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Fixed points for cyclic Rcontractions and solution of nonlinear Volterra integrodifferential equations
Fixed Point Theory and Applications volumeÂ 2016, ArticleÂ number:Â 61 (2016)
Abstract
In this paper, we introduce the notion of cyclic Rcontraction mapping and then study the existence of fixed points for such mappings in the framework of metric spaces. Examples and application are presented to support the main result. Our result unify, complement, and generalize various comparable results in the existing literature.
1 Introduction and preliminaries
Let \((X,d)\) be any metric space, Y a subset of X, and \(f:X\rightarrow Y\). A point x in X that remains invariant under f is called a fixed point of f. The set of all fixed points of f is denoted by \(F(f)\). A sequence \(\{x_{n}\}\) in X defined by \(x_{n+1}=f(x_{n})=f^{n}(x_{0})\), \(n=0,1,2,\ldots\)â€‰, is called a sequence of successive approximations of f starting from \(x_{0}\in X\). If it converges to a unique fixed point of f, then f is called a Picard operator.
Fixed point theory plays a vital role in the study of existence of solutions of nonlinear problems arising in physical, biological, and social sciences. Some fixed point results simply ensure the existence of a solution but provide no information about the uniqueness and determination of the solution. The distinguishing feature of BanachCaccioppoli contraction principle is that it addresses three most important aspects known as existence, uniqueness, and approximation or construction of a solution of linear and nonlinear problems. The simplicity and usefulness of this principle has motivated many researchers to extend it further, and hence there are a number of generalizations and modifications of the principle. One way to extend the Banach theorem is to weaken the contractive condition by employing the concept of comparison functions. For a detailed survey of such extensions obtained in this direction, we refer to [1, 2] and references therein.
We denote by \(P_{cl}(X)\), \(\mathbb{N}\), \(\mathbb{N}_{0}\), \(\mathbb{R}\), and \(\mathbb{R}^{+}\) the collection of nonempty closed subsets of a metric space \((X,d)\), the set of positive integers, the set of nonnegative integers, the set of real numbers, and the set of positive real numbers, respectively.
Let \((X,d)\) be a metric space. A self mapping f on X is called a Ï†contraction if
for all x, y in X, where Ï† is a suitable function on \([0,\infty)\), called a comparison function.
Definition 1.1
A map \(\varphi_{1}:[0,\infty)\rightarrow{}[0,\infty)\) is said to be a Browder function if \(\varphi_{1}\) is right continuous and monotone increasing.
Browder functions are examples of comparison functions. A selfmapping f on X is called a Browder contraction if
for all \(x,y\in X\), where \(\varphi_{1}\) is a Browder function. Every Browder contraction on a complete metric space is a Picard operator [3]. Every Banachcontraction is a Browder contraction if \(\varphi_{1}(t)=\gamma t\) for \(\gamma \in {}[0,1)\).
Boyd and Wong [4] introduced a class of comparison functions as follows.
Definition 1.2
A function \(\varphi_{2}:[0,\infty)\rightarrow{}[0,\infty)\) is called a BoydWong function if \(\varphi_{2}\) is upper semicontinuous from the right and \(\varphi_{2}(t)< t\) for all \(t>0\).
A selfmapping f on X is called a BoydWong contraction if for all \(x,y\in X\),
where \(\varphi_{2}\) is a BoydWong function. Every BoydWong contraction on a complete metric space is a Picard operator [4]. Note that Browder functions are BoydWong functions.
Matkowski [5] initiated another class of comparison functions as follows.
Definition 1.3
A function \(\phi:[0,\infty)\rightarrow{}[0,\infty)\) is called a Matkowski function if Ï• is increasing and \(\lim_{n\rightarrow\infty}\phi^{n}(t)=0\) for all \(t\geq0\).
Every Matkowski function is a BoydWond function ([1]).
Geraghty [6] defined the following class of comparison functions.
Let Î¦ be the class of all mappings \(\beta:[0,\infty)\rightarrow {}[0,1)\) satisfying the condition: \(\beta(t_{n})\rightarrow1\) implies \(t_{n}\rightarrow0\). Elements of Î¦ are called Geraghty functions.
