New fixed point theorem under Rcontractions
 Antonio Francisco Roldán López de Hierro^{1} and
 Naseer Shahzad^{2}Email author
https://doi.org/10.1186/s136630150345y
© Roldán López de Hierro and Shahzad 2015
Received: 19 March 2015
Accepted: 3 June 2015
Published: 1 July 2015
Abstract
In this manuscript we introduce the notions of Rfunction and Rcontractions, and we show an ad hoc fixed point theorem. We prove that this new kind of contractions properly includes the family of all MeirKeeler contractions and other wellknown classes of contractions that have been given very recently (for instance, those using simulation functions and manageable functions). As a consequence, our approach turns out to be appropriate to unify the treatment of different kinds of contractive nonlinear operators.
Keywords
MSC
1 Introduction
Fixed point theory is a branch of nonlinear analysis that can be applied successfully to a wide range of contexts in social and natural sciences. Although some results had been introduced before, it is usually considered that this field of study was born in 1922, when Banach presented a celebrated theorem in order to guarantee that a nonlinear operator had a fixed point. After the appearance of the Banach contractive mapping principle, lots of generalizations, in many different frameworks, have been done. In many cases, new results that currently are being obtained involve contractivity conditions that depend on auxiliary functions (comparison functions, Geraghty functions, altering distance functions, BianchiniGrandolfi gauge functions, etc.).
One of the extensions that have attracted much attention over the last years was due to Meir and Keeler (see [1]) who introduced in 1969 a family of contractive mappings in a new sense. Although their original notion did not depend on auxiliary functions, Lim [2] proved that a selfmapping was a MeirKeeler contraction if, and only if, it satisfied a contractivity condition in a classical sense depending on a new class of functions (that he called Lfunctions). After that, several extensions of MeirKeeler contractions have appeared (see, for instance, [3–11]).
Very recently, Khojasteh et al. (see [12]) introduced the notion of simulation function, which was later modified by RoldánLópezdeHierro et al. in a subtle way (see [13]). The main difference with respect to previous approaches was that simulation functions depend on two variables rather than on a unique variable. And, in order to extend some results in the field of multivalued maps, Du and Khojasteh presented the very close (but independent) notion of manageable function (see [14]). Surprisingly, contractions that use simulation functions turned out to be MeirKeeler contractions (see [15]).
This fact points up the difficulty in finding true extensions of MeirKeeler contractions when we use a simple contractivity condition only involving classical terms as \(d(x,y)\) and \(d(Tx,Ty)\), where d is the distance and \(T:X\rightarrow X\) is the nonlinear operator.
The main aim of the present manuscript is to give a set of auxiliary functions that let us consider a true extension of MeirKeeler contractions. To do that, we present the notion of Rcontraction, which permits us to introduce such a large family of contractions that includes not only MeirKeeler contractions, but Geraghty contractions, contractions depending on simulations functions and manageable functions, etc. We illustrate such kind of contractions with an example in which previous results are not applicable.
2 Preliminaries
Following [16, 17], we say that T is a weakly Picard operator if, for all \(x_{0}\in X\), the Picard sequence of T based on \(x_{0}\) converges to a fixed point of T. Furthermore, T is a Picard operator if it is a weakly Picard operator and it has a unique fixed point. In such a case, if \(z_{0}\) is the unique fixed point of T, then \(\{T^{n}x_{0}\}\rightarrow z_{0}\) for all \(x_{0}\in X\).
2.1 Simulation functions and manageable functions
The notion of simulation function was introduced by Khojasteh et al. in [12] as follows.
Definition 1
(Khojasteh et al. [12])
 (\(\zeta_{1}\)):

\(\zeta(0,0)=0\);
 (\(\zeta_{2}\)):

