# f-Contractive multivalued maps and coincidence points

## Abstract

In this paper, we prove a result on the existence of an f-orbit for generalized f-contractive multivalued maps. Then, we establish main results on the existence of coincidence points and common fixed points for generalized f-contractive maps not involving the extended Hausdorff metric and the continuity condition. Our results either generalize or improve a number of metric fixed point results.

MSC: 47H10, 47H09, 54H25.

## 1 Introduction

Let $\left(X,d\right)$ be a metric space. Let ${2}^{X}$, $Cl\left(X\right)$ and $CB\left(X\right)$ denote the collection of nonempty subsets of X, nonempty closed subsets of X, and nonempty closed bounded subsets of X, respectively. Let H be the Hausdorff metric with respect to d, that is,

$H\left(A,B\right)=max\left\{\underset{x\in A}{sup}d\left(x,B\right),\underset{y\in B}{sup}d\left(y,A\right)\right\}$

for every $A,B\in CB\left(X\right)$, where $d\left(x,B\right)={inf}_{y\in B}d\left(x,y\right)$.

Let $f:X\to X$ be a single-valued map, and let $T:X\to {2}^{X}$ be a multivalued map. A point $x\in X$ is called a fixed point of T if $x\in T\left(x\right)$, and the set of fixed points of T is denoted by $Fix\left(T\right)$. A point $x\in X$ is called a coincidence point of f and T if $f\left(x\right)\in T\left(x\right)$. We denote by $C\left(f\cap T\right)$ the set of coincidence points of f and T.

We say a sequence $\left\{{x}_{n}\right\}$ in X is an f-orbit of T at ${x}_{0}\in X$ if $f{x}_{n}\in T{x}_{n-1}$ for all $n\ge 1$. We say that f and T weakly commute if $fTx\subset Tfx$ for all $x\in X$. Clearly, commuting maps f and T weakly commute.

A multivalued map $T:X\to CB\left(X\right)$ is called

(i) contraction  if for a fixed constant $\lambda \in \left(0,1\right)$ and for each $x,y\in X$,

$H\left(T\left(x\right),T\left(y\right)\right)\le \lambda d\left(x,y\right).$

(ii) f-contraction  if for a fixed constant $\lambda \in \left(0,1\right)$ and for each $x,y\in X$,

$H\left(T\left(x\right),T\left(y\right)\right)\le \lambda d\left(f\left(x\right),f\left(y\right)\right).$

Using the concept of Hausdorff metric, Nadler  established the following fixed point result for multivalued contraction maps, which in turn is a generalization of the well-known Banach contraction principle.

Theorem 1.1 

Let $\left(X,d\right)$ be a complete metric space, and let $T:X\to CB\left(X\right)$ be a contraction map. Then $Fix\left(T\right)\ne \mathrm{\varnothing }$.

This result has been generalized in many directions. Kaneko  extended the corresponding results of Jungck , Nadler  and others as follows.

Theorem 1.2 

Let $\left(X,d\right)$ be a complete metric space, and let $T:X\to CB\left(X\right)$ be a multivalued f-contraction map which commutes with a continuous map f. Then $C\left(f\cap T\right)\ne \mathrm{\varnothing }$.

This result has been generalized in different directions. For example, see .

On the other hand, Kada et al.  introduced the concept of w-distance on a metric space as follows:

Let $\left(X,d\right)$ be a metric space. A function $\omega :X×X\to \left[0,\mathrm{\infty }\right)$ is called a w-distance on X if it satisfies the following for each $x,y,z\in X$:

(${w}_{1}$) $\omega \left(x,z\right)\le \omega \left(x,y\right)+\omega \left(y,z\right)$;

(${w}_{2}$) a map $\omega \left(x,\cdot \right):X\to \left[0,\mathrm{\infty }\right)$ is lower semicontinuous;

(${w}_{3}$) for any $ϵ>0$, there exists $\delta >0$ such that $\omega \left(z,x\right)\le \delta$ and $\omega \left(z,y\right)\le \delta$ imply $d\left(x,y\right)\le ϵ$.

