\right)\), we obtain the estimate Since \(I\left( 0\right) >0,\) then there exists \(\tau < T\) by continuity such that \(I\left( t\right) \geq 0,\). We get from (12) and (13) that Hence we have From (11) and (13) we obvioulsy have \(\forall t\in \left[ 0,\text{ }\tau \right] ,\) \(J\left( u_{k}\left( t\right) \right) \leq E\left( u_{k}\left( t\right) \right) .\) Thus we obtain Since \(E\) is a decreasing function of \(t,\) we have By using Lemma 2, we easily have \begin{eqnarray*} \left\Vert u_{k}\right\Vert _{p}^{p} &\leq &c^{p}\left\Vert \nabla u_{k}\right\Vert _{m}^{p}=c^{p}\left( \underset{\Omega }{\int }\left\vert \nabla u_{k}\right\vert ^{m}dx\right) ^{\frac{p}{m}}\leq c^{p}\left( \underset{\Omega }{\int }\sqrt{1+\left\vert \nabla u_{k}\right\vert ^{2m}} dx\right) ^{\frac{p}{m}} \\ &\leq &c^{p}\left( \underset{\Omega }{\int }\sqrt{1+\left\vert \nabla u_{k}\right\vert ^{2m}}dx\right) ^{\frac{p-m}{m}}\underset{\Omega }{\int } \sqrt{1+\left\vert \nabla u_{k}\right\vert ^{2m}}dx \end{eqnarray*} Using the inequality (25), we deduce \begin{equation*} \left\Vert u_{k}\right\Vert _{p}^{p}\leq c^{p}\left( \frac{mp}{p-m}E\left( 0\right) \right) ^{\frac{p-m}{m}}\underset{\Omega }{\int }\sqrt{1+\left\vert \nabla u_{k}\right\vert ^{2m}}dx. \end{equation*} Now exploiting the inequality (14), we obtain Hence \(\underset{\Omega }{\int }\sqrt{1+\left\vert \nabla u_{k}\right\vert ^{2m}}dx-\) \(\left\Vert u_{k}\right\Vert _{p}^{p}>0,\) \(\forall t\in \left[ 0, \text{ }\tau \right] ,\) this shows that \(I\left( u_{k}\left( t\right) \right) >0,\) by repeating this procedure, \(\tau \) is extended to T.

Since \(\underset{\Omega }{\int }\sqrt{1+\left\vert \nabla u_{k}\right\vert ^{2m}}dx>\lVert \nabla u_{k}\rVert _{m}^{m},\) it follows from (20) and (26) that

From (27), we have Furthermore, we have from Lemma 2 and (28) that By (28) and (29), we infer that there exists a subsequence of \(u_{k}\) (denote still by the same symbol) and a function \(u\) such that By the Aubin-Lions compactness Lemma [7], we conclude from (30) that \begin{equation*} \left\{ \begin{array}{c} u_{k}\rightharpoonup u\ \ \text{strongly in}\ \ C\left( \left[ 0,T\right] ;L^{2}\left( \Omega \right) \right) , \\ u_{k}^{\prime }\rightharpoonup u^{\prime }\mathit{\ }\text{ strongly in }C\left( \left[ 0,T\right] ;L^{2}\left( \Omega \right) \right) , \end{array} \right. \end{equation*} and It follows from Lemma 1.3 in [21] and (31) By the last formula (32) and (30), we obtain \(\mathcal{X}=\lvert u\rvert ^{p-2}u\) On the other hand, taking \(\phi =1\), (17) become We have \begin{equation*} \lvert \left( u_{k}^{\prime \prime }\left( t\right) ,1\right) +\mu \left( u_{k}^{\prime },1\right) \rvert \geq \lVert u_{k}^{\prime \prime }\rVert -\mu \lVert u_{k}^{\prime }\rVert. \end{equation*} Since, the measure of \(\Omega \) is finite, by the embedding theorem, (30) and (33), we obtain \begin{equation*} \lVert u_{k}^{\prime \prime }\rVert \leq C, \end{equation*} then \begin{equation*} \left\{ u_{k}^{\prime \prime }\right\} \mathit{\ }\text{is uniformly bounded in }L^{\infty }\left( \left[ 0,T\right] ;L^{1}\left( \Omega \right) \right) . \end{equation*} Similarly, we have Setting up \(k\longrightarrow \infty \) and passing to the limit in (17), we obtain \begin{equation*} \left( u^{\prime \prime }\left( t\right) ,v_{l}\right) +\left( \frac{\mid \nabla u\mid ^{2m-2}\nabla u}{\sqrt{1+\mid \nabla u_{k}\mid ^{2m}}},\nabla v_{l}\right) +\omega \left( \nabla u^{\prime },\nabla v_{l}\right) +\mu \left( u^{\prime },v_{l}\right) =\left( u\lvert u\rvert ^{p-2},v_{l}\right) , \end{equation*} \(l=1,...,k.\) Since \(\{v_{l}\}_{l=1}^{\infty }\) is a base of \(W_{0}^{1,m}\left( \Omega \right)\), we deduce that \(u\) satisfies (1).

From (30), (34) and Lemma 3.1.7 in [22], with \(B=L^{2}\left( \Omega \right) \) and \(B=L^{1}\left( \Omega \right) ,\) respectively, we infer that

We get from (18) and (35) that \(u\left( 0\right) =u_{0},\) \(u^{\prime }\left( 0\right) =u_{1}.\) Thus, the proof is complete.

Lemma 3. Assume that \(p>m\) and \( u_{0}\in \mathcal{N}^{+},\) \( u_{1}\in L^{2}\left( \Omega \right) .\) If \(0< E\left( 0\right) \) and satisfy (14) then the local solution of the problem (1)-(3) is global in time.

Proof. Since the map \(t\longmapsto E\left( t\right) \) is a decreasing of the time \(t,\) we have

which give thus, \(\forall t\in \left[ 0,\text{ }T\right) ,\) \(\left\Vert u_{t}\right\Vert _{2}^{2}+\underset{\Omega }{\int }\sqrt{1+\left\vert \nabla u\right\vert ^{2m}}dx\) is uniformly bounded by a constant depending only on \(E\left( 0\right) ,\) \(p\) and \(m\) then the solution is global, so \(T_{\max }=\infty .\)

Theorem 2. Assume that \(p>m\). Let \(u_{0}\in \mathcal{N}^{+}\) and \(u_{1}\in L^{2}\left( \Omega \right) .\) Moreover, assume that \(0< E\left( 0\right) \) and satisfy (14). Then there exists two positive constants \( \alpha \) and \(\beta \) independent of \(t\) such that: \(0< E\left( t\right) \leq \beta e^{-\alpha t},\) \(\forall t>0.\)

Proof. Since we have proved that \(t\geq 0,\) \(u\left( t\right) \in \mathcal{N}^{+},\) we already have \begin{equation*} 0< E\left( t\right) ,\forall t\geq 0. \end{equation*} We define a Lyaponov function, for \(\epsilon >0.\)

We prove that \(L\left( t\right) \) and \(E\left( t\right) \) are equivalent in the sens that there exist two constants \(B_{1}\) and \(B_{2}\) depending on \( \epsilon\) such that for \(t\geq 0\) By the Lemma 1, we have \begin{equation*} L\left( t\right) =E\left( t\right) +\epsilon \underset{\Omega }{\int } u_{t}udx+\frac{\epsilon \omega }{2}\left\Vert \nabla u\right\Vert _{2}^{2} \leq E\left( t\right) +\epsilon \left( \frac{1}{4\delta }\left\Vert u_{t}\right\Vert _{2}^{2}+\delta \left\Vert u\right\Vert _{2}^{2}\right) + \frac{\epsilon \omega }{2}\left\Vert \nabla u\right\Vert _{2}^{2}. \end{equation*} Thanks of the Poincaré inequality and since \(\delta\) is an arbitrary constant, we choose \(\delta \) small suffisant for that, Then, we get \begin{equation*} L\left( t\right) \leq E\left( t\right) +\epsilon \frac{1}{4\delta } \left\Vert u_{t}\right\Vert _{2}^{2}+\epsilon \left( \delta C+\frac{\omega }{ 2}\right) \left\Vert \nabla u\right\Vert _{2}^{2} \leq E\left( t\right) +\epsilon \frac{1}{4\delta }\left\Vert u_{t}\right\Vert _{2}^{2}+\epsilon \left( 1+\frac{\omega }{2}\right) \underset{\Omega }{\int }\sqrt{1+\left\vert \nabla u\right\vert ^{2m}}dx. \end{equation*} By (37), we get where \(B_{2}=\left( 1+\epsilon \frac{1}{2\delta }+\epsilon \left( 1+\frac{\omega }{2}\right) \frac{mp}{p-m}\right)\).

On the other hand, we have

\begin{align*} L\left( t\right) &\geq E\left( t\right) -\epsilon \left( \frac{1}{4\delta } \left\Vert u_{t}\right\Vert _{2}^{2}+\delta \left\Vert u\right\Vert _{2}^{2}\right) +\frac{\epsilon \omega }{2}\left\Vert \nabla u\right\Vert _{2}^{2}\\ &\geq E\left( t\right) -\epsilon \frac{1}{4\delta }\left\Vert u_{t}\right\Vert _{2}^{2}-\epsilon \delta \left\Vert u\right\Vert _{2}^{2}\\ &\geq E\left( t\right) -\epsilon \frac{1}{2\delta }E\left( t\right) -\epsilon \delta \left\Vert u\right\Vert _{2}^{2}\\ &\geq \left( 1-\epsilon \frac{1}{2\delta }\right) E\left( t\right) -\epsilon \delta \left\Vert u\right\Vert _{2}^{2}. \end{align*} From (37) and (40), we obtain where \(B_{1}=\left( 1-\epsilon \frac{1}{2\delta }-\epsilon \frac{mp}{p-m}% \right)\).

Now, we have

\begin{align*} \frac{d}{dt}L\left( t\right) &=-\omega \left\Vert \nabla u_{t}\right\Vert _{2}^{2}-\mu \left\Vert u_{t}\right\Vert _{2}^{2}+\epsilon \left\Vert u_{t}\right\Vert _{2}^{2} +\epsilon \underset{\Omega }{\int }{div}\left( \frac{\left\vert \nabla u\right\vert ^{2m-2}\nabla u}{\sqrt{1+\left\vert \nabla u\right\vert ^{2m}}}% \right) udx+\epsilon \left\Vert u\right\Vert _{p}^{p}-\epsilon \mu \underset{% \Omega }{\int }u_{t}udx\\ &=-\omega \left\Vert \nabla u_{t}\right\Vert _{2}^{2}-\mu \left\Vert u_{t}\right\Vert _{2}^{2}+\epsilon \left\Vert u_{t}\right\Vert _{2}^{2}-\epsilon \underset{\Omega }{\int }\frac{\left\vert \nabla u\right\vert ^{2m}}{\sqrt{1+\left\vert \nabla u\right\vert ^{2m}}}dx +\epsilon \left\Vert u\right\Vert _{p}^{p}-\epsilon \mu \underset{\Omega }{% \int }u_{t}udx. \end{align*} So that So Using the inequality (37) and (44), we deduce \begin{eqnarray*} \frac{d}{dt}L\left( t\right) &\leq &2\left( \epsilon \left( \frac{\mu }{ 4\delta }+1\right) -\mu \right) E\left( t\right) +\epsilon \left( 1+\mu \right) \frac{mp}{p-m}E\left( t\right) \\ &\leq &-\left( 2\mu -\epsilon \left( \left( \frac{\mu }{2\delta }+2\right) +\left( 1+\mu \right) \frac{mp}{p-m}\right) \right) E\left( t\right). \end{eqnarray*} We choosing \(\epsilon \) small enough such that So From (39), we have Integrating the provious differential inequality (47) between \(0\) and \(t\) gives the following estimate for the function \(L:\) Consequently, by using (39) once again, we conclude By using (26) and (37) we easily have The proof is complete.

4. Conclusion

In this paper, we have studied a class of hyperbolic equation supplemented with Dirichlet boundary conditions as a model of wave equation with damping and source nonlinear terms. We showed that the solution with positive initial energy exponentially decay, this is mainly due to the presence of one of term of weak or strong damping.

Acknowledgments:

The authors wish to thank deeply the anonymous referee for useful remarks and careful reading of the proofs presented in this paper.

Author Contributions:

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

Conflict of Interests:

The authors declare no conflict of interest.