Note that \(\Phi\neq\phi\). For example, if a mapping \(\beta:[0,\infty )\rightarrow{}[0,1)\) is defined by \(\beta(x)=\frac{1}{1+x^{2}}\), \(x\in{}[0,\infty)\), then \(\beta\in\Phi\).
Let \((X,d)\) be a complete metric space, and \(f:X\rightarrow X\). If there exists a Geraghty function Î² such that for any \(x,y\in X\), we have
then f is a Picard operator.
A selfmapping f on X is called a MeirKeeler mapping if for any \(\epsilon>0\), there exists \(\delta_{\epsilon}>0\) such that for all \(x,y\in X\) with \(\epsilon\leq d(x,y)<\epsilon+\delta\), we have \(d(fx,fy)<\epsilon \).
Lim [7] defined the notion of L function to characterize the MeirKeeler mappings.
Definition 1.4
A mapping \(\eta:[0,\infty)\rightarrow{}[0,\infty)\) is called a Lim function or Lfunction if \(\eta(0)=0\), \(\eta(t)>0\) for all \(t>0\) and for any \(\epsilon>0\), there exists \(\delta_{\epsilon}>0\) such that \(\eta(t)\leq\epsilon\) for all \(t\in{}[\epsilon,\epsilon +\delta]\).
A selfmap f on a metric space \((X,d)\) is a MeirKeeler mapping iff there exists an Lfunction Î· such that \(d(fx,fy)<\eta (d(x,y))\) for all \(x,y\in X\) with \(d(x,y)>0\).
The notion of simulation functions was introduced by Khojasteh et al. [8] and then modified in [9] and [10].
Definition 1.5
A mapping \(\zeta:[0,\infty)\times{}[0,\infty)\rightarrow \mathbb{R}\) is called a simulation function if the following conditions hold:
 (\(\zeta_{1}\)):

\(\zeta(t,s)< st\) for all \(t,s>0\);
 (\(\zeta_{2}\)):

if \(\{t_{n}\}\) and \(\{s_{n}\}\) are sequences in \((0,\infty)\) such that \(\lim_{n\rightarrow\infty}t_{n}=\lim_{n\rightarrow\infty}s_{n} \in(0,\infty)\) and \(t_{n}< s_{n}\) for all \(n\in\mathbb{N}\) then \(\lim\sup_{n\rightarrow\infty} \zeta (t_{n},s_{n})<0\).
Note that BoydWong functions are simulation functions.
Consistent with RodanLopezdeHierro and Shahzad [10], the following definitions, examples, and results will be needed in the sequel.
Definition 1.6
Let \(A\subset\mathbb{R}\) be a nonempty set. A function \(\varrho :A\times A\rightarrow\mathbb{R}\) is called an Rfunction if:
 (\(\varrho_{1}\)):

for any sequence \(\{a_{n}\}\subset(0,\infty)\cap A\) with \(\varrho(a_{n+1},a_{n})>0\) \(\forall n\in\mathbb{N}\), we have \(\lim_{n\rightarrow\infty}a_{n}=0\);
 (\(\varrho_{2}\)):

for any sequences \(\{a_{n}\}\), \(\{b_{n}\}\) in \((0,\infty )\cap A\) satisfying \(\varrho(a_{n},b_{n})>0\) \(\forall n\in\mathbb {N}\), \(\lim_{n\rightarrow\infty}a_{n}=\lim_{n\rightarrow \infty}b_{n}=L\geq0\) and \(L< a_{n}\) imply that \(L=0\).
Example 1.7
([10], ExampleÂ 18)
Define \(\varrho:[0,\infty)\times{}[0,\infty)\rightarrow\mathbb{R}\) by
Then Ï± is an Rfunction that is not a simulation function.
RodanLopezdeHierro and Shahzad [10] also considered the following condition:
 (\(\varrho_{3}\)):

If \(\{a_{n}\}\) and \(\{b_{n}\}\) are sequences in \((0,\infty)\cap A\) such that \(\lim_{n\rightarrow\infty}b_{n}=0\) and \(\varrho (a_{n},b_{n})>0\) \(\forall n\in\mathbb{N}\), then \(\lim_{n\rightarrow\infty}a_{n}=0\).
