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 Banach-Caccioppoli 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 φ-contraction 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 self-mapping 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 Banach-contraction 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 Boyd-Wong function if \(\varphi_{2}\) is upper semicontinuous from the right and \(\varphi_{2}(t)< t\) for all \(t>0\).
A self-mapping f on X is called a Boyd-Wong contraction if for all \(x,y\in X\),
$$ d(fx,fy)\leq\varphi_{2} \bigl(d(x,y)\bigr), $$
where \(\varphi_{2}\) is a Boyd-Wong function. Every Boyd-Wong contraction on a complete metric space is a Picard operator [4]. Note that Browder functions are Boyd-Wong 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 Boyd-Wond 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 self-mapping f on X is called a Meir-Keeler 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 Meir-Keeler mappings.
Definition 1.4
A mapping \(\eta:[0,\infty)\rightarrow{}[0,\infty)\) is called a Lim function or
L-function 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 self-map f on a metric space \((X,d)\) is a Meir-Keeler mapping iff there exists an L-function
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:
- (\(\zeta_{1}\)):
-
\(\zeta(t,s)< s-t\) 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 Boyd-Wong functions are simulation functions.
Consistent with Rodan-Lopez-de-Hierro 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 R-function 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
$$ \varrho(t,s)=\textstyle\begin{cases} \frac{1}{2}s-t&\text{if }t< s, \\ 0&\text{if }t\geq s.\end{cases} $$
Then ϱ is an R-function that is not a simulation function.
Rodan-Lopez-de-Hierro 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 R-function 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
$$ \varrho_{\phi}(t,s)=\phi(s)s-t $$
is an R-function satisfying (\(\varrho_{3}\)).
Example 1.10
([10])
If \(\phi:[0,\infty)\rightarrow {}[0,\infty)\) is an L-function, then \(\varrho_{\phi}:[0,\infty)\times{}[ 0,\infty )\rightarrow\mathbb{R}\) defined by \(\varrho_{\phi}(t,s)=\phi(s)-t\) is an R-function satisfying (\(\varrho_{3}\)).
Definition 1.11
Let \((X,d)\) be a metric space. A self-map f of X is called an R-contraction 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 self-map on X. If \(\{B_{i}:i=1,2,\ldots,p\}\) is a finite family of nonempty subsets of X such that \(f(B_{1})\subset B_{2}, f(B_{2})\subset B_{3},\ldots, f(B_{p-1})\subset B_{p}, f(B_{p})\subset B_{1}\). 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 φ-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
φ-contraction 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 Boyd-Wong function.
Kirk et al. [11] established the following fixed point results for Geraghty, Boyd-Wong, and Caristi cyclic φ-contractions.
Theorem 1.14
Let
\((X,d)\)
be a complete metric space, and
p
a natural number. Suppose that a self-mapping
f
is a cyclic
φ-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
φ-contraction. 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
R-contraction 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
R-contraction 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 \(B_{p+1}=B_{1}\).
Meir-Keeler, Geraghty, and simulation contractions are typical examples of R-contractions that satisfy (\(\varrho_{3}\)). Consequently, the cyclic-R-contractions are a generalization of cyclic Meir-Keeler, cyclic Geraghty, cyclic manageable, and cyclic simulative contractions.
In this paper, we prove a fixed point result for cyclic
R-contractions. Our result extends and unifies fixed point results involving Boyd-Wong cyclic contractions, Meir-keeler cyclic contractions, and Geraghty cyclic contraction mappings. Applying our result, we obtain the existence of solutions of nonlinear Volterra integro differential equations.