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*f*-Contractive multivalued maps and coincidence points

*Journal of Inequalities and Applications*
**volume 2013**, Article number: 141 (2013)

## 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 (X,d) be a metric space. Let {2}^{X}, Cl(X) and CB(X) 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,

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

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(x), and the set of fixed points of *T* is denoted by Fix(T). A point x\in X is called a coincidence point of *f* and *T* if f(x)\in T(x). We denote by C(f\cap T) the set of coincidence points of *f* and *T*.

We say a sequence \{{x}_{n}\} 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(X) is called

(i) *contraction* [1] if for a fixed constant \lambda \in (0,1) and for each x,y\in X,

(ii) *f*-*contraction* [2] if for a fixed constant \lambda \in (0,1) and for each x,y\in X,

Using the concept of Hausdorff metric, Nadler [1] 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** [1]

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

This result has been generalized in many directions. Kaneko [2] extended the corresponding results of Jungck [3], Nadler [1] and others as follows.

**Theorem 1.2** [2]

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

This result has been generalized in different directions. For example, see [4–10].

On the other hand, Kada *et al.* [11] introduced the concept of *w*-distance on a metric space as follows:

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

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

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

({w}_{3}) for any \u03f5>0, there exists \delta >0 such that \omega (z,x)\le \delta and \omega (z,y)\le \delta imply d(x,y)\le \u03f5.

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

**Lemma 1.1** [11]

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

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

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

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

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

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

(iii) *T* is *w*-*contractive* [12] if there exist a *w*-distance *ω* on *X* and \lambda \in (0,1) such that for any x,y\in X and u\in T(x), there is v\in T(y) with

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

where

Using the concept of *w*-distance, Suzuki and Takahashi [12] improved Nadler’s fixed point result as follows.

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

This result has been generalized by many authors, for example, see [13–16]. In this paper, first we establish a lemma with respect to a *w*-distance, which is an improved version of the lemma given in [3], 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 [3] as follows.

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

*then the sequence* \{{x}_{n}\} *converges in* *X*.

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

Thus

Consequently, for m\ge n, we get

and thus

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

The following lemma is crucial for our main results.

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

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

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

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

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

that is,

Note that

and we get

Also, note that

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

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

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

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

and for m\ge n

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 [2], Theorem 3.3 in [17] and Theorem 2 in [7].

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

*Then* C(f\cap T)\ne \mathrm{\varnothing}.

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

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

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

which is impossible, and thus f(y)\in T(y), 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* (X,d) *be a complete metric space*, *let* *ω* *be a* *w*-*distance on* *X*, *and let* T:X\to Cl(X) *be a multivalued map satisfying the following*:

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

*where*

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

*Then* Fix(T)\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(x)\ne {f}^{2}(x), *which implies* f(x)\notin T(x), *then* *f* *and* *T* *have a common fixed point*.

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

that is, f(y) is a fixed point of *T*. Also note that f(y) is a fixed point of *f* and thus f(y) 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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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