\(\zeta(t,s)< st\) for all \(t,s>0\);
 (\(\zeta_{3}\)):

if \(\{t_{n}\}\), \(\{s_{n}\}\) are sequences in \((0,\infty)\) such that \(\lim_{n\rightarrow\infty}t_{n}=\lim_{n\rightarrow \infty}s_{n}>0\), then$$\limsup_{n\rightarrow\infty}\zeta(t_{n},s_{n})< 0. $$
The third condition is symmetric in both arguments of ζ but, in proofs, this property is not necessary. In fact, in practise, the arguments of ζ have different meanings and they play different roles. Then, RoldánLópezdeHierro et al. slightly modified the previous definition in order to highlight this difference and to enlarge the family of all simulation functions.
Definition 2
(RoldánLópezdeHierro et al. [13])
 (\(\zeta_{1}\)):

\(\zeta(0,0)=0\);
 (\(\zeta_{2}\)):

\(\zeta(t,s)< st\) for all \(t,s>0\);
 (\(\zeta_{3}\)):

if \(\{t_{n}\}\), \(\{s_{n}\}\) are sequences in \((0,\infty)\) such that \(\lim_{n\rightarrow\infty}t_{n}=\lim_{n\rightarrow \infty}s_{n}>0\) and \(t_{n}< s_{n}\) for all \(n\in\mathbb{N}\), then$$\limsup_{n\rightarrow\infty}\zeta(t_{n},s_{n})< 0. $$
Let \(\mathcal{Z}\) be the family of all simulation functions \(\zeta :[0,\infty)\times{}[0,\infty)\rightarrow\mathbb{R}\).
Every simulation function in the original Khojasteh et al.’s sense (Definition 1) is also a simulation function in our sense (Definition 2), but the converse is not true (see [13]).
Definition 3
(Khojasteh et al. [12], RoldánLópezdeHierro et al. [13])
In 2014, Du and Khojasteh [14] introduced the concept of manageable functions. They showed that many known results can be deduced of some local constraints related to manageable functions.
Definition 4
(Du and Khojasteh [14])
 (\(\eta_{1}\)):

\(\eta(t,s)< st\) for all \(s,t>0\).
 (\(\eta_{2}\)):

For any bounded sequence \(\{t_{n}\}\subset(0,\infty)\) and any nonincreasing sequence \(\{s_{n}\}\subset (0,\infty)\), it holdsWe denote the set of all manageable functions by \(\widehat{\operatorname{Man}(\mathsf{R})}\).$$\limsup_{n\rightarrow\infty}\frac{t_{n}+\eta(t_{n},s_{n})}{s_{n}}< 1. $$
Several examples of simulation functions and manageable functions can be found on [12–14].
Example 5
If \(k\in(0,1)\), then the function \(\eta:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}\) given by \(\eta(t,s)=k st\) for all \(t,s\in\mathbb{R}\), is a manageable function, and its restriction to \([0,\infty)\times{}[0,\infty)\) is a simulation function.
The notion of manageable function was introduced in order to study multivalued contractions. Next, we particularize such a notion to singlevalued mappings.