1. Introduction

This paper deals with the initial boundary value problem of the dispersive wave equation with memory and source terms where \(\Omega\) is a bounded domain in \(\mathbb{R}^{d}\) \((d\geq1)\) with a smooth boundary \(\partial\Omega\), \(\alpha\) is a positive constant and \(g(t)\) is a positive function that represents the kernel of the memory term, which will be specified in Section 2. Here, we understand \(\Delta^{2}u\) to be the dispersive term. In the absence of the viscoelastic term and the dispersive term (that is, if \(g=\alpha= 0\)), the model (1) reduces to the weakly damped wave equation The interaction between the weak damping term and the source term are considered by many authors. We refer the reader to, Haraux and Zuazua [1], Ikehata [2] and Levine [3, 4]. If \(\alpha=0\) and \(g\) is not trivial on \(\mathbb{R}\), but replacing the fourth order memory term in (1) by a weaker memory of the form \(\int_{0}^{t}g(t-\tau)\Delta u(\tau)d\tau\), then (1) can be rewritten as follows The Equation (3) has been considered by Wang et al [5]. Under some appropriate assumptions on \(g\), by introducing potential wells they obtained the existence of global solution and the explicit exponential energy decay estimates. Our main goal in the present paper is to discuss the global solutions and general decay to the following weakly damped wave equation with dispersive term, the fourth order memory term and the nonlinear source term with simply supported boundary condition and initial conditions where \(\Omega\) is a bounded domain of \(\mathbb{R}^{d}\) with a smooth boundary \(\partial\Omega\) and \(p>1\). Here, \(\nu\) is the unit outward normal to \(\partial\Omega\), and \(g(t)\) is a positive function that represents the kernel of the memory term, which will be specified in Section 2. We prove that Problem (4)-(6) has a global weak solution assuming small initial data. In addition, we show the general decay of solutions. The global solutions are constructed by means of the Galerkin approximations and the general decay is obtained by employing the technique used in [6].

2. Preliminaries

Before proceeding to our analysis, we use the following abbreviations \(\|\cdot\|_{q}=\|\cdot\|_{L^{q}(\Omega)}\) \((1\leq q\leq+\infty)\) denotes usual \(L^{q}\) norm, \((\cdot,\cdot)\) denotes the \(L^{2}\)-inner product, and consider the Sobolev spaces \(H^{1}_{0}(\Omega)\) and \(H^{2}_{0}(\Omega)\) with their usual scalar products and norms. We also use the embedding \(H^{1}_{0}(\Omega)\hookrightarrow L^{q}(\Omega)\) for \(2< q< \frac{2d}{d-2}\) if \(d\geq3\) or \(2< q< \infty\) if \(d=1,2\). In this case, the embedding constant is denoted by \(C_{*}\), that is \( \|u\|_{q}\leq C_{*}\|\nabla u\|_{2}. \) We define the polynomial \(Q\) by \( Q(z)=\frac{1}{2}z^{2}-\frac{C_{*}^{p+1}}{p+1}z^{p+1}, \) which is increasing in \([0,z_{0}]\), where \( z_{0}=C_{*}^{\frac{p+1}{1-p}} \) is its unique local maximum. Next, we give the assumptions for Problem (4)-(6).
(G1) The relaxation function \(g:\mathbb{R}_{+}\rightarrow \mathbb{R}_{+}\) is a bounded \(C^{1}\) function such that \( g(0)>0,\quad 0< \eta=1-\int_{0}^{\infty}g(\tau)d\tau\leq1-\int_{0}^{t}g(\tau)d\tau=\eta(t). \)
(G2) There exist positive constants \(\xi_{1}\) and \(\xi_{2}\) such that \( -\xi_{1}g(t)\leq g'(t)\leq-\xi_{2}g(t)\quad \forall t\geq0. \)
(G3) We also assume that \( 1< p\leq \frac{d+2}{d-2}\ \ \mbox {if} \ \ d\geq3 \ \ \mbox {and} \ \ \ p>1 \ \ \ \mbox {if} \ \ d=1,2. \) where \(\lambda_{1}\) is the first eigenvalue of the following problem

Remark 1. [7] Assuming \(\lambda_{1}\) is the first eigenvalue of the problem (7), we have

Now, we define the following energy function associated with a solution \(u\) of the Problem (4)-(6) for \(u\in H^{2}_{0}(\Omega)\), and is the initial total energy. To facilitate further on our analysis, we use the following notation \begin{equation*} (g\circ\Delta u)(t)=\int_{0}^{t}g(t-\tau)\|\Delta u(\tau)-\Delta u(t)\|_{2}^{2}d\tau. \end{equation*} Now, we are in a position to state our main results.

3. Main results

Theorem 1. Assume that \((G1)-(G3)\) hold, \(u_{0}\in H^{2}_{0}(\Omega)\), \(u_{1}\in L^{2}(\Omega)\). Further assume that \(\|\nabla u_{0}\|_{2}< z_{0}\) and \(E(0)< Q(z_{0})\), then the Problem (4)-(6) possesses a global weak solution satisfying; \(u\in L^{\infty}(0,\infty;H^{2}_{0}(\Omega)),\quad u_{t}\in L^{\infty}(0,\infty;L^{2}(\Omega))\) for \(0\leq t< \infty\), and the energy identity

holds for \(0\leq t< \infty\). Moreover, for \(\zeta:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}\) a increasing \(C^{2}\) function satisfying and, if \(\|g\|_{L^{1}(0,\infty)}\) is sufficiently small, we have for \(\kappa>0\); \( E(t)\leq E(0)e^{-\kappa \zeta(t)},\quad \forall t\geq0. \)

Remark 2. From (11) and \((G2)\), we can easily obtain

Remark 3. For \(\zeta(t)=t+\frac{t}{t+1}\), we can get the exponential decay rate \( E(t)\leq E(0)e^{-\kappa t},\quad \forall t\geq0\). For \(\zeta(t)=ln(1+t)\), we can get polynomial decay rate \( E(t)\leq E(0)(1+t)^{-\kappa },\quad \forall t\geq0. \)

4. Proof of main results

In this section, we shall divide the proof into two steps. In Step 1, we prove the global existence of weak solutions by using Galerkin's approximations. In Step 2, we establish the general decay of energy employing the method used in [6].

Step 1 Global existence of weak solutions

Let \(\left\{\omega_{j}\right\}_{j=1}^{\infty}\) be an orthogonal basis of \(H^{2}_{0}(\Omega)\) with \(\omega_{j}\) being the eigenfunction of the problem \( -\Delta \omega_{j}=\lambda_{j}\omega_{j},\quad x\in\Omega,\quad \omega_{j}=0,\quad x\in\partial\Omega. \) Let \(V^{n}=\text{Span}\left\{\omega_{1},\omega_{2},\cdot\cdot\cdot,\omega_{n}\right\}\). By the standard method of ODE, we know that \( u^{n}(t)=\sum_{j=1}^{n}b^{n}_{j}(t)\omega_{j}(x) \) of the Cauchy problem as follows By the standard theory of ODE system, we prove the existence of solutions of Problem (14)-(15) on some interval \([0, t_{n})\), \(0< t_{n}< T\) for arbitrary \(T>0\), then, this solution can be extended to the whole interval \([0,T]\) using the first estimate given below. Taking \(\omega=u^{n}_{t}(t)\) in (14), we obtain For the last term on the left hand side of (16) we have Inserting (17) into (16) and integrating over \([0,t]\subset[0, T]\), we obtain Now from assumption \((G3)\) and the Sobolev embedding, we have that and then we have By using the fact that \( -\int_{0}^{t}(g'\circ\Delta u^{n})(\tau)d\tau+\int_{0}^{t}g(\tau)\|\Delta u^{n}(\tau)\|^{2}_{2}d\tau\geq0, \) estimate (20) yields From \(E(0)< \mathcal{Q}(z_{0})\) and (15), it follows that for sufficiently large \(n\). We claim that there exists an integer \(N\) such that Suppose the claim is proved. Then \(\mathcal{Q}(\|\nabla u^{n}\|^{2}_{2})\geq0\) and from (21) and (22), for sufficiently large \(n\) and \(0\leq t< \infty\).

Proof. [Proof of the claim] Suppose that (23) false. Then for each \(n>N\), there exists \(t\in[0,t_{n})\) such that \(\|\nabla u^{n}(t)\|_{2}\geq z_{0}\). We note that from \(\|\nabla u_{0}\|_{2}< z_{0}\) and (15) there exists \(N_{0}\) such that \( \|\nabla u^{n}(0)\|_{2}< z_{0}\quad \forall n>N_{0}. \) Then by continuity there exits a first \(t_{n}^{*}\in[0,t_{n})\) such that

from where \( \mathcal{Q}(\|\nabla u^{n}(t)\|_{2})\geq0 \quad \forall t\in[0,t_{n}^{*}]. \) Now from \(E(0)< \mathcal{Q}(z_{0})\) and (24), there exists \(N>N_{0}\) and \(\gamma\in(0,z_{0})\) such that \( 0\leq\frac{1}{2}\|u^{n}_{t}(t)\|^{2}_{2}+\frac{\eta(t)}{2}\|\Delta u^{n}(t)\|^{2}_{2}+\frac{1}{2}(g\circ\Delta u^{n})(t)+\mathcal{Q}(\|\nabla u^{n}(t)\|^{2}_{2})\leq\mathcal{Q}(\gamma)\quad \forall t\in[0,t_{n}^{*}]\quad \forall n>N. \) Then the monotonicity of \(\mathcal{Q}\) in \([0,z_{0}]\) implies that \( 0\leq\|\nabla u^{n}(t)\|^{2}_{2}\leq\gamma< z_{0}\quad \forall t\in[0,t_{n}^{*}], \) and in particular, \(\|\nabla u^{n}(t)\|^{2}_{2}< z_{0}\), which is a contradiction to (24). From (24), we have Using Sobolev inequality, (8) and (26), it follows that Furthermore, by (29), we get The estimates (26)-(30) permit us to obtain a subsequences of \(\left\{u_{n}\right\}\) which from now on will be also denoted by \(\left\{u_{n}\right\}\) and functions \(u\), \(\chi\) such that Besides, from Lions-Aubin Lemma we also have and consequently, making use of the Lemma 1.3 in [8], we deduce Thus, we obtain that \(u\) is a global weak of problem (4)-(6). Next, we shall prove that \(u\) satisfies (11). From the discussion above, we obtain for each fixed \(t>0\) that We obtain for each fixed \(t>0\) that as \(n\rightarrow +\infty\), and as \(n\rightarrow +\infty\), where \(0< \theta_{n}< 1\). Hence, we have From (15), it follows that \(E^{n}(0)\rightarrow E(0)\) as \(n\rightarrow+\infty\). Finally, taking \(n\rightarrow +\infty\) in (18), we deduce that the energy identity (11) holds for \(0\leq t< \infty\).

Step 2 General decay of the energy

Firstly, we state several Lemmas to prove the decay rate estimate of the energy.

Lemma 1. Let \(u\in L^{\infty}(0,\infty;H^{2}_{0}(\Omega))\) be the solution of (4)-(6) and \(E(0)< \mathcal{Q}(z_{0})\), \(\|\nabla u_{0}\|_{2}< z_{0}\), then we have

where \(C_{1}=\frac{1}{2}+(2\lambda_{1})^{-1}\).

Proof. From \(E(0)< \mathcal{Q}(z_{0})\) and \(\|\nabla u_{0}\|_{2}< z_{0}\), we can obtain \(\mathcal{Q}(\|\nabla u(t)\|_{2})\geq0\) for \(0\leq t< \infty\). Thus we have

and

Lemma 2. The energy \(E(t)\) satisfies

Proof. From (13), we have

From assumptions \((G2)\) and since \(\int_{0}^{t}g'(\tau)d\tau=g(t)-g(0)\), we obtain Then, Combining (45) and (44) our conclusion holds. Multiplying (43) by \(e^{\kappa\zeta(t)}\) \((\kappa>0)\) and using (40), we have Using the fact that \(\zeta_{t}\) is decreasing by (12), we arrive at Choosing \(\|g\|_{L^{1}(0,\infty)}\) sufficiently small so that \(g(0)-\xi_{1}\|g\|_{L^{1}(0,\infty)}=B>0\) and defining \(\kappa_{0}=\min\left\{\frac{2}{\zeta_{t}(0)},\frac{\xi_{2}}{\zeta_{t}(0)},\frac{B}{2C_{1}\zeta_{t}(0)}\right\},\) we conclude by taking \(\kappa\in(0, \kappa_{0}]\) in (47) that \( \frac{d}{dt}\left(e^{\kappa\zeta(t)}E(t)\right)\leq0,\quad t>0. \) Integrating the above inequality over \((0,t)\), it follows that \( E(t)\leq E(0)e^{-\kappa\zeta(t)},\quad t>0. \)

Competing Interests

The author declares no conflict of interest.

