- Research Article
- Open Access

# Approximate Endpoints for Set-Valued Contractions in Metric Spaces

- N Hussain
^{1}, - A Amini-Harandi
^{2}and - YJ Cho
^{3}Email author

**2010**:614867

https://doi.org/10.1155/2010/614867

© N. Hussain et al. 2010

**Received:**18 March 2010**Accepted:**26 April 2010**Published:**30 May 2010

## Abstract

The existence of approximate fixed points and approximate endpoints of the multivalued almost -contractions is established. We also develop quantitative estimates of the sets of approximate fixed points and approximate endpoints for multivalued almost -contractions. The proved results unify and improve recent results of Amini-Harandi (2010), M. Berinde and V. Berinde (2007), Ćirić (2009), M. Păcurar and R. V. Păcurar (2007) and many others.

## Keywords

- Nash Equilibrium
- Topological Space
- Multivalued Mapping
- Convex Space
- Fixed Point Theory

## 1. Introduction and Preliminaries

In fixed point theory, one of the main directions of investigation concerns the study of the fixed point property in topological spaces. Recall that a topological space is said to have the fixed point property if every continuous mapping has a fixed point. The major contribution to this subject has been provided by Tychonoff. Every compact convex subset of a locally convex space has the fixed point property. Another important branch of fixed point theory is the study of the approximate fixed point property. Recently, the interest in approximate fixed point results arise in the study of some problems in economics and game theory, including, for example, the Nash equilibrium approximation in games; see [1–3] and references therein.

We establish some existence results concerning approximate fixed points, endpoints, and approximate endpoints of multivalued contractions. We also develop quantitative estimates of the sets of approximate fixed points and approximate endpoints for set-valued almost -contractions. The results presented in this paper extend and improve the recent results of [4–10] and many others.

Now, we give some notions and definitions.

Let be a metric space and let and denote the families of all nonempty subsets and nonempty closed subsets of , respectively. Let and be two Hausdorff topological spaces and a multivalued mapping with nonempty values. Then is said to be

(1)*upper semicontinuous* (u.s.c.) if, for each closed set
,
is closed in
;

(2)*lower semicontinuous* (l.s.c.) if, for each open set
,
is open in
;

(3)*continuous* if it is both u.s.c. and l.s.c.;

(4)*closed* if its graph
is closed;

(5)*compact* if
is a compact subset of
.

For any subsets , of a metric space , we consider the following notions:

: the distance between the sets and ;

: the diameter of the sets and ;

: the diameter of the set ;

: the Hausdorff metric on induced by the metric .

Let be a multivalued mapping. An element such that is called a fixed point of . We denote by the set of all fixed points of , that is,

A mapping is called

*multivalued almost contraction*[6] or a

*multivalued*

*-almost contraction*if there exist two constants and such that

*generalized multivalued almost contraction*[6] if there exists a function satisfying for every such that

It is important to note that any mapping satisfying Banach, Kannan, Chatterjea, Zamfirescu, or Ćirić (with the constant in ) type conditions is a single-valued almost contraction; see [5, 6, 8, 11].

## 2. Approximate Fixed Points of Multivalued Contractions

Definition 2.1.

where denotes a closed ball of radius centered at .

We first prove that every generalized multivalued almost contraction has the approximate fixed point property.

Lemma 2.2.

Every generalized multivalued almost contraction has the approximate fixed point property.

Proof.

Since , we get . This completes the proof.

Corollary 2.3 (see [5, Theorem ], [10, Theorem ]).

Let be a metric space and a single-valued almost contraction. Then has the approximate fixed point property.

The authors in [5, 10] obtained the following quantitative estimate of the diameter of the set, , of approximate fixed points of single-valued almost contraction .

Theorem 2.4 (see [5, Theorem ], [10, Theorem ]).

The following simple example shows that the conclusion of Theorem 2.4 is not valid for set-valued almost contractions.

Example 2.5.

Let be defined by . Then and so is multivalued almost contraction with . Further, and so . This shows that conclusion of Theorem 2.4 is not true whenever is multivalued almost contraction.

Theorem 2.6.

Let be a metric space. If is a generalized multivalued almost contraction, then has a fixed point provided either is compact and the function is lower semicontinuous or is closed and compact.

Proof.

By Lemma 2.2, we have . The lower semicontinuity of the function and the compactness of imply that the infimum is attained. Thus there exists an such that and so .

Now, since is compact, we may assume that converges to a point as . Consequently, also converges to as . Since is closed, then This completes the proof.