Note that, in general, for $x,y\in X$, $\omega \left(x,y\right)\ne \omega \left(y,x\right)$ and not either of the implications $\omega \left(x,y\right)=0⇔x=y$ necessarily hold. We say the w-distance ω on X is a ${w}_{0}$-distance if $x=y$ implies $\omega \left(x,y\right)=0$. Clearly, the metric d is a w-distance on X. Let $\left(Y,\parallel \cdot \parallel \right)$ be a normed space. Then the functions ${\omega }_{1},{\omega }_{2}:Y×Y\to \left[0,\mathrm{\infty }\right)$ defined by ${\omega }_{1}\left(x,y\right)=\parallel y\parallel$ and ${\omega }_{2}\left(x,y\right)=\parallel x\parallel +\parallel y\parallel$ for all $x,y\in Y$ are w-distances . Many other examples and properties of the w-distance can be found in [11, 12].

The following useful lemma concerning a w-distance is given in .

Lemma 1.1 

Let $\left(X,d\right)$ be a metric space, and let ω be a w-distance on X. Let $\left\{{x}_{n}\right\}$ and $\left\{{y}_{n}\right\}$ be sequences in X, and let $\left\{{\alpha }_{n}\right\}$ and $\left\{{\beta }_{n}\right\}$ be sequences in $\left[0,\mathrm{\infty }\right)$ converging to zero. Then, for the w-distance ω on X, the following hold for every $x,y,z\in X$:

(a) if $\omega \left({x}_{n},y\right)\le {\alpha }_{n}$ and $\omega \left({x}_{n},z\right)\le {\beta }_{n}$ for any $n\in \mathbb{N}$, then $y=z$; in particular, if $\omega \left(x,y\right)=0$ and $\omega \left(x,z\right)=0$, then $y=z$;

(b) if $\omega \left({x}_{n},{y}_{n}\right)\le {\alpha }_{n}$ and $\omega \left({x}_{n},z\right)\le {\beta }_{n}$ for any $n\in \mathbb{N}$, then $\left\{{y}_{n}\right\}$ converges to z;

(c) if $\omega \left({x}_{n},{x}_{m}\right)\le {\alpha }_{n}$ for any $n,m\in \mathbb{N}$ with $m>n$, then $\left\{{x}_{n}\right\}$ is a Cauchy sequence;

(d) if $\omega \left(y,{x}_{n}\right)\le {\alpha }_{n}$ for any $n\in \mathbb{N}$, then $\left\{{x}_{n}\right\}$ is a Cauchy sequence.

For $x\in X$ and $A\in {2}^{X}$, we denote, $\omega \left(x,A\right)={inf}_{y\in A}\omega \left(x,y\right)$. Now, let $T:X\to Cl\left(X\right)$ be a multivalued map, and let $f:X\to X$ be a single-valued map. We say

(iii) T is w-contractive  if there exist a w-distance ω on X and $\lambda \in \left(0,1\right)$ such that for any $x,y\in X$ and $u\in T\left(x\right)$, there is $v\in T\left(y\right)$ with

$\omega \left(u,v\right)\le \lambda \omega \left(x,y\right).$

(iv) T is generalized f-contractive if there exist a ${w}_{0}$-distance ω on X and $\lambda \in \left(0,1\right)$ such that for any $x,y\in X$, $u\in T\left(x\right)$, there is $v\in T\left(y\right)$ with

$\omega \left(u,v\right)\le \lambda {M}_{f}\left(x,y\right),$

where

$\begin{array}{rcl}{M}_{f}\left(x,y\right)& =& max\left\{\omega \left(f\left(x\right),f\left(y\right)\right),\omega \left(f\left(x\right),T\left(x\right)\right),\omega \left(f\left(y\right),T\left(y\right)\right),\\ \frac{1}{2}\left[\omega \left(f\left(x\right),T\left(y\right)\right)+\omega \left(f\left(y\right),T\left(x\right)\right)\right]\right\}.\end{array}$

Using the concept of w-distance, Suzuki and Takahashi  improved Nadler’s fixed point result as follows.