Definition 6
(Khojasteh et al. [12])
Let \((X,d)\) be a metric space and let \(T:X\rightarrow X\) be a selfmapping. We say that T is a \(\widehat{\operatorname{Man}(\mathsf{R})}\) contraction if there exists \(\eta \in\widehat{\operatorname{Man}(\mathsf{R})}\) such that \(\eta ( d ( Tx,T^{2}x ) ,d ( x,Tx ) ) \geq0\) for all \(x\in X\).
2.2 MeirKeeler contractions
Meir and Keeler generalized the Banach theorem in the following way.
Definition 7
(Meir and Keeler [1])
A MeirKeeler contraction is a mapping \(T:X\rightarrow X\) from a metric space \((X,d)\) into itself such that for all \(\varepsilon>0\), there exists \(\delta>0\) verifying that if \(x,y\in X\) and \(\varepsilon \leq d(x,y)<\varepsilon+\delta\), then \(d(Tx,Ty)<\varepsilon\).
MeirKeeler contractions have attracted much attention in the last years (see, for instance, [3–11]). Lim characterized this kind of mappings in terms of a contractivity condition using the following class of auxiliary functions.
Definition 8
(Lim [2])
 (a)
\(\phi(0)=0\),
 (b)
\(\phi(t)>0\) for all \(t>0\), and
 (c)
for all \(\varepsilon>0\), there exists \(\delta>0\) such that \(\phi(t)\leq\varepsilon\) for all \(t\in [ \varepsilon,\varepsilon+\delta]\).
Theorem 9
(Lim [2], Theorem 1)
Using a result of Chu and Diaz [19], Meir and Keeler [1] proved that every MeirKeeler contraction from a complete metric space into itself has a unique fixed point.
For our purposes, we highlight the following properties of Lfunctions and MeirKeeler contractions.
Lemma 10
 (1)
\(\phi(t)\leq t\) for all \(t\in [ 0,\infty ) \).
 (2)
For all \(\varepsilon>0\), there exists \(\delta>0\) such that \(\phi(t)\leq\varepsilon\) for all \(t\in [ 0,\varepsilon+\delta ] \).
Proof
(1) If \(t=0\), then \(\phi(0)=0\). And if \(t>0\), using \(\varepsilon=t>0\), we deduce, from (c), that \(\phi(t)=\phi(\varepsilon)\leq\varepsilon=t\).
(2) Let \(\varepsilon>0\) be arbitrary and let \(\delta>0\) be given by (c). Then, for all \(t\in [ 0,\varepsilon ) \), we have that \(\phi(t)\leq t<\varepsilon\). □
The following result is useful to guarantee that a selfmapping is not a MeirKeeler contraction.
Proposition 11
Proof
3 RFunctions and Rcontractions
In this section we introduce the family of auxiliary functions we will use to present a new kind of contractive mappings. We will also show that this family contains several classes of contractive mappings, including MeirKeeler contractions.
3.1 The family of Rfunctions
Definition 12
 (\(\varrho_{1}\)):

