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Fixed Point Theory and Applications

, 2010:614867 | Cite as

Approximate Endpoints for Set-Valued Contractions in Metric Spaces

Open Access
Research Article

Abstract

The existence of approximate fixed points and approximate endpoints of the multivalued almost Open image in new window -contractions is established. We also develop quantitative estimates of the sets of approximate fixed points and approximate endpoints for multivalued almost Open image in new window -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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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 Open image in new window is said to have the fixed point property if every continuous mapping Open image in new window 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, 2, 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 Open image in new window -contractions. The results presented in this paper extend and improve the recent results of [4, 5, 6, 7, 8, 9, 10] and many others.

Now, we give some notions and definitions.

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

(1)upper semicontinuous (u.s.c.) if, for each closed set Open image in new window , Open image in new window is closed in Open image in new window ;

(2)lower semicontinuous (l.s.c.) if, for each open set Open image in new window , Open image in new window is open in Open image in new window ;

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

(4)closed if its graph Open image in new window is closed;

(5)compact if Open image in new window is a compact subset of Open image in new window .

For any subsets Open image in new window , of a metric space Open image in new window , we consider the following notions:

Open image in new window : the distance between the sets Open image in new window and Open image in new window ;

Open image in new window : the diameter of the sets Open image in new window and Open image in new window ;

Open image in new window : the diameter of the set Open image in new window ;

Open image in new window : the Hausdorff metric on Open image in new window induced by the metric Open image in new window .

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

A mapping Open image in new window is called

a multivalued contraction (or multivalued Open image in new window -contraction) if there exists a number Open image in new window such that
a multivalued almost contraction [6] or a multivalued Open image in new window -almost contraction if there exist two constants Open image in new window and Open image in new window such that
a generalized multivalued almost contraction [6] if there exists a function Open image in new window satisfying Open image in new window for every Open image in new window such that

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

2. Approximate Fixed Points of Multivalued Contractions

Definition 2.1.

A multivalued mapping Open image in new window is said to have the approximate fixed point property [2] provided
or, equivalently, for any Open image in new window , there exists Open image in new window such that
or, equivalently, for any Open image in new window , there exists Open image in new window such that

where Open image in new window denotes a closed ball of radius Open image in new window centered at Open image in new window .

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.

Let Open image in new window be an arbitrary metric space and Open image in new window a generalized multivalued almost contraction. Let Open image in new window and Open image in new window be such that
By passing to the subsequences, if necessary, we may assume that the sequence Open image in new window is convergent. Then we have

Since Open image in new window , we get Open image in new window . This completes the proof.

Corollary 2.3 (see [5, Theorem Open image in new window ], [10, Theorem Open image in new window ]).

Let Open image in new window be a metric space and Open image in new window a single-valued almost contraction. Then Open image in new window has the approximate fixed point property.

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

Theorem 2.4 (see [5, Theorem Open image in new window ], [10, Theorem Open image in new window ]).

Let Open image in new window be a metric space. If Open image in new window is a single-valued almost contraction with Open image in new window , then

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

Example 2.5.

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

Theorem 2.6.

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

Proof.

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

Suppose that Open image in new window is closed and compact. According to Lemma 2.2, Open image in new window has the approximate fixed point property. Therefore, for any Open image in new window , there exist Open image in new window and Open image in new window such that

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

Let Open image in new window be a single-valued mapping and Open image in new window a multivalued mapping. Then Open image in new window is called a multivalued almost Open image in new window -contraction [6, 8] if there exist constants Open image in new window and Open image in new window such that
We say that Open image in new window is a generalized multivalued almost Open image in new window -contraction if there exists a function Open image in new window satisfying Open image in new window for every Open image in new window such that
The mappings Open image in new window and Open image in new window are said to have an approximate coincidence point property provided
or, equivalently, for any Open image in new window , there exists Open image in new window such that

A point Open image in new window is called a coincidence (common fixed) point of Open image in new window and Open image in new window if Open image in new window ( Open image in new window ).

Theorem 2.7.

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

Proof.

Let Open image in new window be a generalized multivalued almost Open image in new window -contraction and let Open image in new window and Open image in new window be such that
By passing to the subsequences, if necessary, we may assume that the sequence Open image in new window is convergent. Then we have

we get Open image in new window .

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

Corollary 2.8.

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

Let Open image in new window be a metric space and Open image in new window a multivalued mapping from Open image in new window into Open image in new window Suppose that there exist a function Open image in new window such that
and Open image in new window and Open image in new window satisfying the following two conditions:

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

Proof.

Let Open image in new window and Open image in new window be the sequences that satisfy (2.16). By passing to subsequences, if necessary, we may assume that both of the sequences Open image in new window and Open image in new window are convergent (note that Open image in new window is bounded since Open image in new window ). Then we have

Since Open image in new window , we get Open image in new window .

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

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

Theorem 2.10.

Let Open image in new window be a metric space and Open image in new window a multivalued mapping from Open image in new window into Open image in new window Suppose that there exist a function Open image in new window such that
and Open image in new window and Open image in new window satisfying the following two conditions:

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

Proof.