Example 1.8
([10], LemmaÂ 15)
Every simulation function is an Rfunction that satisfies (\(\varrho_{3}\)).
Example 1.9
([10])
If \(\phi:[0,\infty)\rightarrow{}[0,1 )\) is a Geraghty function, then \(\varrho_{\phi}:[0,\infty)\times{}[ 0,\infty )\rightarrow\mathbb{R}\) defined by
is an Rfunction satisfying (\(\varrho_{3}\)).
Example 1.10
([10])
If \(\phi:[0,\infty)\rightarrow {}[0,\infty)\) is an Lfunction, then \(\varrho_{\phi}:[0,\infty)\times{}[ 0,\infty )\rightarrow\mathbb{R}\) defined by \(\varrho_{\phi}(t,s)=\phi(s)t\) is an Rfunction satisfying (\(\varrho_{3}\)).
Definition 1.11
Let \((X,d)\) be a metric space. A selfmap f of X is called an Rcontraction if there exists \(\varrho\in R_{A}\) such that \(\operatorname {ran}(d)\subseteq A\) and \(\varrho(d(fx,fy),d(x,y))>0\) for all \(x,y\in X\) with \(x\neq y\), where \(R_{A}\) is the family of all functions \(\varrho:A\times A\rightarrow \mathbb{R}\) satisfying the conditions (\(\varrho_{1}\)) and (\(\varrho_{2}\)), and \(\operatorname {ran}(d)\) is the range of the metric d defined by \(\operatorname {ran}(d)=\{ d(x,y):x,y\in X\}\subseteq{}[0,\infty)\).
Definition 1.12
Let X be a nonempty set, p a positive integer, and f a selfmap on X. If \(\{B_{i}:i=1,2,\ldots,p\}\) is a finite family of nonempty subsets of X such that \(f(B_{1})\subset B_{2}, f(B_{2})\subset B_{3},\ldots, f(B_{p1})\subset B_{p}, f(B_{p})\subset B_{1}\). Then the set \(\bigcup_{i=1}^{p}B_{i}\) is called a cyclic representation of X with respect to f.
Kirk et al. [11] introduced the notion of cyclic Ï†contraction mappings as follows.
Definition 1.13
Let \((X,d)\) be a metric space, and \(\{B_{i}:i=1,2,\ldots,p\}\) be a finite family of nonempty closed subsets of X. An operator \(f:\bigcup_{i=1}^{p}B_{i} \rightarrow\bigcup_{i=1}^{p}B_{i}\) is said to be a cyclic Ï†contraction if \(\bigcup_{i=1}^{p}B_{i}\) is a cyclic representation of X with respect to f and
for all \(x\in B_{i}\), \(y\in B_{i+1}\), \(1\leq i\leq p\), where \(B_{p+1}=B_{1}\), and Ï† is a BoydWong function.
Kirk et al. [11] established the following fixed point results for Geraghty, BoydWong, and Caristi cyclic Ï†contractions.
Theorem 1.14
Let \((X,d)\) be a complete metric space, and p a natural number. Suppose that a selfmapping f is a cyclic Ï†contraction on \(\bigcup_{i=1}^{p}B_{i}\). Then there exists a unique element \(z\in\bigcap_{i=1}^{p}B_{i}\) such that \(f(z)=z\).
Later, Pacurar and Rus [12] introduced the notion of weakly cyclic Ï†contraction. Karapinar [13] improved the results in [12] dropping the requirement of continuity. For more results in this direction, we refer to [14â€“16] and references therein.
We now introduce the following notion of cyclic Rcontraction mapping.