Theorem 1.3 Let $\left(X,d\right)$ be a complete metric space. Then for each w-contractive map $T:X\to Cl\left(X\right)$, the set $Fix\left(T\right)\ne \mathrm{\varnothing }$.

This result has been generalized by many authors, for example, see . In this paper, first we establish a lemma with respect to a w-distance, which is an improved version of the lemma given in , and then we prove a key lemma on the existence of an f-orbit for generalized f-contractive maps. Finally, we present our main results on the existence of coincidence points and common fixed points for generalized f-contractive maps not involving the extended Hausdorff metric. As a consequence, we obtain a fixed point result. Our results either generalize or improve a number of known results.

## 2 Results

Using the concept of w-distance, first we improve a corresponding result of Jungck  as follows.

Lemma 2.1 Let $\left(X,d\right)$ be a complete metric space with a w-distance ω. If there exist a sequence $\left\{{x}_{n}\right\}$ in X and a constant λ, $0<\lambda <1$, such that for all $n\in \mathbb{N}$,

$\omega \left({x}_{n},{x}_{n+1}\right)\le \lambda \omega \left({x}_{n-1},{x}_{n}\right),$

then the sequence $\left\{{x}_{n}\right\}$ converges in X.

Proof It is enough to show that $\left\{{x}_{n}\right\}$ is a Cauchy sequence in X. Note that for each $n\in \mathbb{N}$, we have

$\begin{array}{rcl}\omega \left({x}_{n},{x}_{n+1}\right)& \le & \lambda \omega \left({x}_{n-1},{x}_{n}\right)\\ \le & {\lambda }^{2}\omega \left({x}_{n-2},{x}_{n-1}\right)\\ ⋮\\ \le & {\lambda }^{n}\omega \left({x}_{0},{x}_{1}\right).\end{array}$

Thus

$\omega \left({x}_{n},{x}_{n+1}\right)\le {\lambda }^{n}\omega \left({x}_{0},{x}_{1}\right).$

Consequently, for $m\ge n$, we get

$\begin{array}{rcl}\omega \left({x}_{n},{x}_{m}\right)& \le & \omega \left({x}_{n},{x}_{n+1}\right)+\omega \left({x}_{n+1},{x}_{n+2}\right)+\cdots +\omega \left({x}_{m-1},{x}_{m}\right)\\ \le & {\lambda }^{n}\omega \left({x}_{0},{x}_{1}\right)+{\lambda }^{n+1}\omega \left({x}_{0},{x}_{1}\right)+\cdots +{\lambda }^{m-1}\omega \left({x}_{0},{x}_{1}\right),\end{array}$

and thus

$\omega \left({x}_{n},{x}_{m}\right)\le \frac{{\lambda }^{n}}{1-\lambda }\omega \left({x}_{0},{x}_{1}\right).$

Since $0<\lambda <1$, we have ${\lambda }^{n}\to 0$ as $n\to \mathrm{\infty }$. And thus by Lemma 1.1, $\left\{{x}_{n}\right\}$ is a Cauchy sequence in X. Since X is complete, the sequence $\left\{{x}_{n}\right\}$ converges to a point in X. □

The following lemma is crucial for our main results.

Lemma 2.2 Let $\left(X,d\right)$ be a complete metric space, and let $T:X\to Cl\left(X\right)$ be a generalized f-contractive map such that $T\left(X\right)\subset f\left(X\right)$. Then there exists an f-orbit $\left\{{x}_{n}\right\}$ of T at ${x}_{0}\in X$ such that $\left\{f\left({x}_{n}\right)\right\}$ converges in X.