If \(\{a_{n}\}\subset ( 0,\infty ) \cap A\) is a sequence such that \(\varrho(a_{n+1},a_{n})>0\) for all \(n\in\mathbb{N}\), then \(\{a_{n}\}\rightarrow0\).
 (\(\varrho_{2}\)):

If \(\{a_{n}\},\{b_{n}\}\subset ( 0,\infty ) \cap A\) are two sequences converging to the same limit \(L\geq0\) and verifying that \(L< a_{n}\) and \(\varrho(a_{n},b_{n})>0\) for all \(n\in\mathbb{N}\), then \(L=0\).
We denote by \(R_{A}\) the family of all Rfunctions whose domain is \(A\times A\).
 (\(\varrho_{3}\)):

If \(\{a_{n}\},\{b_{n}\}\subset ( 0,\infty ) \cap A\) are two sequences such that \(\{b_{n}\} \rightarrow0\) and \(\varrho(a_{n},b_{n})>0\) for all \(n\in\mathbb{N}\), then \(\{a_{n}\}\rightarrow0\).
Remark 13
Notice that conditions (\(\varrho_{1}\)), (\(\varrho_{2}\)) and (\(\varrho_{3}\)) establish that if there exist sequences verifying some assumptions, then a thesis must hold. However, we point out that if such kind of sequences does not exist, then conditions (\(\varrho_{1}\)), (\(\varrho_{2}\)) and (\(\varrho_{3}\)) hold.
Proposition 14
If \(\varrho ( t,s ) \leq st\) for all \(t,s\in A\cap ( 0,\infty ) \), then (\(\varrho_{3}\)) holds.
Proof
Assume that \(\{a_{n}\},\{b_{n}\}\subset ( 0,\infty ) \cap A\) are two sequences such that \(\{b_{n}\}\rightarrow0\) and \(\varrho(a_{n},b_{n})>0\) for all \(n\in\mathbb{N}\). Since \(a_{n},b_{n}\in ( 0,\infty ) \cap A\), then \(0<\varrho(a_{n},b_{n})\leq b_{n}a_{n}\) for all \(n\in\mathbb {N}\). As a consequence, \(0< a_{n}< b_{n}\) for all \(n\in\mathbb{N}\), which means that \(\{a_{n}\}\rightarrow0\). □
Firstly, we show some examples.
Lemma 15
Every simulation function is an Rfunction that also verifies (\(\varrho_{3}\)).
Proof
Let \(\zeta:[0,\infty)\times{}[0,\infty)\rightarrow\mathbb{R}\) be a simulation function.
(\(\varrho_{3}\)) Let \(\{a_{n}\},\{b_{n}\}\subset ( 0,\infty ) \) be two sequences such that \(\{b_{n}\}\rightarrow0\) and \(\zeta(a_{n},b_{n})\geq0\) for all \(n\in\mathbb{N}\). Since ζ is a simulation function, \(0\leq\zeta(a_{n},b_{n})< b_{n}a_{n}\) for all \(n\in\mathbb{N}\). Hence, \(0< a_{n}< b_{n}\) for all \(n\in\mathbb{N}\), which means that \(\{a_{n}\}\rightarrow0\). □
Lemma 16
Every manageable function is an Rfunction that also verifies (\(\varrho_{3}\)).
Proof
Let \(\eta:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}\) be a manageable function.
(\(\varrho_{3}\)) Let \(\{a_{n}\},\{b_{n}\}\subset ( 0,\infty ) \) be two sequences such that \(\{b_{n}\}\rightarrow0\) and \(\eta(a_{n},b_{n})\geq0\) for all \(n\in\mathbb{N}\). Since η is a manageable function, \(0\leq\eta(a_{n},b_{n})< b_{n}a_{n}\) for all \(n\in\mathbb{N}\). Hence, \(0< a_{n}< b_{n}\) for all \(n\in\mathbb{N}\), which means that \(\{a_{n}\}\rightarrow0\). □
Lemmas 15 and 16 provide us with a wide range of Rfunctions taking into account examples given in [12, 14]. In the following examples we show that the notion of Rfunction is more general than the previous ones.
Example 17
Given \(\lambda\in ( 0,1 ) \), let \(\varrho: [ 0,1 ] \times [ 0,1 ] \rightarrow\mathbb{R}\) be the function given by \(\varrho ( t,s ) =\lambda st\) for all \(t,s\in [ 0,1 ] \). Then ϱ is an Rfunction, but it is not a simulation function neither a manageable function because its domain is neither \([0,\infty)\times {}[0,\infty)\) nor \(\mathbb{R}\times\mathbb{R}\).