Definition 1.15
Let \((X,d)\) be a metric space, and \(B_{1}, B_{2},\ldots,B_{p}\in P_{cl}(X)\). A mapping \(f:\bigcup_{i=1}^{p}B_{i}\rightarrow\bigcup_{i=1}^{p}B_{i}\) is said to be a cyclic Rcontraction if

(i)
there exists \(\varrho\in R_{A}\) with \(\operatorname {ran}(d)\subseteq A\);

(ii)
\(\bigcup_{i=1}^{p}B_{i}\) is a cyclic representation of X with respect to f, and

(iii)
\(\varrho(d(fx,fy),d(x,y))>0\) for all \(x\in B_{i}\), \(y\in B_{i+1}\), \(1\leq i\leq p\), where \(B_{p+1}=B_{1}\).
MeirKeeler, Geraghty, and simulation contractions are typical examples of Rcontractions that satisfy (\(\varrho_{3}\)). Consequently, the cyclicRcontractions are a generalization of cyclic MeirKeeler, cyclic Geraghty, cyclic manageable, and cyclic simulative contractions.
In this paper, we prove a fixed point result for cyclic Rcontractions. Our result extends and unifies fixed point results involving BoydWong cyclic contractions, Meirkeeler cyclic contractions, and Geraghty cyclic contraction mappings. Applying our result, we obtain the existence of solutions of nonlinear Volterra integro differential equations.
2 Main results
We start with the following result.
Theorem 2.1
Let \((X,d)\) be a complete metric space, and \(B_{1}, B_{2},\ldots ,B_{p}\in P_{cl}(X)\). Suppose that a mapping f is a cyclic Rcontraction on \(\bigcup_{i=1}^{p} B_{i}\). Then there exists a unique element \(z\in\bigcap_{i=1}^{p} B_{i}\) such that \(f(z)=z\).
Proof
Let \(x_{0}\) be a given point in \(\bigcup_{i=1}^{p}B_{i}\). Then there exists \(i_{0}\) in \(\{1,2,\ldots ,p\}\) such that \(x_{0}\in B_{i_{0}}\). Since \(f(B_{i_{0}})\subset B_{i_{0}+1}\), we have that \(f(x_{0})\in B_{i_{0}+1}\). Thus, there exists \(x_{1}\in B_{i_{0}+1}\) with \(f(x_{0})=x_{1}\). Similarly, there exists \(x_{2}\in B_{i_{0}+2}\) with \(f(x_{1})=x_{2}\). Continuing in this way, we can construct a sequence in \(\bigcup_{i=1}^{p}B_{i}\) by \(x_{n}=f(x_{n1})=f^{n}(x_{0})\in B_{i_{0}+n}\) for all \(n\in\mathbb{N}\). Now, if \(x_{n+1}=x_{n}\) for some \(n\in\mathbb{N}\), then the result follows immediately. Suppose that \(x_{n+1}\neq x_{n}\) for all \(n\in\mathbb {N}\). Note that
From property (\(\varrho_{1}\)) of an Rfunction we have
We now show that \(\{x_{n}\}\) is a Cauchy sequence. If not, then there exists \(L>0\) such that for any \(k\in \mathbb{N} \), we can construct two subsequences \(\{x_{m_{k}}\}\) and \(\{x_{n_{k}}\} \) of \(\{x_{n}\}\) with \(n_{k}>m_{k}\geq k\) satisfying
Without any loss of generality, we assume that \(n_{k}\) is the smallest integer greater than \(m_{k}\) for which the last inequality holds. We can choose \(j_{k}\in\{1,2,\ldots,p\}\) such that \(n_{k}>m_{k}>\) \(m_{k}j_{k}\) with \(n_{k}\) belonging to the residue class of \(m_{k}j_{k}+1\), and hence \(x_{m_{k}j_{k}}\) and \(x_{n_{k}}\) lie in different adjacently labeled sets \(B_{i}\) and \(B_{i+1}\) for some \(i\in\{1,2,\ldots,p\}\). Thus,