Proof Let ${x}_{0}\in X$ and choose ${y}_{0}\in T\left({x}_{0}\right)$. Since $T\left({x}_{0}\right)\subset f\left(X\right)$, then there exists ${x}_{1}\in X$ such that $f\left({x}_{1}\right)={y}_{0}\in T\left({x}_{0}\right)$, and thus, by the definition of T, there exists ${y}_{1}\in T\left({x}_{1}\right)$ such that

$\omega \left(f\left({x}_{1}\right),{y}_{1}\right)\le \lambda {M}_{f}\left({x}_{0},{x}_{1}\right),$

where $0<\lambda <1$. Since $T\left({x}_{1}\right)\subset f\left(X\right)$, there exists ${x}_{2}\in X$ such that $f\left({x}_{2}\right)={y}_{1}\in T\left({x}_{1}\right)$. Thus

$\omega \left(f\left({x}_{1}\right),f\left({x}_{2}\right)\right)\le \lambda {M}_{f}\left({x}_{0},{x}_{1}\right).$

Similarly, using the definition of T and the fact that $T\left(X\right)\subset f\left(X\right)$, there exists ${x}_{3}\in X$ such that $f\left({x}_{3}\right)\in T\left({x}_{2}\right)$ and

$\omega \left(f\left({x}_{2}\right),f\left({x}_{3}\right)\right)\le \lambda {M}_{f}\left({x}_{1},{x}_{2}\right).$

Continuing this process, we get a sequence $\left\{{x}_{n}\right\}$ in X such that for all n, $f\left({x}_{n+1}\right)\in T\left({x}_{n}\right)$ and

$\omega \left(f\left({x}_{n}\right),f\left({x}_{n+1}\right)\right)\le \lambda {M}_{f}\left({x}_{n-1},{x}_{n}\right),$

that is,

$\begin{array}{rcl}\omega \left(f\left({x}_{n}\right),f\left({x}_{n+1}\right)\right)& \le & \lambda max\left\{\omega \left(f\left({x}_{n-1}\right),f\left({x}_{n}\right)\right),\omega \left(f\left({x}_{n-1}\right),T\left({x}_{n-1}\right)\right),\omega \left(f\left({x}_{n}\right),T\left({x}_{n}\right)\right),\\ \frac{1}{2}\left[\omega \left(f\left({x}_{n-1}\right),T\left({x}_{n}\right)\right)+\omega \left(f\left({x}_{n}\right),T\left({x}_{n-1}\right)\right)\right]\right\}.\end{array}$

Note that

$\begin{array}{rcl}\omega \left(f\left({x}_{n}\right),f\left({x}_{n+1}\right)\right)& \le & \lambda max\left\{\omega \left(f\left({x}_{n-1}\right),f\left({x}_{n}\right)\right),\omega \left(f\left({x}_{n-1}\right),f\left({x}_{n}\right)\right),\omega \left(f\left({x}_{n}\right),f\left({x}_{n+1}\right)\right),\\ \frac{1}{2}\left[\omega \left(f\left({x}_{n-1}\right),f\left({x}_{n+1}\right)\right)+\omega \left(f\left({x}_{n}\right),f\left({x}_{n}\right)\right)\right]\right\}\\ =& \lambda max\left\{\omega \left(f\left({x}_{n-1}\right),f\left({x}_{n}\right)\right),\omega \left(f\left({x}_{n}\right),f\left({x}_{n+1}\right)\right),\frac{1}{2}\left[\omega \left(f\left({x}_{n-1}\right),f\left({x}_{n+1}\right)\right)\right]\right\},\end{array}$

and we get

$\omega \left(f\left({x}_{n}\right),f\left({x}_{n+1}\right)\right)\le \lambda max\left\{\omega \left(f\left({x}_{n-1}\right),f\left({x}_{n}\right)\right),\frac{1}{2}\left[\omega \left(f\left({x}_{n-1}\right),f\left({x}_{n+1}\right)\right)\right]\right\}.$