Example 18
(\(\varrho_{3}\)) Let \(\{a_{n}\},\{b_{n}\}\subset ( 0,\infty ) \cap A\) be two sequences such that \(\{b_{n}\}\rightarrow0\) and \(\varrho(a_{n},b_{n})>0\) for all \(n\in\mathbb{N}\). Therefore, \(0< a_{n}< b_{n}\), which implies that \(\{a_{n}\}\rightarrow0\).
As a consequence, ϱ is an Rfunction on \([ 0,\infty ) \) which also satisfies condition (\(\varrho_{3}\)). However, ϱ is not a simulation function because if we take \(t_{n}=s_{n}=1\) for all \(n\in\mathbb{N}\), then \(\{t_{n}\}\rightarrow1\), \(\{s_{n}\} \rightarrow1\) but \(\varrho(t_{n},s_{n})=0\) for all \(n\in\mathbb{N}\), which implies that ϱ does not verify condition (\(\zeta_{3}\)). The same argument guarantees that ϱ, defined from \(\mathbb{R}\times \mathbb{R}\) into \(\mathbb{R}\), is not a manageable function.
Proposition 19
If \(\varrho\in R_{A}\), then \(\varrho(a,a)\leq0\) for all \(a\in ( 0,\infty ) \cap A\).
Proof
By contradiction, assume that there exists \(a\in ( 0,\infty ) \cap A\) such that \(\varrho(a,a)>0\). Let us define \(a_{n}=a\) for all \(n\in \mathbb{N}\). Therefore \(\varrho(a_{n+1},a_{n})=\varrho(a,a)>0\) for all \(n\in\mathbb{N}\). Condition (\(\varrho_{1}\)) implies that \(\{a_{n}\} \rightarrow0\), which contradicts the fact that \(a>0\). □
Functions taking values greater than or equal to an Rfunction can be an Rfunction.
Proposition 20
If \(\varrho\in R_{A}\) and \(\lambda>0\), then \(\varrho_{\lambda}:A\times A\rightarrow\mathbb{R}\), defined by \(\varrho_{\lambda}(t,s)=\lambda \varrho(t,s)\) for all \(t,s\in A\), is also an Rfunction. And if ϱ satisfies (\(\varrho_{3}\)), then \(\varrho_{\lambda}\) also satisfies it.
An interesting subclass of the family of Rfunctions can be considered involving Lfunctions as follows.
Theorem 21
Proof
(\(\varrho_{3}\)) Let \(\{a_{n}\},\{b_{n}\}\subset ( 0,\infty ) \cap A\) be two sequences such that \(\{b_{n}\}\rightarrow0\) and \(\varrho_{\phi}(a_{n},b_{n})>0\) for all \(n\in\mathbb{N}\). Therefore, by item (1) of Lemma 10, \(0<\varrho _{\phi }(a_{n},b_{n})=\phi(b_{n})a_{n}\leq b_{n}a_{n}\), so \(0< a_{n}< b_{n}\) for all \(n\in\mathbb{N}\), which implies that \(\{a_{n}\}\rightarrow0\). □
Theorem 22
Proof
(\(\varrho_{3}\)) Let \(\{a_{n}\},\{b_{n}\}\subset ( 0,\infty ) \cap A\) be two sequences such that \(\{b_{n}\}\rightarrow0\) and \(\varrho_{\phi}(a_{n},b_{n})>0\) for all \(n\in\mathbb{N}\). By condition (4), \(0< a_{n}< b_{n}\) for all \(n\in\mathbb{N}\), which implies that \(\{a_{n}\}\rightarrow0\). □
Another example, involving Geraghty functions, is the next statement.
Lemma 23
Proof
(\(\varrho_{3}\)) Let \(\{a_{n}\},\{b_{n}\}\subset ( 0,\infty ) \cap A\) be two sequences such that \(\{b_{n}\}\rightarrow0\) and \(\varrho_{\phi}^{\prime}(a_{n},b_{n})>0\) for all \(n\in\mathbb{N}\). Therefore \(0<\varrho_{\phi}^{\prime}(a_{n},b_{n})=\phi(b_{n}) b_{n}a_{n}\leq b_{n}a_{n}\), so \(0< a_{n}< b_{n}\) for all \(n\in\mathbb{N}\), which implies that \(\{a_{n}\}\rightarrow0\). □
3.2 RContractions
In this section we introduce the notion of Rcontraction and we show several examples of such kind of contractions.
Definition 24
The following result shows an extensive family of Rcontractions.
Theorem 25
Every MeirKeeler contraction is an Rcontraction with respect to an Rfunction ϱ which satisfies (\(\varrho _{3} \)).