1. Introduction

The dynamics of evolution processes is often subjected to abrupt changes such as shocks, harvesting, and natural disasters. Often these short-term perturbations are treated as having acted instantaneously or in the form of impulses [1]. The study of dynamical systems with impulsive effects is of great importance. Impulsive differential equations have become more important in recent years in some mathematical models of real processes and phenomena studied in control, physics, chemistry, population dynamics, aero- nautics and engineering. The concept of controllability plays an important role in many areas of applied mathematics. In recent years, significant progress has been made in the controllability of linear and nonlinear deterministic infinite dimensional systems, see for instance [2, 3, 4, 5, 6, 7, 8, 9, 10, 11] and the references therein. Many authors studied the controllability problem of nonlinear systems with delay in infinite dimensional Banach spaces; see for instance [2, 6, 9, 10, 11] etc and the references contained in them.

The controllability problem for nonlinear impulsive systems in infinite dimensional Banach spaces has been studied by several authors, see e.g., [6, 7, 11]. In [11], Selvi and Arjunan considered the following impulsive differential systems with finite delay Using the Hausdorff measure of noncompactness and the Mönch fixed-point theorem and under some sufficient conditions, they obtained a controllability result for Equation (1), without assuming the compactness of the semigroup. In [6], Machado et al., considered the following impulsive mixed-type functional integro-differential system with finite delay and nonlocal conditions of the form Using the Mönch fixed-point theorem via measures of noncompactness and semigroup theory, they obtained a controllability result for Equation (2) without assuming the compactness of the evolution system. However, the result obtained in [11] and [6] are only in connection with finite delay and the impulsive functions \(I_k\ (k = 1,\cdots , m)\) are assumed to be bounded. But since most often many systems arising from realistic models can be described as functional differential and integrodifferential systems with infinite delay [12], it would be natural and interesting to discuss this kind of problems. In an attempt to address this kind of problems, Chang [13] considered the following impulsive functional differential systems with infinite delay; where \(\mathcal{BM}_h\) is an abstract phase space. Assuming the compactness of the C\(_0\)-semigroup generated by \(A\) and using Schauder's fixed point theorem together with some sufficient conditions, the author obtained a controllability result for Equation (3).
Motivated by the above works, we study in this paper the controllability for some systems that take the form of the following abstract model of impulsive partial functional integrodifferential equation with infinite delay in a Banach space \((X,\ \|\cdot\|)\); where \(A:\mathcal{D}(A)\rightarrow X\) is the infinitesimal generator of a \(C_0\)-semigroup \(\big(T(t)\big)_{t\geq0}\) on a Banach space \(X\); for \(t\geq0\), \(\gamma(t)\) is a closed linear operator with domain \(\mathcal{D}(\gamma(t))\supset\mathcal{D}(A)\). The control \(u\) belongs to \(L^2(J,U)\) which is a Banach space of admissible controls, where \(U\) is a Banach space. The operator \( C\in\mathcal{L}(U,X)\), where \(\mathcal{L}(U,X)\) denotes the Banach space of bounded linear operators from \(U\) into \(X\), and the phase space \(\mathcal{P}\) is a linear space of functions mapping \(]-\infty,0]\) into \(X\) satisfying axioms which will be described later, for every \(t\geq0\), \(x_t\) denotes the history function of \(\mathcal{P}\) defined by \(x_t(\theta)=x(t+\theta)\ \ \text{for}\ -\infty\leq\theta\leq0.\) Here \(0< t_1< \cdots< t_m< t_{m+1}< b\) are prefixed numbers, \(f:\, J\times \mathcal{P}\rightarrow X,\ I_k:\mathcal{P}\rightarrow X\) are appropriate functions satisfying some conditions, and the symbol \(\Delta\xi(t)\) represent the jump of the function \(\xi\) at \(t\), which is defined by \(\Delta\xi(t)=\xi(t^{+})-\xi(t^{-})\). In the literature devoted to equations with finite delay, the phase space is the space of continuous functions on \([-r,0]\), for some \(r>0\), endowed with the uniform norm topology. But when the delay is unbounded, the selection of the phase space \(\mathcal{P}\) plays an important role in both qualitative and quantitative theories. A usual choice is a normed space satisfying some suitable axioms, which was introduced by Hino et al., [14]. In this work, we use resolvent operators for integral equations, the Mönch fixed-point theorem and the measure of noncompactness, without any compactness assumption on the resolvent operators. In [15], Grimmer proved the existence and uniqueness of resolvent operators for this type of functional integrodifferential equations that give the variation of parameters formula for the solution. In [16], Desch, Grimmer and Schappacher proved the equivalence of the compactness of the resolvent operator and that of the operator semigroup. In this work, we use the equivalence between the operator-norm continuity of the associated resolvent operator and that of the operator semigroup. This property allows us to drop the compactness assumption on the operator semigroup, considered by the authors in [2, 10], and prove that the operator solution satisfies the Mönch condition. The variation of parameters formula for the mild solutions of Equation (4) is given by the resolvent operator, and we prove the controllability result using the Mönch fixed-point theorem and the measure of noncompactness. This method enables us overcome the resolvent operator case considered in this work. In contrary to the evolution semigroup case considered in [6, 11], here the semigroup property can not be used because resolvent operators in general do not form semigroups. To the best of our knowledge, up to now no work has reported on controllability of impulsive partial functional integrodifferential Equation (4) with infinite delay. It has been an untreated topic in the literature, and this fact is the main aim and motivation of the present work. The work is organized as follows; Section 2 is devoted to preliminary results. In this Section, we give the definition of resolvent operator. This allows us to define the mild solution of Equation (4). In Section 3, we study the controllability of Equation (4). In Section 4, we give an example to illustrate this work.

2. Integrodifferential equations, measure of noncompactness and Mönch's theorem

In this Section, we introduce some definitions and lemmas that will be used throughout the paper. Let \(J=[0,b],\ \ b>0\) and let \(X\) be a Banach space. A measurable function \(x:J\rightarrow X\) is Bochner integrable if and only if \(\|x\|\) is Lebesgue integrable. We denote by \(L^1(J,X)\) the Banach space of Bochner integrable functions \(x:J\rightarrow X\) normed by $$ \|x\|_{L^1}=\int_0^b\|x(t)\|dt.$$ In considering the impulsive condition, it is important to introduce some additional concepts and notations. We say that a function \(x:[\mu,\eta]\rightarrow X\) is a normalized piecewise continuous function on \([\mu,\eta]\) if \(x\) is piecewise continuous, and left continuous on \((\mu,\eta]\). Let \(\mathcal{PC}([\mu,\eta],X)\) denote the space of normalized piecewise continuous functions from \([\mu,\eta]\) to \(X\). The notation \(\mathcal{PC}\) stands for the space of all functions \(x:[\mu,\eta]\rightarrow X\) such that \(x\) is continuous at \(t\neq t_k,\ x(t_k^{-})=x(t_k)\) and \(x(t_k^{+})\) exists for all \(k=1,2,\cdots,m\). In this Section, \((\mathcal{PC},\|\cdot\|_{\mathcal{PC}})\) is a Banach space endowed with the norm \(\|x\|_{\mathcal{PC}}=\sup_{s\in J}\|x(s)\|\). In this work, we will employ an axiomatic definition of the phase space \(\mathcal{P}\) introduced by Hino et al., in [14]. Thus, \((\mathcal{P},\|\cdot\|_{\mathcal{P}})\) will be a normed linear space of functions mapping \(]-\infty,0]\) into \(X\) and satisfying the following axioms;

• (\(A_1\)) For \(\sigma>0\), if \(x:\,]-\infty,\mu+\sigma]\rightarrow X\) is such that \(x_{\mu}\in\mathcal{P}\) and \(x|_{[\mu,\mu+\sigma]}\in \mathcal{PC}([\mu,\mu+\sigma];X)\) then, for every \(t\in[\mu,\mu+\sigma]\), the following conditions hold; There exist positive constant \(H\) and functions \(K:\,\mathbb{R}^{+}\rightarrow [1,\infty)\) continuous and \(M:\,\mathbb{R}^{+}\rightarrow[1,\infty)\) locally bounded, and all independent of \(x\), such that
  • (i) \(x_t\in\mathcal{P}\),
  • (ii) \(\|x(t)\|\leq H\|x_t\|_{\mathcal{P}}\), which is equivalent to \(\|\varphi(0)\|\leq H\|\varphi\|_{\mathcal{P}}\) for every \(\varphi\in\mathcal{P}\),
  • (iii) \(\|x_t\|_{\mathcal{P}}\leq K(t-\mu)\displaystyle\sup_{\mu\leq s\leq t}\|x(s)\|+M(t-\mu)\|x_{\mu}\|_{\mathcal{P}}\).
• \((A_2)\) For the function \(x\) in \(A_1\), \(t\rightarrow x_t\) is a \(\mathcal{P}\)-valued continuous function for \(t\in[\mu,\mu+\sigma]\).
• (\(A_3\)) The space \(\mathcal{P}\) is complete.

Consider the following linear homogeneous equation; where \(A\) and \(\gamma(t)\) are closed linear operators on a Banach space \(X\). In the sequel, we assume \(A\) and \(\big(\gamma(t)\big)_{t\geq0 }\) satisfy the following conditions;

• \((H_1)\)] \(A\) is a densely defined closed linear operator in \(X\), hence \(\mathcal{D}(A)\) is a Banach space equipped with the graph norm defined by, \(|y|=\|Ay\|+\|y\|\) which will be denoted by \((X_1,|\cdot|)\).
• \((H_2)\)] \(\big(\gamma(t)\big)_{t\geq0 }\) is a family of linear operators on \(X\) such that \(\gamma(t)\) is continuous when regarded as a linear map from \((X_1,|\cdot|)\) into \((X,\|\cdot\|)\) for almost all \(t\geq0\) and the map \(t\mapsto \gamma(t)y\) is measurable for all \(y\in X_1\) and \(t\geq0\), and belongs to \(W^{1,1}(\mathbb{R}^{+},X)\). Moreover there is a locally integrable function \(b:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}\) such that \(\|\gamma(t)y\|\ \leq\ b(t)|y|\ \ {\text and}\ \left\|\frac{d}{dt}\gamma(t)y\right\|\ \leq\ b(t)|y|\ .\)

Remark 1. Note that \((H_2)\) is satisfied in the modelling of Heat Conduction in materials with memory and viscosity. More details can be found in [17].

Let \(\mathcal{L}(X)\) be the Banach space of bounded linear operators on \(X\).

Definition 1.[18] A resolvent operator \(\big(R(t)\big)_{t\geq0 }\) for Equation \((5)\) is a bounded operator valued function $$R\, :\ [0,+\infty)\ \longrightarrow \ \mathcal{L}(X)$$ such that

• (i) \(R(0)=Id_X\) \ and \ \(\|R(t)\|\leq Ne^{\beta t}\) for some constants \(N\) and \(\beta\).
• (ii) For all \(x\in X\),\ the map \(t\mapsto R(t)x\) is continuous for \(t\geq0\).
• (iii) Moreover for \ \(x\in X_1\),\ \ \(R(\cdot)x\,\in\,\mathcal{C}^1(\mathbb{R}^{+};X)\cap\mathcal{C}(\mathbb{R}^{+};X_1)\) \ and \(R'(t)x = AR(t)x+\int_0^t\gamma(t-s)R(s)xds = R(t)Ax+\int_0^tR(t-s)\gamma(s)xds.\)

\label{defresolvent}

Observe that the map defined on \(\mathbb{R}^{+}\) by \ \(t\mapsto R(t)x_0\)\ solves Equation \((5)\) for \(x_0\in\mathcal{D}(A)\).

Theorem 1.[15] Assume that \((H_{1})\) and \((H_2)\) hold. Then, the linear Equation (5) has a unique resolvent operator \(\big(R(t)\big)_{t\geq0 }\).

Remark 2. In general, the resolvent operator \(\big(R(t)\big)_{t\geq0}\) for Equation (5) does not satisfy the semigroup law, namely, \(R(t+s) \ \neq \ R(t)R(s) \ \mbox{for some} \ \ t,\, s >0\, . \)

We have the following theorem that establishes the equivalence between the operator-norm continuity of the \(C_0\)-semigroup and the resolvent operator for integral equations.