Let Open image in new window and Open image in new window satisfy (2.19). By passing to subsequences, if necessary, we may assume that the sequence Open image in new window is convergent. Then we have

Since Open image in new window , we get Open image in new window .

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

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

3. Endpoints of Multivalued Nonlinear Contractions

Let Open image in new window be a multivalued mapping. An element Open image in new window is said to be a endpoint (or stationary point) [13] of Open image in new window if Open image in new window . We say that a multivalued mapping Open image in new window has the approximate endpoint property [4] if
Let Open image in new window be a single-valued mapping and Open image in new window a multivalued contraction. We say that the mappings Open image in new window and Open image in new window have an approximate endpoint property provided

A point Open image in new window is called an endpoint of Open image in new window and Open image in new window if Open image in new window .

Lemma 3.1.

Let Open image in new window be a metric space. Let Open image in new window be a single-valued mapping such that Open image in new window for all Open image in new window , where Open image in new window is a constant. If Open image in new window is a multivalued almost Open image in new window -contraction with Open image in new window , then

Proof.

Since Open image in new window , from (3.6), we have

The following simple example shows that under the assumptions of Lemma 3.1, Open image in new window may be empty.

Example 3.2.

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

Lemma 3.3.

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

Proof.

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

Theorem 3.4.

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

Proof.

It is clear that, if Open image in new window and Open image in new window have an endpoint, then Open image in new window and Open image in new window have the approximate endpoint property. Conversely, suppose that Open image in new window and Open image in new window have the approximate endpoint property. Then
Also it is clear that, for each Open image in new window , Open image in new window . By Lemma 3.3, Open image in new window is closed for each Open image in new window . Since Open image in new window and Open image in new window have the approximate endpoint property, then Open image in new window for each Open image in new window . Now, we show that Open image in new window . To show this, let Open image in new window . Then, from Lemma 3.1,
and so Open image in new window . It follows from the Cantor intersection theorem that

Thus Open image in new window is the unique endpoint of Open image in new window and Open image in new window .

If Open image in new window is the identity mapping on Open image in new window , then the above result reduces to the following.

Corollary 3.5.

Let Open image in new window be a metric space. If Open image in new window is a multivalued almost contraction with Open image in new window , then

where Open image in new window .

Corollary 3.6.

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

Corollary 3.7 (see [4, Corollary Open image in new window ]).

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

Theorem 3.8.

Let Open image in new window be a complete metric space and Open image in new window a multivalued mapping from Open image in new window into Open image in new window . Suppose that there exist a function Open image in new window such that
and Open image in new window and Open image in new window satisfying the two following conditions:

where Open image in new window . Then Open image in new window has the approximate endpoint property. Further, Open image in new window has an endpoint provided Open image in new window is compact and the function Open image in new window is lower semicontinuous.

Proof.

We first prove that Open image in new window has the approximate endpoint property. Let Open image in new window and Open image in new window that satisfy (3.13). By passing to subsequences, if necessary, we may assume that the sequence Open image in new window is convergent. Then we have

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

The following theorem extends and improves Theorem Open image in new window in [4].

Theorem 3.9.

Let Open image in new window be a complete metric space. Let Open image in new window be a continuous single-valued mapping such that Open image in new window , where Open image in new window is a constant. Let Open image in new window be a multivalued mapping satisfying

where Open image in new window is a function such that Open image in new window and Open image in new window for each Open image in new window . Then Open image in new window and Open image in new window have a unique endpoint if and only if Open image in new window and Open image in new window have the approximate endpoint property.

Proof.

It is clear that, if Open image in new window and Open image in new window have an endpoint, then Open image in new window and Open image in new window have the approximate endpoint property. Conversely, suppose that Open image in new window and Open image in new window have the approximate endpoint property. Then
Also it is clear that, for each Open image in new window , Open image in new window . Since the mapping Open image in new window is continuous (note that Open image in new window and Open image in new window are continuous), we have that Open image in new window is closed. Now we show that Open image in new window . On the contrary, assume that Open image in new window . Since Open image in new window , then Open image in new window (note that the sequences Open image in new window and Open image in new window are nonincreasing and bounded below and then they have the limits). Let Open image in new window be such that Open image in new window . Given Open image in new window , from (3.16) and triangle inequality, we have
Therefore, we have
From (3.19), we have Open image in new window for each Open image in new window and so we get
Hence we have
From (3.21), we obtain
which is a contradiction and so Open image in new window . It follows from the Cantor intersection theorem that

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

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

Corollary 3.10.

Let Open image in new window be a complete metric space. Let Open image in new window be a multivalued mapping satisfying

where Open image in new window is a function such that Open image in new window and Open image in new window for each Open image in new window . Then Open image in new window has a unique endpoint if and only if Open image in new window has the approximate endpoint property.

Example 3.11.

Let Open image in new window with the usual metric Open image in new window . Let Open image in new window be a multivalued mapping defined by Open image in new window and Open image in new window be a function defined by

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

Notes

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).

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© N. Hussain et al. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Department of MathematicsKing Abdulaziz UniversityJeddahSaudi Arabia
  2. 2.Department of MathematicsUniversity of ShahrekordShahrekordIran
  3. 3.Department of Mathematics Education and the RINSGyeongsang National UniversityChinjuRepublic of Korea

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