\(\varrho(t,s)< st\) for all \(t,s\in ( 0,\infty ) \).

\(\varrho(t,s)\leq st\) for all \(t,s\in [ 0,\infty ) \).
Proof
The previous statement implies that every fixed point theorem that can be proved for Rcontractions (such as Theorem 27) also holds for MeirKeeler contractions. However, the converse is false as we shall see in the next section.
Corollary 26
Every Geraghty contraction is an Rcontraction with respect to an Rfunction ϱ which satisfies (\(\varrho_{3}\)).
Proof
4 Some fixed point theorems under Rcontractivity conditions
This section is dedicated to obtaining fixed point theorems under Rcontractivity conditions. Later, we will show that some wellknown results can be deduced as simple consequences of our main result, which is the following one.
Theorem 27
 (a)
T is continuous.
 (b)
The function ϱ satisfies condition (\(\varrho_{3}\)).
 (c)
\(\varrho ( t,s ) \leq st\) for all \(t,s\in A\cap ( 0,\infty ) \).
Then T is a Picard operator. In particular, it has a unique fixed point.
Proof
Case 1. Assume that T is continuous. In this case, \(\{x_{n+1}=Tx_{n}\}\rightarrow Tz\), so \(Tz=z\).
Subcase 2.1. Assume that Ω is finite. In this case, there exists \(n_{0}\in\mathbb{N}\) such that \(d(x_{n+1},Tz)=a_{n}>0\) for all \(n\geq n_{0}\). By (7), \(d(x_{n},z)=b_{n}>0\) for all \(n\geq n_{0}\). Taking into account (6), condition (\(\varrho_{3}\)), applied to \(\{a_{n}\}_{n\geq n_{0}}\) and \(\{b_{n}\}_{n\geq n_{0}}\), implies that \(\{d(x_{n+1},Tz)=a_{n}\}\rightarrow0\), which means that \(\{x_{n+1}\}\rightarrow Tz\). By the uniqueness of the limit, \(Tz=z\).
Case 3. Assume that \(\varrho ( t,s ) \leq st\) for all \(t,s\in A\cap ( 0,\infty )\). Proposition 14 implies that Case 2 is applicable.
In any case, z is a fixed point of T. Then T is a weakly Picard operator.
Corollary 28
Every continuous Rcontraction from a complete metric space into itself has a unique fixed point.
Corollary 29
Every \(\mathcal{Z}\)contraction from a complete metric space into itself has a unique fixed point.
Corollary 30
Every \(\widehat{\operatorname{Man}(\mathsf{R})}\)contraction from a complete metric space into itself has a unique fixed point.
Corollary 31
Corollary 32
Every Geraghty contraction from a complete metric space into itself has a unique fixed point.
Corollary 33
Every MeirKeeler contraction from a complete metric space into itself has a unique fixed point.
(\(\varrho_{1}\)) We claim that it is impossible to have a sequence \(\{a_{n}\}\subset ( 0,\infty ) \cap A\) such that \(\varrho(a_{n+1},a_{n})>0\) for all \(n\in\mathbb{N}\). To prove it, assume that such sequence exists. As \(a_{n}>0\) for all \(n\in\mathbb{N}\), description (11) leads to three cases.
Case 1. There exists \(n_{0}\in\mathbb{N}\) such that \(a_{n_{0}}=1\). As \(t=a_{n_{0}+1}>0\) and \(s=a_{n_{0}}=1<5\), then the inequality \(\varrho(a_{n_{0}+1},1)=\varrho(a_{n_{0}+1},a_{n_{0}})>0\) is impossible following (13).
Case 2. There exists \(n_{0}\in\mathbb{N}\) such that \(a_{n_{0}}=1+\frac{1}{m}\) for some \(m\in\mathbb{N}^{\ast}\). As \(t=a_{n_{0}+1}>0\) and \(s=a_{n_{0}}=1+\frac{1}{m}<5\), then necessarily \(a_{n_{0}+1}=1\), but this is impossible by Case 1.
In any case, it is impossible to have a sequence \(\{a_{n}\}\subset ( 0,\infty ) \cap A\) such that \(\varrho(a_{n+1},a_{n})>0\) for all \(n\in\mathbb{N}\), which means that (\(\varrho_{1}\)) holds.
(\(\varrho_{3}\)) We claim that it is impossible to have two sequences \(\{a_{n}\},\{b_{n}\}\subset ( 0,\infty ) \cap A\) such that \(\{b_{n}\}\rightarrow0\) and \(\varrho(a_{n},b_{n})>0\) for all \(n\in\mathbb{N}\). To prove it, assume that such sequences exist. Since \(b_{n}>0\), then \(b_{n}\geq1\) for all \(n\in\mathbb{N}\) by (12), which contradicts the fact that \(\{b_{n}\}\rightarrow0\). Hence (\(\varrho_{3}\)) holds.
As a consequence, Theorem 27 guarantees that T has a unique fixed point. However, other previous results about MeirKeeler contractions are not applicable.
Declarations
Acknowledgements
This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The second author, therefore, acknowledges with thanks DSR for technical and financial support. The authors are grateful to three anonymous referees for their useful suggestions and comments. AF Roldán López de Hierro is grateful to the Department of Quantitative Methods for Economics and Business of the University of Granada. The same author has been partially supported by Junta de Andalucía by project FQM268 of the Andalusian CICYE.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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