By (2.2) we have
Taking the limit as \(k\rightarrow\infty\) on both sides of this inequality, we have
Similarly,
Also,
Taking the limit as \(k\rightarrow\infty\) on both sides of (2.5) and (2.6), we obtain that
Now, since
then by property (\(\varrho_{2}\)) of an Rfunction, we conclude that \(0=L>0\), a contradiction. Hence, \(\{x_{n}\}\) is a Cauchy sequence in X. Since \((X,d)\) is complete, there exists \(\gamma \in X\) such that \(\lim_{n\rightarrow\infty}x_{n}=\gamma\). Since \(\bigcup_{i=1}^{p}B_{i}\) is a cyclic representation of X with respect to f, there exist subsequences \(\{x_{n_{p}}\}, \{x_{n_{p+1}}\}, \{ x_{n_{p+2}}\}, \ldots, \{x_{n_{p+p2}}\}, \{x_{n_{p+p1}}\}\), and \(\{x_{n_{p+p}}\}\) of \(\{x_{n}\}\) such that \(\{x_{n_{p}}\}\subset B_{1}, \{x_{n_{p+1}}\}\subset B_{2}, \{x_{n_{p+2}}\}\subset B_{3}, \ldots, \{x_{n_{p+p2}}\}\subset B_{p1}, \{x_{n_{p+p1}}\}\subset B_{p}\), and \(\{x_{n_{p+p}}\}\subset B_{p+1}= B_{1}\). Since each \(B_{i}\), \(i\in\{1,2,3,\ldots,p\}\), is a closed subset of X and \(\lim_{n\rightarrow\infty}x_{n}=\gamma\), we deduce that \(\gamma\in\bigcap_{i=1}^{p}B_{i}\).
Note that for each \(n\in\mathbb{N}\), there exists \(i_{n}\in\{ 1,2,\ldots ,p\} \) such that \(x_{n1}\in B_{i_{n1}}\), \(x_{n}\in B_{i_{n}}\), and \(\gamma\in B_{i_{n}}\). Thus,
Using property (\(\varrho_{1}\)) of an Rfunction, we obtain that \(\lim_{n\rightarrow\infty}d(f\gamma,x_{n})=d(f\gamma ,\gamma )=0\).
Therefore, Î³ is a fixed point of f in \(\bigcap_{i=1}^{p}B_{i} \).
Uniqueness: Suppose that there exists another fixed point \(x^{\ast}\) of f in \(\bigcap_{i=1}^{p}B_{i}\), that is, \(d(x^{\ast},\gamma)>0\) and \(d(f\gamma ,fx^{\ast})=d(\gamma,x^{\ast})\). Since f is a cyclic Rcontraction, we have
By property (\(\varrho_{1}\)) of an Rfunction we have \(0< d(x^{\ast},\gamma)=\lim_{n\rightarrow\infty}d(x^{\ast },\gamma)=0\), a contradiction. This establishes the result.â€ƒâ–¡
Example 2.2
Let \(X=\mathbb{R}\) be endowed with the Euclidean metric \(d(x,y)=\vert xy\vert \) for all \({x,y\in X}\). Suppose that \(B_{1}=[1,0]\), \(B_{2}=[0,1]\), and \(A=\operatorname {ran}(d)\subset[0,\infty)\). Define \(f:\bigcup_{i=1}^{2}B_{i}\rightarrow\bigcup_{i=1}^{2}B_{i}\) and \(\varrho:A\times A\rightarrow \mathbb{R}\) as
Note that \((X,d)\) is a complete space and \(B_{1}\) and \(B_{2}\) are closed in X. If \(x\in B_{1}\), that is, \(1\leq x\leq0\), then \(0\leq\frac {x}{5}\leq \frac{1}{5}\) implies that \(f(x)\in B_{2}\). Similarly, if \(x\in B_{2}\), that is, \(0\leq x\leq1\), then \(\frac{1}{5}\leq\frac{x}{5}\leq0\) implies that \(f(x)\in B_{1}\).
Further, \(\varrho(d(fx,fy),d(x,y))=\frac{1}{2}d(x,y)d(fx,fy)=\frac {3}{10}\vert xy\vert >0\) for all \(x\in B_{1}\), \(y\in B_{2}\). Thus, all conditions of TheoremÂ 2.1 are satisfied. Moreover, \(z=0\in\bigcap_{i=1}^{2}B_{i}\) is a fixed point of f.