Also, note that

$\begin{array}{rcl}\omega \left(f\left({x}_{n}\right),f\left({x}_{n+1}\right)\right)& \le & \lambda max\left\{\omega \left(f\left({x}_{n-1}\right),f\left({x}_{n}\right)\right),\frac{1}{2}\left[\omega \left(f\left({x}_{n-1}\right),f\left({x}_{n}\right)\right)+\omega \left(f\left({x}_{n}\right),f\left({x}_{n+1}\right)\right)\right]\right\}\\ \le & \lambda max\left\{\left[\omega \left(f\left({x}_{n-1}\right),f\left({x}_{n}\right)\right),\omega \left(f\left({x}_{n}\right),f\left({x}_{n+1}\right)\right)\right]\right\}.\end{array}$

Thus, for each $n\in \mathbb{N}$, we get

$\omega \left(f\left({x}_{n}\right),f\left({x}_{n+1}\right)\right)\le \lambda \omega \left(f\left({x}_{n-1}\right),f\left({x}_{n}\right)\right).$
(1)

Since the sequence $\left\{f\left({x}_{n}\right)\right\}$ is in the complete metric space X satisfying the inequality (1), it follows from Lemma 2.1 that $\left\{f\left({x}_{n}\right)\right\}$ converges in X. □

Remark 2.1 Since for each $n\in \mathbb{N}$ we have

$\omega \left(f\left({x}_{n}\right),f\left({x}_{n+1}\right)\right)\le \lambda \omega \left(f\left({x}_{n-1}\right),f\left({x}_{n}\right)\right),$

following the proof of Lemma 2.1, we obtain the following two useful inequalities.

$\omega \left(f\left({x}_{n}\right),f\left({x}_{n+1}\right)\right)\le {\lambda }^{n}\omega \left(f\left({x}_{0}\right),f\left({x}_{1}\right)\right)$
(2)

and for $m\ge n$

$\omega \left(f\left({x}_{n}\right),f\left({x}_{m}\right)\right)\le \frac{{\lambda }^{n}}{1-\lambda }\omega \left(f\left({x}_{0}\right),f\left({x}_{1}\right)\right).$
(3)

Without using the extended Hausdorff metric and continuity conditions, we prove a coincidence result which improves many known results including Theorem 1.2 due to , Theorem 3.3 in  and Theorem 2 in .

Theorem 2.1 Suppose that all the hypotheses of Lemma  2.2 hold. Furthermore, if for every $y\in X$ with $f\left(y\right)\notin T\left(y\right)$

$inf\left\{\omega \left(f\left(x\right),y\right)+\omega \left(f\left(x\right),T\left(x\right)\right):x\in X\right\}>0.$

Then $C\left(f\cap T\right)\ne \mathrm{\varnothing }$.

Proof By Lemma 2.2, there exists an f-orbit $\left\{{x}_{n}\right\}$ of T at ${x}_{0}\in X$ such that $\left\{f\left({x}_{n}\right)\right\}$ converges in X. Also note that for each $n\in \mathbb{N}$, we have

$\omega \left(f\left({x}_{n}\right),f\left({x}_{n+1}\right)\right)\le \lambda \omega \left(f\left({x}_{n-1}\right),f\left({x}_{n}\right)\right),$

where $0<\lambda <1$. Let $f\left({x}_{n}\right)\to y\in X$. Now since $\omega \left(f\left({x}_{n}\right),\cdot \right)$ is lower semicontinuous, from Remark 2.1 (2), we have

$\begin{array}{rcl}\omega \left(f\left({x}_{n}\right),y\right)& \le & \underset{m\to \mathrm{\infty }}{lim}inf\omega \left(f\left({x}_{n}\right),f\left({x}_{m}\right)\right)\\ \le & \frac{{\lambda }^{n}}{1-\lambda }\omega \left(f\left({x}_{0}\right),f\left({x}_{1}\right)\right).\end{array}$

Since $\lambda <1$, we get $\omega \left(f\left({x}_{n}\right),y\right)\to 0$ as $n\to \mathrm{\infty }$. Assume that $f\left(y\right)\notin T\left(y\right)$, then from the hypothesis and Remark 2.1, we get