*multivalued almost*

*-contraction*[6, 8] if there exist constants and such that

*generalized multivalued almost*

*-contraction*if there exists a function satisfying for every such that

A point
is called a *coincidence (common fixed) point* of
and
if
(
).

Theorem 2.7.

Every generalized multivalued almost -contraction in a metric space has the approximate coincidence point property provided each is -invariant. Further, if is compact and the function is lower semicontinuous, then and have a coincidence point.

Proof.

we get .

Further, the lower semi-continuity of the function and the compactness of imply that the infimum is attained. Thus there exists such that and so as required. This completes the proof.

Corollary 2.8.

Every multivalued almost -contraction in a metric space has the approximate coincidence point property provided each is -invariant. Further, if is compact and the function is lower semicontinuous, then and have a coincidence point.

Recently, Ćirić [7] has introduced multivalued contractions and obtained some interesting results which are proper generalizations of the recent results of Klim and Wardowski [9], Feng and Liu [12], and many others. In the results to follow, we obtain approximate fixed point property for these multivalued contractions.

Theorem 2.9.

where . Then has the approximate fixed point property. Further, has a fixed point provided either is compact and the function is lower semicontinuous or is closed and compact.

Proof.

Since , we get .

Further, the lower semi-continuity of the function and the compactness of imply that the infimum is attained. Thus there exists such that and so .

The second assertion follows as in the proof of Theorem 2.6. This completes the proof.

Theorem 2.10.

where . Then has the approximate fixed point property. Further, has a fixed point provided either is compact and the function is lower semicontinuous or is closed and compact.

Proof.

Since , we get .

Further, the lower semi-continuity of the function and the compactness of imply that the infimum is attained. Thus there exists such that and so .

The second assertion follows as in the proof of Theorem 2.6. This completes the proof.

## 3. Endpoints of Multivalued Nonlinear Contractions

*endpoint (or stationary point)*[13] of if . We say that a multivalued mapping has the

*approximate endpoint property*[4] if

*approximate endpoint property*provided

A point
is called an *endpoint* of
and
if
.

Lemma 3.1.

Proof.

The following simple example shows that under the assumptions of Lemma 3.1, may be empty.

Example 3.2.

Let be a multivalued mapping defined by for each and the identity mapping. Then and so is a multivalued almost -contraction with . However, for each .

Lemma 3.3.

Let be a metric space. Let be a continuous single-valued mapping and a lower semicontinuous multivalued mapping. Then, for each , is closed.

Proof.

Let be such that with as . Let . Since is lower semicontinuous, then there exists such that . Since , then and so . Since is continuous, . Therefore, , that is, . This completes the proof.

Theorem 3.4.

Let be a complete metric space. Let be a continuous single-valued mapping such that , where is a constant. Let be a lower semicontinuous multivalued almost -contraction. Then and have a unique endpoint if and only if and have the approximate endpoint property.

Proof.

Thus is the unique endpoint of and .

If is the identity mapping on , then the above result reduces to the following.

Corollary 3.5.

where .

Corollary 3.6.

Let be a complete metric space. Let be a lower semicontinuous multivalued almost contraction with . Then has a unique endpoint if and only if has the approximate endpoint property.

Corollary 3.7 (see [4, Corollary ]).

Let be a complete metric space. Let be a multivalued -contraction. Then has a unique endpoint if and only if has the approximate endpoint property.

Theorem 3.8.

where . Then has the approximate endpoint property. Further, has an endpoint provided is compact and the function is lower semicontinuous.

Proof.

Thus has the approximate endpoint property. The lower semi-continuity of the function and the compactness of imply that the infimum is attained. Thus there exists such that . Therefore, . This completes the proof.

The following theorem extends and improves Theorem in [4].

Theorem 3.9.

where is a function such that and for each . Then and have a unique endpoint if and only if and have the approximate endpoint property.

Proof.

Thus and hence . To prove the uniqueness of the endpoints of and , let be an arbitrary endpoint of and . Then =0 and so . Thus . This completes the proof.

From Theorem 3.9, we obtain the following improved version of the main result of [4].

Corollary 3.10.

where is a function such that and for each . Then has a unique endpoint if and only if has the approximate endpoint property.

Example 3.11.

Then and satisfy the conditions of Corollary 3.10, but the conditions of Theorem in [4] are not satisfied (note that ).

## Declarations

### Acknowledgments

The authors would like to thank the referees for their valuable suggestions to improve the paper. This work was supported by the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-313-C00050).

## Authors’ Affiliations

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