Theorem 2.[5] Let \(A\) be the infinitesimal generator of a \(C_0\)-semigroup \(\big(T(t)\big)_{t\geq0}\) and let \(\big(\gamma(t)\big)_{t\geq0}\) satisfy \((H_2)\). Then the resolvent operator \(\big(R(t)\big)_{t\geq0}\) for Equation \((5)\) is operator-norm continuous (or continuous in the uniform operator topology) for \(t>0\) if and only if \(\big(T(t)\big)_{t\geq0}\) is operator-norm continuous for \(t>0\). \label{normcontinuity}

Definition 2. Let \(u\in L^2(J,U)\) and \(\varphi\in\mathcal{P}\). A function \(x:\,]-\infty,b]\rightarrow X\) is called a mild solution of equation (4) if \(x(t)=\varphi(t)\ \ \text{for}\ \ t\in(-\infty,0],\ \Delta x(t_k)=I_k(x_{t_k}),\ \ k=1,2,\cdots,m\), the restriction of \(x\) to intervals \(J_k=(t_k,t_{k+1}]\ (k=0,\cdots,m)\) is continuous and the following integral equation is satisfied

Definition 3. Equation \((4)\) is said to be controllable on the interval \(J\) if for every \(\varphi\in\mathcal{P}\) and \( x_1\in X\), there exists a control \(u\in L^2(J,U)\) such that a mild solution \(x\) of Equation \((4)\) satisfies the condition \(x(b)=x_1\).

For proving the main result of the paper we recall some properties of the measure of noncompactness and the Mönch fixed-point theorem.

Definition 4. [19] Let \(D\) be a bounded subset of a normed space \(Y\). The Hausdorff measure of noncompactness ( shortly MNC) is defined by $$\beta(D)=\inf\Big\{\epsilon>0: D\ has\ a\ finite\ cover\ by\ balls\ of\ radius\ less\ than\ \epsilon\Big\}.$$

Theorem 3. [19] Let \(D,\ D_1,\ D_2\) be bounded subsets of a Banach space \(Y\). The Hausdorff MNC has the following properties:

• (i) If \(D_1\subset D_2\), \ then \(\beta(D_1)\leq\beta(D_2)\), \ (monotonicity).
• (ii) \(\beta(D)=\beta(\overline{D})\).
• (iii) \(\beta(D)=0\) if and only if \(D\) is relatively compact.
• (iv) \(\beta(\lambda D)=|\lambda|\beta(D)\) for any \(\lambda\in\mathbb{R}\), (Homogeneity).
• (v) \(\beta(D_1+D_2)\leq\beta(D_1)+\beta(D_2)\), where \(D_1+D_2=\{d_1+d_2:\ d_1\in D_1,\ d_2\in D_2\}\), (subadditivity).
• (vi) \(\beta(\{a\}\cup D)=\beta(D)\) for every \(a\in Y\).
• (vii) \(\beta(D)=\beta(\overline{co}(D))\), where \(\overline{co}(D)\) is the closed convex hull of \(D\).
• (viii) For any map \ \(G:\mathcal{D}(G)\subseteq X\rightarrow Y\) which is Lipschitz continuous with a Lipschitz constant \(k\), we have \(\beta(G(D))\ \leq\ k\beta(D),\) for any subset \(D\subseteq\mathcal{D}(G)\).

Let \(R_b=\displaystyle\sup_{t\in[0,b]}\|R(t)\|,\ \ K_b=\sup_{t\in[0,b]}\|K(t)\|,\ \ M_b=\sup_{t\in[0,b]}\|M(t)\|.\) We now state the following useful result for equicontinuous subsets of \(\mathcal{C}([a,b];X)\), where \(X\) is a Banach space. %The proof of the following Lemma 1 can be found in [19].

Lemma 1. [19] Let \(M\subset \mathcal{PC}([a,b];X)\) be bounded and piecewise equicontinuous on \([a,b]\). Then \(\beta(M(t))\) is piecewise continuous for \(t\in[a,b]\) and \(\beta(M)=\sup\{\beta(M(t));\,t\in [a,b]\},\ \ \ \text{where}\ M(t)=\{x(t);\,x\in M\}.\)

Lemma 2. [19] Let \(M\subset\mathcal{C}([a,b];X)\) be bounded and equicontinuous. Then the set \(\overline{co}(M)\) is also bounded and equicontinuous.

To prove the controllability for Equation (4), we need the following results.

Lemma 3.[4] If \((u_n)_{n\geq1}\) is a sequence of Bochner integrable functions from \(J\) into a Banach space \(Y\) with the estimation \(\|u_n(t)\|\leq\mu(t)\) for almost all \(t\in J\) and every \(n\geq1\), where \(\mu\in L^1(J,\mathbb{R})\), then the function \(\psi(t)\ =\ \beta(\{u_n(t):n\geq1\})\) belongs to \(L^1(J,\mathbb{R}^{+})\) and satisfies the following estimation \(\beta\left(\left\{\int_0^tu_n(s)ds\,:\ n\geq1\right\}\right)\ \leq\ 2\int_0^t\psi(s)ds.\)

We now state the following nonlinear alternative of Mönch's type for selfmaps, which we shall use in the proof of the controllability of Equation \((4)\).

Theorem 4. {[20](Mönch, 1980) Let \(\mathcal{K}\) be a closed and convex subset of a Banach space \(Z\) and \(0\in \mathcal{K}\). Assume that \(F:\mathcal{K}\rightarrow \mathcal{K}\) is a continuous map satisfying Mönch's condition, namely, \(D\subseteq \mathcal{K}\) be countable and \( D\subseteq \overline{co}(\{0\}\cup F(D))\) implies \( \overline{D}\) is compact. Then \(F\) has a fixed point.

3. Controllability result

In this Section, we give sufficient conditions ensuring the controllability of Equation (4). For that goal, we need to assume that;

• \((H_3)\)]
  • (i) The following linear operator \(W\,:\, L^2(J,U)\rightarrow X\)\ defined by \(Wu\ =\ \int_0^bR(b-s)Cu(s)\,ds,\) is surjective so that it induces an isomorphism between \(L^2(J,U)\,/_{{\text Ker}W}\)\ and \ \(X\) again denoted by \(W\) with inverse \(W^{-1}\) taking values in \(L^2(J,U)\,/_{{\text Ker}W}\) [21].
  • (ii)There exists a function \(L_W\in L^1(J,\mathbb{R}^{+})\) such that for any bounded set \(Q\subset X\) we have \(\beta((W^{-1}Q)(t))\leq L_W(t)\beta(Q),\) where \(\beta\) is the Hausdorff MNC.
• \((H_4)\) The function \(f\,:\, J\times \mathcal{P}\longrightarrow X\) satisfies the following two conditions;
  • (i) \(f(\cdot,\varphi)\) is measurable for \(\varphi\in \mathcal{P}\) and \(f(t,\cdot)\) is continuous for a.e \(t\in J\),
  • (ii)for every positive integer \(q\), there exists a function \(l_q\in L^1(J,\mathbb{R}^{+})\) such that \(\displaystyle\sup_{\|\varphi\|_{\mathcal{P}}\leq q}\|f(t,\varphi)\|\leq l_q(t)\) for a.e. \(t\in J\) and \(\liminf_{q\rightarrow+\infty}\int_0^b\frac{l_q(t)}{q}dt=l< +\infty,\)
  • (iii) there exists a function \(h\in L^1(J,\mathbb{R}^{+})\) such that for any bounded set \(D\subset\mathcal{P}\), \ \(\beta(f(t,D))\leq h(t)\sup_{-\infty< \theta\leq0}\beta(D(\theta))\) for a.e. \(t\in J,\) where \(D(\theta)=\{\phi(\theta):\,\phi\in D\}.\)
• \((H_5)\) \(I_k:\mathcal{P}\rightarrow X,\ k=1,2,\cdots,m\) are continuous such that;
  • (i) There are nondecreasing functions \(L_k:\mathbb{R}^{+}\rightarrow \mathbb{R}^{+}\) such that \(\|I_k(x)\|\leq L_k(\|x\|_{\mathcal{P}}),\ \ k=1,2,\cdots,m,\ \ x\in\mathcal{P},\) and \(\liminf_{\rho\rightarrow+\infty}\frac{L_k(\rho)}{\rho}=\lambda_k< +\infty,\ \ k=1,2,\cdots,m.\)
  • (ii) There exist constants \(\alpha_k\geq0\) such that, \(\beta(I_k(D))\leq\alpha_k\sup_{-\infty< \theta\leq0}\beta(D(\theta)),\ \ k=1,2,\cdots,m,\) for every bounded subset \(D\) of \(\mathcal{P}\). \(\tau=\Big(1+2R_bM_2\|L_W\|_{L^1}\Big)\left(2R_b\|h\|_{L^1}+R_b\sum_{k=0}^m\alpha_k\right)< 1,\) where \(R_b=\displaystyle\sup_{0\leq t\leq b}\|R(t)\|\) and \(M_2\) is such that \(M_2=\|C\|\).

Theorem 5. Suppose that hypotheses \((H_3)-(H_5)\) hold and Equation (5) has a resolvent operator \(\big(R(t)\big)_{t\geq0}\) that is continuous in the operator-norm topology for \(t>0\). Then Equation \((4)\) is controllable on \(J\) provided that