Example 2.3
Let \(X=\mathbb{R}\) and \(d(x,y)=\vert xy\vert \) for all \(x,y\in X\). Suppose that \(B_{1}=\{\frac{1}{2n}\}_{n\in\mathbb{N}\cup\{0\}}\), \(B_{2}=\{ \frac{1}{2n1}\}_{n\in\mathbb{N}\cup\{0\}}\), and \(A=\operatorname {ran}(d)\subset[ 0,\infty )\). Define \(f:\bigcup_{i=1}^{2}B_{i}\rightarrow\bigcup_{i=1}^{2}B_{i}\) and \(\varrho:A\times A\rightarrow\mathbb{R}\) as
It is clear that \(B_{1}\) and \(B_{2}\) are closed subsets of a complete metric space \((X,d)\) such that \(f(B_{1})\subset B_{2}\) and \(f(B_{2})\subset B_{1}\). Note that
for all \(x\in B_{1}\), \(y\in B_{2}\). Hence, all conditions of TheoremÂ 2.1 are satisfied, and \(z=0\in\bigcap_{i=1}^{2}B_{i}\) is a fixed point of f.
Remark 2.4
In this example, the mapping is a cyclic Rcontraction that is neither a MeirKeeler cyclic contraction nor a simulative cyclic contraction and hence neither a BoydWong nor a Geraghty cyclic contraction. Indeed, if we take \(t=s=1\), then (\(\zeta_{2}\)) fails.
Corollary 2.5
Let \((X,d)\) be a complete metric space, and \(B_{1},B_{2},\ldots,B_{p}\in P_{cl}(X)\). Suppose that a mapping f is a manageable cyclic contraction, or a simulative cyclic contraction, or a Geraghty cyclic contraction, or a BoydWong cyclic contraction, or a MeirKeeler cyclic contraction on \(\bigcup_{i=1}^{p}B_{i}\). Then there exists a unique element \(z\in \bigcap_{i=1}^{p}B_{i}\) such that \(f(z)=z\).
3 Application to nonlinear Volterra integral equations
Motivated by the work in [17], we obtain the existence and uniqueness of solutions for nonlinear Volterra integral differential equations.
Consider the following problem:
where \(f\in C(\mathbb{R}^{+}\times\mathbb{R}^{+}, \mathbb{R})\), \(g\in C(E_{1}\times\mathbb{R}^{+}, \mathbb{R})\), \(h\in C(E_{2}\times\mathbb {R}^{+}, \mathbb{R})\), \(E_{1}=\{f(x,y,s):s\leq x\in{}[0,\infty), y\in {}[ 0,\infty)\}\), and \(E_{2}=\{f(x,y,s,t):s\leq x\in{}[0,\infty), t\leq y\in{}[0,\infty)\}\).
Let X be the space of functions \(z\in C(\mathbb{R}^{+}\times\mathbb {R}^{+},\mathbb{R})\) satisfying \(\vert z(x, t)\vert =O(e^{\lambda(x+y)})\), where Î» is a positive constant, that is, \(\vert z(x, y)\vert \leq M_{0} e^{\lambda(x+y)}\) for some constant \(M_{0}>0\).
Define the norm on X by \(\Vert z\Vert _{X}=\sup_{(x,y) \in (\mathbb{R}^{+}\times\mathbb{R}^{+})}\{ \vert z(x, y)\vert e^{\lambda(x+y)} \}\).
Note that \((X,\Vert \cdot \Vert _{X})\) is a Banach space. Define the mapping \(T:X\rightarrow X\) by
for every \(u\in X\). It is easy to see that \(u^{\ast}\in X\) is a solution of problem (3.1) if \(T(u^{\ast})=u^{\ast}\).