$\begin{array}{rcl}0& <& inf\left\{\omega \left(f\left(x\right),y\right)+\omega \left(f\left(x\right),T\left(x\right)\right):x\in X\right\}\\ \le & inf\left\{\omega \left(f\left({x}_{n}\right),y\right)+\omega \left(f\left({x}_{n}\right),T\left({x}_{n}\right)\right):n\in \mathbb{N}\right\}\\ \le & inf\left\{\omega \left(f\left({x}_{n}\right),y\right)+\omega \left(f\left({x}_{n}\right),f\left({x}_{n+1}\right)\right):n\in \mathbb{N}\right\}\\ \le & inf\left\{\frac{{\lambda }^{n}}{1-\lambda }\omega \left(f\left({x}_{o}\right),f\left({x}_{1}\right)\right)+{\lambda }^{n}\omega \left(f\left({x}_{0}\right),f\left({x}_{1}\right)\right):n\in \mathbb{N}\right\}\\ =& \left\{\frac{2-\lambda }{1-\lambda }\right\}\omega \left(f\left({x}_{0}\right),f\left({x}_{1}\right)\right)inf\left\{{\lambda }^{n}:n\in \mathbb{N}\right\}=0,\end{array}$

which is impossible, and thus $f\left(y\right)\in T\left(y\right)$, that is, y is a coincidence point of f and T. □

If we take $f=I$ (an identity map on X) in Theorem 2.1, we obtain the following improved version of the corresponding fixed point results in [12, 17, 18].

Corollary 2.1 Let $\left(X,d\right)$ be a complete metric space, let ω be a w-distance on X, and let $T:X\to Cl\left(X\right)$ be a multivalued map satisfying the following:

(I) for fixed $\lambda \in \left(0,1\right)$, for each $x,y\in X$ and $u\in T\left(x\right)$, there exists $v\in T\left(y\right)$ such that

$\omega \left(u,v\right)\le \lambda {M}_{\omega }\left(x,y\right),$

where

${M}_{\omega }\left(x,y\right)=max\left\{\omega \left(x,y\right),\omega \left(x,T\left(x\right)\right),\omega \left(y,T\left(y\right)\right),\frac{1}{2}\left[\omega \left(x,T\left(y\right)\right)+\omega \left(y,T\left(x\right)\right)\right]\right\},$

(II) $inf\left\{\omega \left(x,y\right)+\omega \left(x,T\left(x\right)\right):x\in X\right\}>0$.

Then $Fix\left(T\right)\ne \mathrm{\varnothing }$.

Finally, we obtain a common fixed point result.

Theorem 2.2 Suppose that all the hypotheses of Theorem  2.1 hold. Further, if the maps f and T commute weakly and satisfy the condition that $f\left(x\right)\ne {f}^{2}\left(x\right)$, which implies $f\left(x\right)\notin T\left(x\right)$, then f and T have a common fixed point.

Proof From Theorem 2.1 we have $f\left(y\right)\in T\left(y\right)$, and thus we get $f\left(y\right)={f}^{2}\left(y\right)$. Note that

$f\left(y\right)=f\left(f\left(y\right)\right)\in f\left(T\left(y\right)\right)\subseteq T\left(f\left(y\right)\right),$

that is, $f\left(y\right)$ is a fixed point of T. Also note that $f\left(y\right)$ is a fixed point of f and thus $f\left(y\right)$ is a common fixed point of T and f. □

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## Acknowledgements

This paper is funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant No. 491/130/1432. The author therefore acknowledges with thanks DSR for technical and financial support.

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Correspondence to Marwan A Kutbi.

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Kutbi, M.A. f-Contractive multivalued maps and coincidence points. J Inequal Appl 2013, 141 (2013). https://doi.org/10.1186/1029-242X-2013-141

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• DOI: https://doi.org/10.1186/1029-242X-2013-141

### Keywords

• metric space
• fixed point
• multivalued contractive map
• coincidence point 