where \(M_3\) is such that \(M_3=\|W^{-1}\|\). \label{mainthm}

Proof Using \((H_3)\) and given an arbitrary function \(x\), we define the control as usual by the following formula; $$u_x(t)\ =\ W^{-1}\left\{x_1-R(b)\varphi(0)-\int_0^bR(b-s)f(s,x_s)\,ds-\sum_{0< t_k< t}R(b-t_k)I_k(x_{t_k})\right\}(t)\qquad \text{for}\ t\in I.$$ For each \(x\in \mathcal{PC}\) such that \(x(0)=\varphi(0)\), we define its extension \(\widetilde{x}\) from \(]-\infty,b]\) to \(X\) as follows \begin{equation*} \widetilde{x}(t)=\left\{ \begin{array}{l} x(t)\ \ \text{if}\ \ t\in[0,b],\\ \varphi(t) \ \ \text{if}\ \ t\in]-\infty,0]. \end{array} \right.%\eqno(2.2) \end{equation*} We define the space \(E_b=\Big\{x:]-\infty,b]\rightarrow X\ \text{such that}\ x|_{J}\in \mathcal{PC}\ \text{and}\ x_0\in\mathcal{P}\Big\},\) where where \(x|_{J}\) is the restriction of \(x\) to \(J\). We show, by using this control that the operator \(P:\,E_b\rightarrow E_b\) defined by \begin{equation*} (Px)(t)= R(t)\varphi(0)+\displaystyle{\int_0^tR(t-s)\big[f(s,\widetilde{x}_s)+Cu_x(s)\big]\,ds}+\sum_{0< t_k< t}R(t-t_k)I_k(x_{t_k})\, \ \ \text{for}\ \ t\in I=[0,b] \end{equation*} has a fixed-point. This fixed point is then a mild solution of Equation \((4)\). Observe that \((Px)(b)=x_1\). This means that the control \(u_x\) steers the integrodifferential equation from \(\varphi\) to \(x_1\) in time \(b\) which implies that the Equation \((4)\) is controllable on \(J\). For each \(\varphi\in\mathcal{P}\), we define the function \(y\in \mathcal{PC}\) by \(y(t)= R(t)\varphi(0)\) and its extension \(\widetilde{y}\) on \(]-\infty,0]\) by \begin{equation*} \widetilde{y}(t)=\left\{ \begin{array}{l} y(t)\ \ \text{if}\ \ t\in[0,b], \\ \varphi(t)\ \ \text{if}\ \ t\in]-\infty,0]. \end{array} \right. \end{equation*} For each \(z\in \mathcal{PC}\), set \(\widetilde{x}(t)=\widetilde{z}(t)+\widetilde{y}(t)\), where \(\widetilde{z}\) is the extension by zero of the function \(z\) on \(]-\infty,0]\). Observe that \(x\) satifies \((\ref{eqn3})\) if and only if \(z(0)=0\) and $$z(t)= \int_0^tR(t-s)\big[f(s,\widetilde{z}_s+\widetilde{y}_s)+Cu_z(s)\big]\,ds+\sum_{0< t_k< t}R(t-t_k)I_k(z_{t_k}+\widetilde{y}_{t_k})\ \ \text{for}\ t\in[0,b],$$ where \(u_z(t)\ =\ W^{-1}\left\{x_1-R(b)\varphi(0)-\int_0^bR(b-s)f(s,\widetilde{z}_s+\widetilde{y}_s)\,ds-\sum_{0< t_k< t}R(b-t_k)I_k(z_{t_k}+\widetilde{y}_{t_k})\right\}(t).\) Now let \(E_b^0=\Big\{z\in E_b\ \text{such that}\ z_0=0\Big\}.\) Thus \(E_b^0\) is a Banach space provided with the supremum norm. Define the operator \(\Gamma\, :\,E_b^0\rightarrow E_b^0\) by \begin{equation*} (\Gamma z)(t)= \displaystyle{\int_0^tR(t-s)\big[f(s,\widetilde{z}_s+\widetilde{y}_s)+Cu_z(s)\big]\,ds}+\sum_{0< t_k< t}R(t-t_k)I_k(z_{t_k}+\widetilde{y}_{t_k})\ \ \text{for}\ \ t\in[0,b]. \end{equation*} Note that the operator \(P\) has a fixed point if and only if \(\Gamma\) has one. So to prove that \(P\) has a fixed point, we only need to prove that \(\Gamma\) has one. For each positive number \(q\), let \(B_q=\{z\in E_b^0:\|z\|\leq q\}\). Then, for any \(z\in B_q\), we have by axiom \((A_1)\) that \begin{eqnarray} \|z_s+y_s\|&\leq &\|z_s\|_{\mathcal{P}}+\|y_s\|_{\mathcal{P}}\\ &\leq&K(s)\|z(s)\|+M(s)\|z_0\|_{\mathcal{P}}+K(s)\|y(s)\|+M(s)\|y_0\|_{\mathcal{P}}\\ &\leq&K_b\|z(s)\|+K_b\|R(t)\|\|\varphi(0)\|+M_b\|\varphi\|_{\mathcal{P}}\\ &\leq&K_b\|z(s)\|+K_bR_bH\|\varphi\|_{\mathcal{P}}+M_b\|\varphi\|_{\mathcal{P}}\\ &\leq&K_b\|z(s)\|+\Big(K_bR_bH+M_b\Big)\|\varphi\|_{\mathcal{P}}\\ &\leq&K_b\,q+\Big(K_bR_bH+M_b\Big)\|\varphi\|_{\mathcal{P}}. \end{eqnarray} Thus, \(\|z_s+y_s\|\leq K_b\,q+\Big(K_bR_bH+M_b\Big)\|\varphi\|_{\mathcal{P}}=:q'.\) We shall prove the theorem in the following steps;
Step 1. We claim that there exists \(q>0\) such that \(\Gamma(B_q)\subset B_q\). We proceed by contradiction. Assume that it is not true. Then for each positive number \(q\), there exists a function \(z^q\in B_q\), such that \(\Gamma(z^q)\notin B_q,\ i.e.,\ \|(\Gamma z^q)(t)\|>q\) for some \(t\in [0,b]\). Now we have that \begin{eqnarray} q &< & \Big\|(\Gamma z^q)(t)\Big\|\\ &\leq& R_b\int_0^b\Big\|f(s,\tilde{z}_s^q+\widetilde{y}_s)\Big\|\,ds+R_b\int_0^b\|Cu_{z^q}(s)\|\,ds+R_b\sum_{k=0}^mL_k(\|z_{t_k}+\widetilde{y}_{t_k}\|)\\ &\leq& R_b\int_0^b\Big\|f(s,\tilde{z}_s^q+\widetilde{y}_s)\Big\|\,ds+R_b\sum_{k=0}^mL_k(q')\\ && +R_b\int_0^b\Big\|BW^{-1}\Big[x_1-R(b)\varphi(0)-\int_0^bR(b-s)f(s,\tilde{z}_s^q)\,ds-\sum_{0< t_k< t}R(b-t_k)I_k(z_{t_k}+\widetilde{y}_{t_k})\Big]\Big\|\,ds\\ &\leq& bR_bM_2M_3\left(\|x_1\|+R_b\|\varphi(0)\|+R_b\int_0^b\|f(s,\tilde{z}_s^q)\|\,ds+R_b\sum_{k=0}^mL_k(q')\right)\\ && +R_b\int_0^b\Big\|f(s,\tilde{z}_s^q+\widetilde{y}_s)\Big\|\,ds+R_b\sum_{k=0}^mL_k(q')\\ &\leq& bR_bM_2M_3\left(\|x_1\|+R_bH\|\varphi\|_{\mathcal{B}}+R_b\int_0^bl_{q'}(s)\,ds+R_b\sum_{k=0}^mL_k(q')\right)+ R_b\int_0^bl_{q'}(s)\,ds+R_b\sum_{k=0}^mL_k(q'), \end{eqnarray} where \(q':=K_b\,q+q_0,\ \ \text{with}\ q_0:=\Big(K_bR_bH+M_b\Big)\|\varphi\|_{\mathcal{B}}.\) Hence $$q\leq \Big(1+R_bM_2M_3b\Big)\left(R_b\int_0^bl_{q'}(s)\,ds+R_b\sum_{k=0}^mL_k(q')\right)+R_bM_2M_3b\Big(\|x_1\|+R_bH\|\varphi\|_{\mathcal{B}}\Big).$$ Dividing both sides by \(q\) and noting that \(q'=K_bq+q_0\rightarrow+\infty\) as \(q\rightarrow+\infty\), we obtain that $$1\leq\Big(1+R_bM_2M_3b\Big)R_b\left(\frac{\displaystyle{\int_0^bl_{q'}(s)\,ds}+\sum_{k=0}^mL_k(q')}{q}\right)+\frac{R_bM_2M_3b\Big(\|x_1\|+R_bH\|\varphi\|_{\mathcal{B}}\Big)}{q}$$ and $$\liminf_{q\rightarrow+\infty}\left(\frac{\displaystyle{\int_0^bl_{q'}(s)\,ds}+\sum_{k=0}^mL_k(q')}{q}\right)=\liminf_{q\rightarrow+\infty}\left(\frac{\displaystyle{\int_0^bl_{q'}(s)\,ds}}{q'}+\frac{\sum_{k=0}^mL_k(q')}{q'}\right)\frac{q'}{q}=\left(l+\sum_{k=0}^m\lambda_k\right)K_b.$$ Thus we have, \(1\leq\Big(1+R_bM_2M_3b\Big)R_b\left(l+\sum_{k=0}^m\lambda_k\right)K_b\), and this contradicts \((7)\). Hence for some positive number \(q\), we must have \(\Gamma(B_q)\subset B_q\).
Step 2. \(\Gamma:\,E_b^0\rightarrow E_b^0\) is continuous. In fact let \(\Gamma:=\Gamma_1+\Gamma_2\), where $$(\Gamma_1z)(t)=\int_0^tR(t-s)f(s,\widetilde{z}_s+\widetilde{y}_s)\,ds+\sum_{k=0}^mR(t-t_k)I_k(z_{t_k}+\widetilde{y}_{t_k})\ \ \ \text{and}\ \ \ \ \ (\Gamma_2z)(t)=\int_0^tR(t-s)Cu_z(s)\,ds.$$ Let \(\{z^n\}_{n\geq1}\subset E_b^0\) with \(z^n\rightarrow z\) in \(E_b^0\). Then there exists a number \(q>1\) such that \(\|z^n(t)\|\leq q\) for all \(n\) and \(a.e.\ \,t\in J\). So \(z^n,\,z\in B_q\). By \((H_4)-(i),\,\, f(t,\tilde{z}_t^n+\widetilde{y}_t)\rightarrow f(t,\widetilde{z}_t+\widetilde{y}_t)\) for each \(t\in [0,b]\). Also, by \((H_5)-(i),\,\,I_k(z_{t_k}^n+\widetilde{y}_{t_k})\rightarrow I_k(z_{t_k}+\widetilde{y}_{t_k})\) for each \(t\in [0,b]\). And by \((H_4)-(ii)\), \(\|f(t,\tilde{z}_t^n+\widetilde{y}_t)-f(t,\widetilde{z}_t+\widetilde{y}_t)\|\leq 2l_{q'}(t).\) Then we have $$\|\Gamma_1z^n-\Gamma_1z\|_{\mathcal{P}}\leq R_b\int_0^b\|f(s,\tilde{z}_s^n+\widetilde{y}_s)-f(s,\widetilde{z}_s+\widetilde{y}_s)\|\,ds+R_b\sum_{k=0}^m\|I_k(z_{t_k}^n+\widetilde{y}_{t_k})-I_k(z_{t_k}+\widetilde{y}_{t_k})\|\longrightarrow0,\ as\ n\rightarrow+\infty$$ by dominated convergence Theorem. Also we have that $$\|\Gamma_2z^n-\Gamma_2z\|_{\mathcal{P}}\leq R_b^2M_2M_3b\left(\int_0^b\|f(s,\tilde{z}_s^n+\widetilde{y}_s)-f(s,\widetilde{z}_s+\widetilde{y}_s)\|\,ds+\sum_{k=0}^m\|I_k(z_{t_k}^n+\widetilde{y}_{t_k})-I_k(z_{t_k}+\widetilde{y}_{t_k})\right)\longrightarrow0,$$ by dominated convergence Theorem. Thus \(\|\Gamma z^n-\Gamma z\|\leq\|\Gamma_1z^n-\Gamma_1z\|+\|\Gamma_2z^n-\Gamma_2z\|\longrightarrow0,\ as\ n\rightarrow+\infty.\) Hence \(\Gamma\) is continuous on \(E_b^0\).
Step 3. \(\Gamma(B_q)\) is equicontinuous on \([0,b]\). In fact let \(t_1,\,t_2\in J_k,\ \ t_1< t_2\) and \(z\in B_q\), we have \begin{eqnarray} &&\|(\Gamma z)(t_2)-(\Gamma z)(t_1)\|\leq\int_0^{t_1}\|R(t_2-s)-R(t_1-s)\|\|f(s,\widetilde{z}_s+\widetilde{y}_s)+Cu_z(s)\|\,ds\\ && +\sum_{0< t_k< t_1}\|R(t_2-t_k)-R(t_1-t_k)\|\|I_k(z_{t_k}+\widetilde{y}_{t_k})\|+\sum_{t_1\leq t_k< t_2}\|R(t_1-t_k)\|\|I_k(z_{t_k}+\widetilde{y}_{t_k})\|\\ && +\int_{t_1}^{t_2}\|R(t_2-s)\|\|f(s,\widetilde{z}_s+\widetilde{y}_s)+Cu_z(s)\|\,ds\\ &&\leq\int_0^{t_1}\|R(t_2-s)-R(t_1-s)\|l_{q'}(s)\,ds\\ && +\int_0^{t_1}\|R(t_2-s)-R(t_1-s)\|M_2M_3\left(\|x_1\|+R_bH\|\varphi\|_{\mathcal{B}}+R_b\int_0^bl_{q'}(\tau)\,d\tau+\sum_{k=0}^mL_k(q')\right)\,ds\\ && +\sum_{0< t_k< t_1}\|R(t_2-t_k)-R(t_1-t_k)\|L_k(q')+R_b\sum_{t_1\leq t_k< t_2}L_k(q')+\int_{t_1}^{t_2}\|R(t_2-s)\|l_{q'}(s)\,ds\\ && +\int_{t_1}^{t_2}\|R(t_2-s)\|M_2M_3\left(\|x_1\|+R_bH\|\varphi\|_{\mathcal{B}}+R_b\int_0^bl_{q'}(\tau)\,d\tau+\sum_{k=0}^mL_k(q')\right)\,ds. \end{eqnarray} By the continuity of \(\big(R(t)\big)_{t\geq0}\) in the operator-norm toplogy, the dominated convergence Theorem, we conclude that the right hand side of the above inequality tends to zero and independent of \(z\) as \(t_2\rightarrow t_1\). Hence \(\Gamma(B_q)\) is equicontinuous on \(J\).
Step 4. We show that the Mönch's condition holds. Suppose that \(D\subseteq B_q\) is countable and \(D\subseteq\overline{co}(\{0\}\cup \Gamma(D))\). We shall show that \(\beta(D)=0\), where \(\beta\) is the Hausdorff MNC. Without loss of generality, we may assume that \(D=\{z^n\}_{n\geq1}\). Since \(\Gamma\) maps \(B_q\) into an equicontinuous family, \(\Gamma(D)\) is also equicontinuous on \(J\). By \((H_3)-(ii)\), \((H_4)-(iii)\) and Lemma 3, we have that \begin{align*} & \beta\Big(\{u_{z^n}(t)\}_{n\geq1}\Big) = \beta\left(W^{-1}\left\{x_1-R(b)\varphi(0)-\int_0^bR(t-b)f\Big(s,\{\tilde{z}_s^n+\widetilde{y}_s\}_{n\geq1}\Big)\,ds-\sum_{0< t_k< t}R(b-t_k)I_k(z_{t_k}^n+\widetilde{y}_{t_k})\right\}\right)\\ &\leq L_W(t)\beta\left(\left\{x_1-R(b)\varphi(0)\right\}\right)+L_W(t)\beta\left(\left\{\int_0^bR(t-b)f\Big(s,\{\tilde{z}_s^n+\widetilde{y}_s\}_{n\geq1}\Big)\,ds\right\}_{n\geq1}\right) \end{align*} \begin{align*} &\;\;\;+L_W(t)\beta\left(\left\{\sum_{0< t_k< t}R(b-t_k)I_k(z_{t_k}^n+\widetilde{y}_{t_k})\right\}_{n\geq1}\right)\\ &\leq 2R_bL_W(t)\left(\int_0^bh(s)\beta\left(\left\{\tilde{z}_s^n\right\}_{n\geq1}+\{\widetilde{y}_s\}\right)ds\right)+R_bL_W(t)\sum_{k=0}^m\beta\left(\left\{I_k(z_{t_k}^n+\widetilde{y}_{t_k})\right\}_{n\geq1}\right)\\ &\leq 2R_bL_W(t)\left(\int_0^bh(s)\Big[\beta\left(\left\{\tilde{z}_s^n\right\}_{n\geq1}\right)+\beta\Big(\{\widetilde{y}_s\}\Big)\Big]\,ds\right)+R_bL_W(t)\sum_{k=0}^m\alpha_k\sup_{-\infty< \theta\leq0}\beta\left(\left\{z_{t_k}^n+\widetilde{y}_{t_k}\right\}_{n\geq1}\right)\\ &\leq 2R_bL_W(t)\left(\int_0^bh(s)\beta\left(\left\{\tilde{z}_s^n\right\}_{n\geq1}\right)\,ds\right)+R_bL_W(t)\sum_{k=0}^m\alpha_k\sup_{-\infty< \theta\leq0}\beta\left(\left\{z_{t_k}^n\right\}_{n\geq1}\right), \end{align*} since \(\Big\{\widetilde{y}_s:\ s\in[0,b]\Big\}\)is compact, so \begin{align*} &\leq 2R_bL_W(t)\left(\int_0^bh(s)\displaystyle\sup_{-\infty< \theta\leq0}\beta\left(\left\{\tilde{z}_s^n(\theta)\right\}_{n\geq1}\right)\,ds\right)+R_bL_W(t)\sum_{k=0}^m\alpha_k\sup_{-\infty< \theta\leq0}\beta\left(\left\{z_{t_k}^n\right\}_{n\geq1}\right), \end{align*} by Lemma 1, since \(D=\{z^n\}_{n\geq1}\) is equicontinuous, we obtain \begin{align*} &\leq 2R_bL_W(t)\left(\int_0^bh(s)\,ds\right)\displaystyle\sup_{0\leq t\leq b}\beta\left(\left\{z^n(t)\right\}_{n\geq1}\right)+R_bL_W(t)\sum_{k=0}^m\alpha_k\sup_{0\leq \tau_k\leq t_k}\beta\left(\left\{z^n(\tau_k)\right\}_{n\geq1}\right). \end{align*} This implies that \begin{align*} &\beta\Big(\{(\Gamma z^n)(t)\}_{n\geq1}\Big)\leq \beta\left(\left\{\int_0^tR(t-s)f(s,\{\tilde{z}_s^n+\widetilde{y}_s\}_{n\geq1})\,ds\right\}_{n\geq1}\right)+\beta\left(\left\{\int_0^tR(t-s)u_{z^n}(s)\,ds\right\}_{n\geq1}\right)\\ &\;\;\;+\beta\left(\left\{\sum_{0< t_k< t}R(b-t_k)I_k(z_{t_k}^n+\widetilde{y}_{t_k})\right\}_{n\geq1}\right)\\ &\leq2R_b\left(\int_0^bh(s)\,ds\right)\displaystyle\sup_{0\leq t\leq b}\beta\left(\left\{z^n(t)\right\}_{n\geq1}\right)+R_b\sum_{k=0}^m\alpha_k\sup_{0\leq \tau_k\leq t_k}\beta\left(\left\{z^n(\tau_k)\right\}_{n\geq1}\right)\\ & \;\;\;+2R_bM_2\left(\int_0^bL_W(s)\,ds\right)2R_b\left(\int_0^bh(s)\,ds\right)\displaystyle\sup_{0\leq t\leq b}\beta\left(\left\{z^n(t)\right\}_{n\geq1}\right)\\&\;\;\;+2R_b^2M_2\left(\int_0^bL_W(s)\,ds\right)\sum_{k=0}^m\alpha_k\sup_{0\leq \tau_k\leq t_k}\beta\left(\left\{z^n(\tau_k)\right\}_{n\geq1}\right)\\ &\leq 2R_b\|h\|_{L^1}\displaystyle\sup_{0\leq t\leq b}\beta\left(\left\{z^n(t)\right\}_{n\geq1}\right)+R_b\sum_{k=0}^m\alpha_k\sup_{0\leq \tau_k\leq t_k}\beta\left(\left\{z^n(\tau_k)\right\}_{n\geq1}\right)\\&\;\;\; +2R_bM_2\|L_W\|_{L^1}2R_b\|h\|_{L^1}\displaystyle\sup_{0\leq t\leq b}\beta\left(\left\{z^n(t)\right\}_{n\geq1}\right)+2R_b^2M_2\|L_W\|_{L^1}\sum_{k=0}^m\alpha_k\sup_{0\leq \tau_k\leq t_k}\beta\left(\left\{z^n(\tau_k)\right\}_{n\geq1}\right). \end{align*} It follows that \begin{align*} & \beta\Big(\Gamma (D)(t)\Big) \leq 2R_b\|h\|_{L^1}\displaystyle\sup_{0\leq t\leq b}\beta\left(D(t)\right)+R_b\sum_{k=0}^m\alpha_k\sup_{0\leq t\leq b}\beta\left(D(t)\right) +2R_bM_2\|L_W\|_{L^1}2R_b\|h\|_{L^1}\displaystyle\sup_{0\leq t\leq b}\beta\left(D(t)\right)\\ & \;\;\;+2R_b^2M_2\|L_W\|_{L^1}\sum_{k=0}^m\alpha_k\sup_{0\leq t\leq b}\beta\left(D(t)\right)\\ &\leq \left(2R_b\|h\|_{L^1}+R_b\sum_{k=0}^m\alpha_k+2R_bM_2\|L_W\|_{L^1}2R_b\|h\|_{L^1}+2R_b^2M_2\|L_W\|_{L^1}\sum_{k=0}^m\alpha_k\right)\sup_{0\leq t\leq b}\beta\left(D(t)\right)\\ &\leq \Big(1+2R_bM_2\|L_W\|_{L^1}\Big)\left(2R_b\|h\|_{L^1}+R_b\sum_{k=0}^m\alpha_k\right)\sup_{0\leq t\leq b}\beta\left(D(t)\right). \end{align*} Since \(D\) and \(\Gamma(D)\) are equicontinuous on \([0,b]\) and \(D\) is bounded, it follows by Lemma 1 that \(\beta\Big(\Gamma(D)\Big)\leq \tau\beta\Big(D\Big)\), where \(\tau\) is as defined in \((H_5)\). Thus from the Mönch condition, we get that \(\beta\Big(D\Big)\leq\beta\Big(\overline{co}(\{0\}\cup \Gamma(D)\Big)=\beta\Big(\Gamma(D)\Big)\leq \tau\beta\Big(D\Big),\) and since \(\tau< 1\), this implies \(\beta\Big(D\Big)=0\), which implies that \(D\) is relatively compact as desired in \(B_q\) and the Mönch condition is satisfied. We conclude by Theorem 4, that \(\Gamma\) has a fixed point \(z\) in \(B_q\). Then \(x=z+y\) is a fixed point of \(P\) in \(E_b\) and thus equation \((4)\) is controllable on \([0,b]\).