Theorem 3.1
Suppose that problem (3.1) satisfies the following conditions:

(I)
$$ \bigl\vert g(x,y,\xi,u)g(x,y,\xi, \bar{u})\bigr\vert \leq h_{1}(x,y,\xi)\vert u\bar{u}\vert $$
and
$$ \bigl\vert h(x,y, \sigma, \tau,u)h(x,y, \sigma, \tau, \bar{u})\bigr\vert \leq h_{2}(x,y, \sigma,\tau)\vert u\bar{u}\vert , $$where \(h_{1}\in C(E_{1}, [0,\infty))\) and \(h_{2}\in C(E_{2}, [0,\infty ))\);

(II)
There exist Î±, Î² in X and \(\alpha_{0}\), \(\beta_{0}\) in \(\mathbb{R}\) with \(\alpha_{0}\leq\alpha(x,t)\leq\beta(x,t)\leq\beta _{0}(x,t)\) such that
$$ \alpha(x,t)\leq f(x,t)+ \int_{0}^{x}g\bigl(t,s, \xi,\beta(\xi, s)\bigr)\,d \xi + \int_{0}^{x} \int_{0}^{y}h\bigl(t, s, \sigma, \tau, \beta(\sigma, \tau )\bigr)\,d\tau \,d\sigma $$and
$$ \beta(x,t)\geq f(x,t)+ \int_{0}^{x}g\bigl(t, s, \xi,\alpha(\xi, s)\bigr)\,d \xi + \int_{0}^{x} \int_{0}^{y}h\bigl(t, s, \sigma, \tau, \alpha( \sigma, \tau )\bigr)\,d\tau \,d\sigma $$for all \(x,t\in{}[0,\infty)\);

(III)
$$ \int_{0}^{x}h_{1}(x,y,\xi)e^{\lambda(x+y)} \,d\xi + \int_{0}^{x} \int_{0}^{y}h_{2}(x, y, \sigma, \tau)e^{\lambda(\sigma +\tau )}\,d\tau \,d\sigma\leq\delta_{1}e^{\lambda(x+y)} $$
and
$$ \biggl\vert f(x,t)+ \int_{0}^{x}g(x,y,\xi,0)\,d\xi + \int_{0}^{x} \int_{0}^{y}h(x, y, \sigma, \tau, 0)\,d\tau \,d \sigma\biggr\vert \leq \delta_{2}e^{\lambda(x+y)} $$for some nonnegative constants \(\delta_{1},\delta_{2}<1\);

(IV)
There exist Î±, Î² in X such that \(\alpha(t)\leq\beta (t)\), \(T(\alpha(x,t))\leq\beta(x, t)\), and \(T(\beta(x,t))\geq \alpha (x,t)\). Then the integral Eq. (3.1) has a unique solution \(u^{\ast}\) in \(\varpi=\{u\in X:\alpha(x,y)\leq u(x,y)\leq\beta(x,y)\}\).
Proof
Let \(B_{1}=\{u\in X:u(x,t)\leq\beta(x,t)\}\) and \(B_{2}=\{u\in X:u(x,t)\geq \alpha(x,t)\}\). Then \(B_{1}\) and \(B_{2}\) are closed subsets of the complete metric space X. If \(u\in B_{1}\), then by conditions (I), (II), and (IV) we conclude that \(T(u(x,t))\geq\alpha(x,t)\). Hence, \(Tu\in B_{2}\). Similarly, \(u\in B_{2}\) implies that \(Tu\in B_{1}\), and hence \(T(B_{1})\subset B_{2}\) and \(T(B_{2})\subset B_{1}\).
If \(u\in B_{1}\) and \(v\in B_{2}\), then \(u(x,t)\leq\beta(x,t)\leq\beta _{0} \) and \(v(x,t)\geq\alpha(x,t)\geq\alpha_{0}\). From conditions (I) and (III) we obtain that
Thus,
Taking \(\varrho(t,s)=\varsigma st\), we have
Consequently, T is a cyclic Rcontraction on \(\bigcup_{i=1}^{2}B_{i}\). By TheoremÂ 2.1, T has a unique fixed point \(u^{\ast}\) in \(\bigcap_{i=1}^{2}B_{i}\in\varpi\), which is the solution of the integraldifferential Eq. (3.1).â€ƒâ–¡
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Abbas, M., Latif, A. & Suleiman, Y.I. Fixed points for cyclic Rcontractions and solution of nonlinear Volterra integrodifferential equations. Fixed Point Theory Appl 2016, 61 (2016). https://doi.org/10.1186/s1366301605521
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DOI: https://doi.org/10.1186/s1366301605521