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 \(x_{n+1}=f(x_{n})=f^{n}(x_{0})\), \(n=0,1,2,\ldots\) , 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 φcontraction if
$$ d(fx,fy)\leq\varphi \bigl(d(x,y)\bigr) $$
for all x, y in X, where φ 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
$$ d(fx,fy)\leq\varphi_{1} \bigl(d(x,y)\bigr) $$
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 \(x,y\in X\),
$$ d(fx,fy)\leq\varphi_{2} \bigl(d(x,y)\bigr), $$
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 ϕ 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 Φ 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 Φ 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
β such that for any \(x,y\in X\), we have
$$ d(fx,fy)\leq\beta \bigl(d(x,y)\bigr)d(x,y), $$
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
η 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], Example 18)
Define \(\varrho:[0,\infty)\times{}[0,\infty)\rightarrow\mathbb{R}\) by
$$ \varrho(t,s)=\textstyle\begin{cases} \frac{1}{2}st&\text{if }t< s, \\ 0&\text{if }t\geq s.\end{cases} $$
Then ϱ 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], Lemma 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
$$ \varrho_{\phi}(t,s)=\phi(s)st $$
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 φcontraction 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
φcontraction if \(\bigcup_{i=1}^{p}B_{i}\) is a cyclic representation of X with respect to f and
$$ d(fx,fy)\leq\varphi\bigl(d(x,y)\bigr) $$
for all \(x\in B_{i}\), \(y\in B_{i+1}\), \(1\leq i\leq p\), where \(B_{p+1}=B_{1}\), and φ is a BoydWong function.
Kirk et al. [11] established the following fixed point results for Geraghty, BoydWong, and Caristi cyclic φcontractions.
Theorem 1.14
Let
\((X,d)\)
be a complete metric space, and
p
a natural number. Suppose that a selfmapping
f
is a cyclic
φcontraction 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
φcontraction. Karapinar [13] improved the results in [12] dropping the requirement of continuity. For more results in this direction, we refer to [14–16] 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.