Numerical example

Now, we illustrate our main result by the following example.

Example 1. Consider the partial functional integrodifferential system of the form

where \(\ \eta>0,\ \phi\in\mathcal{P},\ I_k>0,\ k=1,2,\cdots,m\), \(u\in L^2((0,\pi))\), \(g:\,[0,1]\times\mathbb{R}\rightarrow\mathbb{R}\) is continuous and Lipschitzian with respect to the second variable, the initial data function \(\phi:\,\mathbb{R}^{-}\times\Omega\rightarrow\mathbb{R}\) is a given function, \(\alpha:\,\mathbb{R}^{-}\rightarrow\mathbb{R}\) is continuous, \(\alpha\in L^1(\mathbb{R}^{-},\mathbb{R})\) and \(\zeta\in \mathcal{C}^{2}([0,b])\) and \(\zeta(0)>0\). Let \(X=L^2(0,\pi)\), and the phase space \(\mathcal{P}= \mathcal{PC}_0\times L^2(\tilde{h},X)\) (\(\tilde{h}:]-\infty,-r]\rightarrow\mathbb{R}\) be a positive function), as introduced in [22]. We define \(A:\mathcal{D}(A)\subset X\rightarrow X\) by $$ \left\{ \begin{array}{l} \mathcal{D}(A)=\Big\{v\in X:\,v\ \text{and}\ v'\ \text{are absolutely continuous},\ v''\in X,\ v(0)=v(\pi)=0\Big\}\\ Av\,=\, v''\ \ \text{for each}\ v\in\mathcal{D}(A). \end{array} \right.%\eqno(4.1) $$ Then, \(Av=\sum_{n=1}^{\infty}n^2\langle v,\,v_n\rangle v,\ \ v\in\mathcal{D}(A),\) where \(v_n(s)=\sqrt{2/\pi}\sin(ns),\ \ n=1,2,3,\cdots\) is the orthogonal set of eigenvectors of \(A\). It is well known that \(A\) is the infinitesimal generator of an analytic semigroup \(\big(T(t)\big)_{t\geq0}\) in \(X\) as is given by \(T(t)v=\sum_{n=1}^{\infty}exp(-n^2t)\langle v,\,v_n\rangle v,\ \ v\in X.\) Moreover, \(\big(T(t)\big)_{t\geq0}\) generated by \(A\) above, is compact for \(t>0\) and operator-norm continuous for \(t>0\). Then by Theorem 2, the corresponding resolvent operator is operator-norm continuous. Now define $$x(t)(\xi)\ =\ v(t,\xi),\ \ \ x'(t)(\xi)\ =\ \frac{\partial v(t,\xi)}{\partial t},\ \ \ u(t,\xi)=u(t)(\xi), \ \ \varphi(\theta)(\xi)=\phi(\theta,\xi)\ \ \text{for}\ \ \theta\in ]-\infty,0]\ \text{and}\ \xi\in(0,\pi).$$ $$I_k(\varphi)(\xi)= \displaystyle\int_{-\infty}^{0}\mu_k(-s)\varphi(s,\xi)ds,\ \ \xi\in(0,\pi),\ k=1,2,\cdots,m.$$ $$f(t,\psi)(\xi)\ =\ \int_{-\infty}^0 \alpha(\theta)g(t,\psi(\theta)(\xi))\,d\theta\ \ \text{for}\ \theta\in ]-\infty,0]\ \text{and}\ \xi\in(0,\pi).$$ Now \(C:X\rightarrow X\) be defined by \(\Big(Cu(t)\Big)(\xi)\,=\,Cu(t)(\xi)=\,\eta u1_{\Gamma}(t,\xi).\) $$(\gamma(t)x)(\xi)\ =\ \zeta(t)\Delta v(t,\xi)\ \ \text{for}\ \ t\in [0,b],\ \ x\in\mathcal{D}(A)\ \text{and}\ \ \xi\in(0,\pi).$$ We suppose that \(\varphi\in \mathcal{P}\). Then, Equation \((8)\) is then transformed into the following form Suppose there exists a continuous function \(p\in L^1(J;\mathbb{R}^{+})\) such that \(|g(t,y_1)-g(t,y_2)|\leq p(t)|y_1-y_2|\ \ \text{for}\ t\in J\ \text{and}\ y_1,\,y_2\in\mathbb{R}\) and \(g(t,0)=0 \ \ \text{for}\ t\in J.\) One can see that \(f\) is Lipschitz continuous with respect to the second variable and moreover for \(\varphi\in\mathcal{P}\), we have we have \( \displaystyle\sup_{\|\varphi\|_{\mathcal{B}}\leq q}\Big\|f(t,\varphi)\Big\|\leq q\,\|\alpha\|\,p(t).\) So \(f\) satisfies \((H_4)-(i)\ \text{and}\ (H_4)-(ii)\) with \(l_q(t)\,=\,q\,\|\alpha\|\,p(t)\). Also \(f\) satisfies \( (H_4)-(iii)\) by condition \( (viii)\) of Theorem 3, since \(f\) is Lipschitz. Now for \(\xi\in(0,\pi)\), the operator \(W\) is given by \((Wu)(\xi)=\eta\int_0^bR(b-s)u(s,\xi)\,ds.\) Assuming that \(W\) satisfies \((H_3)\), then all the conditions of Theorem 5 hold and Equation \((9)\) is controllable.

Conclusion

In this work, we have shown the controllability of some impulsive partial functional integrodifferential differential equation with infinite delay in Banach spaces by using the Hausdorff Measure of Noncompactness and the Mönch fixed point theorem. We achieved this without assuming the compactness of the resolvent operator for the associated undelayed part.

Author Contributions

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

Competing Interests

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

1. Introduction

Let \(\Sigma_{p,n}\) denote the class of functions of the form \[f(z)=\frac{a_{-1}}{z^p}+\sum_{k=n}^\infty a_{k-p}z^{k-p}~(p,n\in \mathbb N=\{1,2,3,\ldots\}),\] which are analytic and \(p\)-valent in the punctured unit disc \( \mathbb E_0=\mathbb E\setminus\{0\},\) where \(\mathbb E = \{z\in\mathbb C:|z|< 1\}\). Define \begin{eqnarray*} D^0f(z)&=&f(z),\\ D^1 f(z)&=&\frac{a_{-1}}{z^p}+2a_0+3a_1z+4a_2z^2+\ldots=\frac{(z^2 f(z))'}{z},\\ D^2f(z)&=&D^1(D^1 f(z)), \end{eqnarray*} and for \(n=1,2,3,\ldots\) \[D^nf(z)=D^1(D^{n-1} f(z))=\frac{(z^2 D^{n-1}f(z))'}{z}.\] Let \(\mathcal {MS}^*_n(p,\alpha)\) denote the class of functions \(f \in \Sigma_{p,n} \) if \[-\Re\frac{1}{p}\left(\frac{D^{n+1}f(z)}{D^nf(z)}-2\right)>\alpha,(\alpha< 1;z \in \mathbb E).\] and let \(\mathcal {MK}_n(p,\alpha)\) denote the class of functions \(f \in \Sigma_{p,n} \) if \[-\Re\frac{1}{p}\left(\frac{(D^{n+1}f(z))'}{(D^nf(z))'}-2\right)>\alpha,(\alpha< 1;z \in \mathbb E).\] The classes of meromorphic starlike functions of order \(\alpha\) and meromorphic convex functions of order \(\alpha\) are denoted by \(\mathcal{MS}^*(\alpha)\) and \(\mathcal{MK}(\alpha)\), respectively and are defined as: \[ \mathcal{MS}^*(\alpha)=\left\{f\in\Sigma:-\Re\left(\frac{zf'(z)}{f(z)}\right)>\alpha,(0\leq\alpha< 1;z \in \mathbb E)\right\},\] and \[\mathcal{MK}(\alpha)=\left\{f\in\Sigma:-\Re\left(1+\frac{zf''(z)}{f'(z)}\right)>\alpha,(0\leq\alpha< 1;z \in \mathbb E)\right\}.\] Note that \(\mathcal{MS}^*(\alpha)=\mathcal{MS}^*_0(1,\alpha)\) and \(\mathcal{MK}(\alpha)=\mathcal{MK}_0(1,\alpha).\) In the theory of meromorphic functions, there exists a variety of results for starlikeness and convexity of meromorphic functions, we state some of them below. Wang et al. [1] proved the following results;

Theorem 1. If \(f(z)\in \Sigma_p\) satisfies the following inequality \[\left|\frac{f(z)}{z f'(z)}\left(1+\frac{z f''(z)}{f'(z)}-\frac{z f'(z)}{f(z)}\right)\right|< \mu~\left(0< \mu< \frac{1}{p}\right),\] then \(f\in\mathcal{MS}_p^*\left(\frac{p}{1+p\mu}\right)\).

Theorem 2. If \(f(z)\in \Sigma_p\) satisfies the inequality \[\left|\frac{z f'(z)}{f(z)}-\frac{z f''(z)}{f'(z)}-1\right|< \delta~(0< \delta< 1),\] then \(f\in\mathcal{MS}_p^*(p(1-\delta))\).

Theorem 3. If \(f(z) \in \Sigma_p\) satisfies the following inequality \[\Re\left(\frac{zf'(z)}{f(z)}+\beta\frac{z^2 f''(z)}{f(z)}\right)< \beta\lambda\left(\lambda+\frac{1}{2}\right)+\frac{1}{2}p\beta-\lambda \hspace{0.5 cm} (\beta\geq 0, p-\frac{1}{2}\leq \lambda \leq p),\] then \(f\in\mathcal{MS}_p^*(\lambda).\)

Goswami et al. [2] proved the following results;

Theorem 4. If \(f(z) \in \Sigma_p,n\) with \(f(z)\neq 0\) for all \(z\in\mathbb E_0\), satisfies the following inequality \[\left| [z^p f(z)]^{\frac{1}{\alpha-p}}\left(\frac{z f'(z)}{f(z)}+\alpha\right)+p-\alpha\right|< \frac{(n+1)(p-\alpha)}{\sqrt{(n+1)^2+1}}, z\in\mathbb E, \] for some real values of \(\alpha~(0\leq\alpha< p)\),then \(f\in\mathcal{MS}_{p,n}^*(\alpha).\)

Theorem 5.If \(f(z) \in \Sigma_p,n\) with \(f(z)\neq 0\) for all \(z\in\mathbb E_0\) satisfies the following inequality \[\left|\frac{ \gamma [z^p f(z)]^\gamma}{z}\left(\frac{z f'(z)}{f(z)}+p\right)\right| \leq\frac{(n+1)}{2\sqrt{(n+1)^2+1}}, z\in\mathbb E, \] for \(\gamma\leq-\frac{1}{p}\),then \(f\in\mathcal{MS}_{p,n}^*\left(p+\frac{1}{\gamma}\right).\)

Theorem 6. If \(f(z) \in \Sigma_p,n\) with \(f(z)\neq 0\) for all \(z\in\mathbb E_0\), satisfies the inequality \[\left| \left(\frac{z^{p+1} f'(z)}{-p}\right)^{\frac{1}{\alpha-p}}\left(1+\frac{z f''(z)}{f'(z)}+\alpha\right)+p-\alpha\right|< \frac{(n+1)(p-\alpha)}{\sqrt{(n+1)^2+1}}, ~z\in\mathbb E ,\] for some real values of \(\alpha~(0\leq\alpha< p)\),then \(f\in\mathcal{MK}_{p,n}(\alpha).\)

Theorem 7. If \(f(z) \in \Sigma_p,n\) with \(f(z)\neq 0\) for all \(z\in\mathbb E_0\), satisfies the inequality \[\left| \frac{1}{z} \left(\frac{z^{p+1} f'(z)}{-p}\right)^{\frac{1}{\alpha-p}}\left(1+\frac{z f''(z)}{f'(z)}+p\right)\right|\leq\frac{(n+1)(p-\alpha)}{2\sqrt{(n+1)^2+1}}, ~z\in\mathbb E ,\] for some real values of \(\alpha~(0\leq\alpha< p)\),then \(f\in\mathcal{MK}_{p,n}(\alpha).\)

From above stated results, we notice that a number of sufficient conditions for meromorphic starlike and meromorphic convex functions have been obtained in terms of differential inequalities in the literature of meromorphic functions. The study of such results is a source of motivation for us to produce the present paper. In the present paper, we study differential inequalities involving a differential operator. As particular cases of our main results, we derive certain sufficient conditions for meromorphic starlike and meromorphic convex functions.

2. Preliminaries

We shall use the following lemma of [3] to prove our result.

Lemma 1. Suppose w is a nonconstant analytic function in \(\mathbb E\) with \(w(0)=0\). If \(|w(z)|\) attains its maximum value at a point \(z_0 \in\mathbb E\) on the circle \(|z|=r< 1,\) then \(z_0 w'(z_0)=m w(z_0),\) where \(m\geq1\), is some real number.

3. Main Results

Theorem 8. Let \(f(z)\in\Sigma_p\) satisfy

for some real numbers \(\alpha, \beta\) and \(\gamma\) such that \(0\leq\alpha< p\), \(\beta\geq0\), \(\gamma\geq0,\beta+\gamma>0,\) then \(f(z)\in\mathcal {MS}_n^*(p,\alpha)\), where \(n\in\mathbb N_0=\mathbb N\cup\{0\} \) and \begin{eqnarray} M(p,\alpha,\beta,\gamma)& = &\begin{cases}\left(1-\frac{\alpha}{p}\right)^\gamma\left(\frac{1}{2}-\frac{\alpha}{p}\right)^{\beta}, 0\leq\alpha< \frac{p}{2},\nonumber \\ \left(1-\frac{\alpha}{p}\right)^{\gamma+\beta}\left(\frac{2}{2-\frac{\alpha}{p}}\right)^{\beta},~\frac{p}{2}\leq\alpha< p. \end{cases} \end{eqnarray}

Proof. We consider the following two cases separately.

Case (i). When \( 0\leq\alpha< \frac{p}{2}\). Writing \(\frac{\alpha}{p}=\mu\), we see that \(0\leq\mu< \frac{1}{2}.\) Define a function \(w\) as

where \(w\) is an analytic function in \(\mathbb E, ~w(0)=0\) and \(w(z)\neq 1\) in \(\mathbb E\). Differentiating (2) logarithmatically, we get From the definition, we have the identity Using (4) in (3), we get \(\frac{D^{n+2}[f](z)}{D^{n+1}[f](z)}=\frac{1+(2\mu-3)w(z)}{1-w(z)}+\frac{2(\mu-1)zw'(z)}{(1-w(z))(1+(2\mu-3)w(z))}\). So, we have We claim that \(|w(z)|< 1,~z\in\mathbb E.\) Suppose, to the contrary, that there exists a point \(z_0\in\mathbb E\) such that \(\max_{|z|\leq|z_0|} |w(z)|=|w(z_0)|=1\). Then by Lemma 1, we have \(w(z_0)=e^{i\theta},0\leq\theta< 2\pi\) and \(z_0w'(z_0)=mw(z_0),m\geq1.\) Therefore \begin{eqnarray} \left|\frac{D^{n+1}[f](z)}{D^{n}[f](z)}-1\right|^\gamma\left|\frac{D^{n+2}[f](z)}{D^{n+1}[f](z)}-1\right|^\beta&=&\left|\frac{2(1-\mu)w(z_0)}{1-w(z_0)}\right|^{\gamma+\beta}\left|1+ \frac{m}{1+(2\mu-3)w(z_0)}\right|^\beta \nonumber\\ &=&\frac{2^{\gamma+\beta}(1-\mu)^{\gamma+\beta}}{|1-e^{i\theta}|^{\gamma+\beta}} \left|1+ \frac{m}{1+(2\mu-3)e^{i\theta}}\right|^\beta \nonumber\\ &\geq&(1-\mu)^{\gamma+\beta}\left(1- \frac{m}{2(1-\mu)}\right)^\beta \nonumber\\ &\geq&(1-\mu)^{\gamma+\beta}\left(1- \frac{1}{2(1-\mu)}\right)^\beta \nonumber\\ &=&(1-\mu)^\gamma \left(\frac{1}{2}-\mu\right)^\beta,\nonumber \end{eqnarray} which contradicts (1) for \( 0\leq\alpha< \frac{p}{2}.\) Therefore we must have \(|w(z)|< 1\) for all \(z\in\mathbb E,\) and hence from (2), we conclude that \(f\in\mathcal {MS}_n^*(p,\alpha)\).

Case (ii). When \(\frac{p}{2}\leq\alpha< p\), therefore we must have \(\frac{1}{2}\leq\mu< 1\), where \(\mu=\frac{\alpha}{p}.\) Let \(w\) be defined by

where \( w(z)\neq\frac{\mu}{1-\mu}\) in \(\mathbb E\). Then w is analytic in \(\mathbb E\) with \(w(0)=0\). Proceeding as in Case (i) above, we obtain We shall prove that \(|w(z)|< 1,~z \in\mathbb E.\) If not, suppose there exists a point \(z_0\in\mathbb E\) such that there exists a point \(z_0\in\mathbb E\) such that \(\max_{|z|\leq|z_0|} |w(z)|=|w(z_0)|=1\). Then by Lemma 1 we have \(w(z_0)=e^{i\theta},0\leq\theta< 2\pi\) and \( z_0w'(z_0)=mw(z_0),m\geq1.\) Therefore \begin{eqnarray*} \left|\frac{D^{n+1}[f](z)}{D^{n}[f](z)}-1\right|^\gamma\left|\frac{D^{n+2}[f](z)}{D^{n+1}[f](z)}-1\right|^\beta&=&\frac{(1-\mu)^{\gamma+\beta}\left(1+\frac{m\mu}{\mu+2-2\mu}\right)^\beta}{|\mu-(1-\mu)w(z_0)|^{\gamma+\beta}}\nonumber\\ &\geq& \left(1-\mu\right)^{\gamma+\beta}\left(\frac{2}{2-\mu}\right)^\beta,\nonumber \end{eqnarray*} which contradicts (1) for \(\frac{p}{2}\leq\alpha< p.\) Therefore, we must have \(|w(z)|< 1\) for all \(z\in\mathbb E\), and hence in view of (6), we conclude that \(f\in\mathcal {MS}_n^*(p,\alpha)\). This completes the proof of the theorem.

Theorem 9. Let \(f(z)\in\Sigma_p\) satisfy

for some real numbers \(\alpha, \beta\) and \(\gamma\) such that \(0\leq\alpha< p\), \(\beta\geq0\), \(\gamma\geq0,\beta+\gamma>0,\) then \(f(z)\in\mathcal {MK}_n(p,\alpha)\), where \(n\in\mathbb N_0=\mathbb N\cup\{0\}\) and \begin{eqnarray} M(p,\alpha,\beta,\gamma)& = &\begin{cases}\left(1-\frac{\alpha}{p}\right)^\gamma\left(\frac{1}{2}-\frac{\alpha}{p}\right)^{\beta}, 0\leq\alpha< \frac{p}{2},\nonumber \\ \left(1-\frac{\alpha}{p}\right)^{\gamma+\beta}\left(\frac{2}{2-\frac{\alpha}{p}}\right)^{\beta},~\frac{p}{2}\leq\alpha< p. \end{cases} \end{eqnarray}

Proof. Again,we consider the following two cases separately.

Case (i). When \( 0\leq\alpha< \frac{p}{2}\). Writing \( \frac{\alpha}{p}=\mu\), we see that \(0\leq\mu< \frac{1}{2}.\) Define a function \(w\) as

where \(w\) is an analytic function in \(\mathbb E, ~w(0)=0\) and \(w(z)\neq 1\) in \(\mathbb E\). Differentiating (9) logarithmatically, we get From the following identity we have Using the identity (12), Equation (10) may be written as \[\frac{(D^{n+2}[f](z))'}{(D^{n+1}[f](z))'}=\frac{1+(2\mu-3)w(z)}{1-w(z)}+\frac{2(\mu-1)zw'(z)}{(1-w(z))(1+(2\mu-3)w(z))}.\] So, we have The remaining part of the proof is similar to that of Theorem 8.

4. Criteria for Starlikeness and Convexity

When we assign particular values to various parameters involved in Theorem 8 and Theorem 9, we obtain following special cases. Setting \(n=0\) in Theorem 8, we obtain the following result.

Corollary 1. Let \(f\in\Sigma_p\) satisfy the condition \begin{eqnarray} \left|\frac{1}{p}\left(\frac{z f'(z)}{ f(z)}\right)-1\right|^\gamma \left|\frac{1}{p}\left(\frac{z f''(z)+3f'(z)}{z f'(z)+2f(z)}\right)-1\right|^\beta &\hspace{-0.3cm}<&\hspace{-0.3cm}\begin{cases}\left(1-\frac{\alpha}{p}\right)^\gamma \left(\frac{1}{2}-\frac{\alpha}{p}\right)^\beta, ~0\leq\alpha< \frac{p}{2},\nonumber \\ \left(1-\frac{\alpha}{p}\right)^{\gamma+\beta} \left(\frac{2}{2-\frac{\alpha}{p}}\right)^\beta,~\frac{p}{2}\leq\alpha< p, \end{cases} \end{eqnarray} for all \(z\in\mathbb E\) and for some real numbers \(\alpha(0\leq\alpha< 1), \beta\geq0\) and \(\gamma\geq0\) with \(\beta+\gamma>0\), then \(f\in\mathcal{MS}^*(p,\alpha)\).

For \(p=1\), Theorem 8 reduces to the following;

Corollary 2. For some real numbers \(\alpha(0\leq\alpha< 1), \beta\geq0\) and \(\gamma\geq0\) with \(\beta+\gamma>0\), if \(f\in\Sigma\) satisfies \begin{eqnarray} \left|\frac{D^{n+1}[f](z)}{D^n[f](z)}-1\right|^\gamma \left|\frac{D^{n+2}[f](z)}{D^{n+1}[f](z)}-1\right|^\beta &\hspace{-0.3cm}<&\hspace{-0.3cm}\begin{cases}\left(1-\alpha\right)^\gamma \left(\frac{1}{2}-\alpha\right)^\beta, ~0\leq\alpha< \frac{1}{2},\nonumber \\ \left(1-\alpha\right)^{\gamma+\beta} \left(\frac{2}{2-\alpha}\right)^\beta,~\frac{1}{2}\leq\alpha< 1, \end{cases} \end{eqnarray} in \(\mathbb E\), then \(\mathcal{MS}^*_n(1,\alpha)\), where \(n\in\mathbb {N}_0.\)

Setting \(n=0\) in above corollary, yields the following result.

Corollary 3. Let \(f(z)\in\Sigma\) satisfy the condition \begin{eqnarray} \left|\frac{z f'(z)}{f(z)}+1\right|^\gamma\left|\frac{z f''(z)+3f'(z)}{z f'(z)+2f(z)}-1\right|^\beta & < &\begin{cases}\left(1-\alpha\right)^\gamma \left(\frac{1}{2}-\alpha\right)^\beta, ~0\leq\alpha< \frac{1}{2},\nonumber \\ \left(1-\alpha\right)^{\gamma+\beta} \left(\frac{2}{2-\alpha}\right)^\beta,~\frac{1}{2}\leq\alpha< 1, \end{cases} \end{eqnarray} where \(z\in\mathbb E, \alpha~(0\leq\alpha< 1), \beta\geq0\) and \(\gamma\geq0\) with \(\beta+\gamma>0\),then \(f\in\mathcal{MS}^*(\alpha)\).

Setting \(\beta=\gamma=1\) and \(\alpha=0\) in above corollary, we obtain the following result.

Remark 1. If \(f(z)\in\Sigma\) satisfies \[\left|\frac{z f'(z)}{f(z)}+1\right|\left|\frac{z f''(z)+3f'(z)}{z f'(z)+2f(z)}-1\right|< \frac{1}{2},~z\in\mathbb E,\] then \(f\in\mathcal{MS}^*\).

By writing \(\beta=1\) and \(\gamma=0\), Theorem 8, we get

Corollary 4. If for all \(z\in\mathbb E\), a function \(f\in\Sigma_p\) satisfies the condition \begin{eqnarray} \frac{D^{n+2}[f](z)}{D^{n+1}[f](z)}&\hspace{-0.3cm}\prec&\begin{cases} 1+\left(\frac{1}{2}-\frac{\alpha}{p}\right)z ,~0\leq\alpha< \frac{p}{2},\nonumber \\ 1+\left[\frac{2\left(1- \frac{\alpha}{p}\right)}{2-\frac{\alpha}{p}}\right]z,~\frac{p}{2}\leq\alpha< p, \end{cases} \end{eqnarray} then \[2-\frac{D^{n+1}[f](z)}{D^n[f](z)}\prec\frac{1+\left(1-\frac{2\alpha}{p}\right)z}{1-z},~z\in\mathbb E, \] i.e. \[\Re\left(2-\frac{D^{n+1}[f](z)}{D^n[f](z)}\right)>\frac{\alpha}{p}.\]

Setting \(n=0\) in Theorem 9, we obtain the following result.

Corollary 5. Let \(f\in\Sigma_p\) satisfy the condition \begin{eqnarray} \left|\frac{1}{p}\left(\frac{z f''(z)+3f'(z)}{ f'(z)}\right)-1\right|^\gamma \left|\frac{1}{p}\left(\frac{z^2 f'''(z)+7zf'(z)+9f'(z)}{z f''(z)+3f(z)}\right)-1\right|^\beta &<&\hspace{-0.3cm}\begin{cases}\left(1-\frac{\alpha}{p}\right)^\gamma \left(\frac{1}{2}-\frac{\alpha}{p}\right)^\beta, ~0\leq\alpha< \frac{p}{2},\nonumber \\ \left(1-\frac{\alpha}{p}\right)^{\gamma+\beta} \left(\frac{2}{2-\frac{\alpha}{p}}\right)^\beta,~\frac{p}{2}\leq\alpha< p, \end{cases} \end{eqnarray} for all \(z\in\mathbb E\) and for some real numbers \(\alpha(0\leq\alpha< 1), \beta\geq0\) and \(\gamma\geq0\) with \(\beta+\gamma>0,\) then \(f\in\mathcal{MK}(p,\alpha)\).

For \(p=1\), Theorem 9 reduces to the following;

Corollary 6. For some real numbers \(\alpha(0\leq\alpha< 1), \beta\geq0\) and \(\gamma\geq0\) with \(\beta+\gamma>0\), if \(f\in\Sigma\) satisfies \begin{eqnarray} \left|\frac{(D^{n+1}[f](z))'}{(D^n[f](z))'}-1\right|^\gamma \left|\frac{(D^{n+2}[f](z))'}{(D^{n+1}[f](z))'}-1\right|^\beta &<&\begin{cases}\left(1-\alpha\right)^\gamma \left(\frac{1}{2}-\alpha\right)^\beta, ~0\leq\alpha< \frac{1}{2},\nonumber \\ \left(1-\alpha\right)^{\gamma+\beta} \left(\frac{2}{2-\alpha}\right)^\beta,~\frac{1}{2}\leq\alpha< 1, \end{cases} \end{eqnarray} in \(\mathbb E\),then \(\mathcal{MK}_n(1,\alpha)\), where \(n\in\mathbb {N}_0.\)

Setting \(n=0\) in above corollary, yields the following result;

Corollary 7. Let \(f(z)\in\Sigma\) satisfy the condition \begin{eqnarray} \left|\frac{z f''(z)}{f'(z)}+2\right|^\gamma\left|\frac{z^2 f'''(z)+7zf''(z)+9f'(z)}{z f''(z)+3f(z)}-1\right|^\beta & < &\begin{cases}\left(1-\alpha\right)^\gamma \left(\frac{1}{2}-\alpha\right)^\beta, ~0\leq\alpha< \frac{1}{2},\nonumber \\ \left(1-\alpha\right)^{\gamma+\beta} \left(\frac{2}{2-\alpha}\right)^\beta,~\frac{1}{2}\leq\alpha< 1, \end{cases} \end{eqnarray} where \(z\in\mathbb E, \alpha~(0\leq\alpha< 1), \beta\geq0\) and \(\gamma\geq0\) with \(\beta+\gamma>0\),then \(f\in\mathcal{MK}(\alpha)\).

Setting \(\beta=\gamma=1\) and \(\alpha=0\) in above corollary, we obtain the following result;

Remark 2. If \(f(z)\in\Sigma\) satisfies \[\left|\frac{z f''(z)}{f'(z)}+2\right|\left|\frac{z^2 f'''(z)+7zf''(z)+9f'(z)}{z f''(z)+3f(z)}-1\right|< \frac{1}{2},~z\in\mathbb E,\] then \(f\in\mathcal{MK}\).

Acknowledgments

The authors are grateful to the referee for careful reading of the paper and valuable suggestions and comments.

Author Contributions

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

Conflict of Interests

The authors declare no conflict of interest.