Open Access

Fixed Point Theorems for Set-Valued Contraction Type Maps in Metric Spaces

Fixed Point Theory and Applications20102010:390183

https://doi.org/10.1155/2010/390183

Received: 13 August 2009

Accepted: 13 January 2010

Published: 14 February 2010

Abstract

We first give some fixed point results for set-valued self-map contractions in complete metric spaces. Then we derive a fixed point theorem for nonself set-valued contractions which are metrically inward. Our results generalize many well-known results in the literature.

1. Introduction and Preliminaries

Let be a metric space and let CB denote the class of all nonempty bounded closed subsets of . Let be the Hausdorff metric with respect to , that is,

(1.1)

for every   CB , where . In 1969, Nadler [1] extended the Banach contraction principle [2] to set-valued mappings.

Theorem 1.1 (Nadler [1]).

Let be a complete metric space and let CB be a set-valued map. Assume that there exists such that
(1.2)

for all . Then has a fixed point.

Mizoguchi and Takahashi [3] proved the following generalization of Theorem 1.1.

Corollary (Mizoguchi and Takahashi [3]).

Let be a complete metric space and let CB be a set-valued map satisfying
(1.3)

where satisfies for each . Then has a fixed point.

Also, Reich [4] has proved that if for each , is nonempty and compact, then the above result holds under the weaker condition for each . To set up our results in the next section, we introduce some definitions and facts.

Definition.

Throughout the paper, let be the family of all functions satisfying the following conditions:

(a) ;

(b) is lower semicontinuous and nondecreasing;

(c) .

Theorem 1.4 (Bae [5]).

Let be a complete metric space, a lower semicontinuous function, and a lower semicontinuous function such that for and
(1.4)
Let be a map such that for any , and
(1.5)

hold. Then has a fixed point in .

Definition.

Let be a complete metric space and be a nonempty closed subset of .

(i)Set
(1.6)

Then is called the metrically inward set of at (see [5]);

(ii)Let be a set-valued map. is said to be metricaly inward, if for each ,

(1.7)

In Section 2 we generalize Corollary 1.2 and Theorem 1.4.

2. Extension of Mizoguchi-Takahashi's Theorem

In the first result of this section, we use the technique in [6] to extend Corollary 1.2.

Theorem 2.1.

Let be a complete metric space and let CB be a set-valued map satisfying
(2.1)

where satisfies for each and . Then has a fixed point.

Proof.

Define a function by . Then and for all . Since is nondecreasing, then from (1.3), for each , we have
(2.2)
Hence for each and , there exists an element such that . Thus we can define a sequence in satisfying
(2.3)
for each . Let us show that is convergent. Since for each , then is a nonincreasing sequence of non-negative numbers and so is convergent to a real number, say . Since and , there exist and such that for all . We can take such that for all with . Since
(2.4)
for all , then we have and so (note that ). If for some , then for each (note that is nonincreasing). Thus is eventually constant, so we have a fixed point of (note that ). Now, we assume that for each . Since is decreasing and is nondecreasing, then the nonnegative sequence converges to some nonnegative real number . Since is nondecreasing and is nonincreasing, then for each . Thus
(2.5)
Thus (note that implies ). Also we have (note for )
(2.6)
Since
(2.7)
then . Hence is a Cauchy sequence. Since is complete, converges to some point . Since is lower semicontinuous and nondecreasing (recall also from above that ), then
(2.8)

and this with closed and (a) of Definition 1.3 implies .

Corollary.

Let be a complete metric space and let CB be a set-valued map satisfying
(2.9)

where and satisfying for each . Then has a fixed point.

Proof.

Let and apply Theorem 2.1.

In the following, we present a fixed point theorem for nonself set-valued contraction type maps which are metrically inward.

Theorem.

Let be a nonempty closed subset of a complete metric space and CB be a set-valued map satisfying
(2.10)
for which is continuous and
(2.11)

Assume that is a lower semicontinuous function satisfying and for . Suppose that is metrically inward on . Then has a fixed point in .

Proof.

We first show that . On the contrary, we assume that there exists a sequence for which
(2.12)
Since , then we get , which contradicts our assumption on . Let be the graph of . Let be given by
(2.13)
We show that is a complete metric space. First note that since then . Clearly, . Now we show the triangle inequality. From (2.11), we have . Hence,
(2.14)
To prove the completeness of , we first need to show that is Hausdorff continuous. To prove this, let be a sequence in such that . Since is continuous at , then . Hence from (2.10), we get . We claim that (and then we are finished). On the contrary, assume that there exist and a subsequence such that , =1,2,3,   . Since is nondecreasing, then , a contradiction. Now, let be a Cauchy sequence in with respect to . Then and are Cauchy sequences in the complete metric space . Then there exist such that and . Since and is Hausdorff continuous, then . Thus and . Therefore, is a complete metric space. Suppose that has no fixed point. Then for each , we have . Since , we can choose such that and
(2.15)
Since satisfies (2.10) and is continuous, then we can choose such that
(2.16)
Let . Then by combining (2.15) and (2.16), we get
(2.17)
From (2.11), we have (note that is nondecreasing)
(2.18)
Thus (2.17) and (2.18) yield
(2.19)

Since , by defining by , from Theorem 1.4, must have a fixed point, say . Then . Hence . This is a contradiction. Therefore, has a fixed point.

Remark.

Note that Theorem 2.3 does not follow from Theorem 3.3 of Bae [5] by replacing the metric by . In Theorem 2.3, we assume is metrically inward with respect to but to apply Theorem 3.3 of [5] with rather than , we need to be metrically inward with respect to .

Letting for each , we get the following corollary due to Bae [5].

Corollary.

Let be a nonempty closed subset of a complete metric space and CB be a set-valued map satisfying
(2.20)

for which is a lower semicontinuous function satisfying . Suppose that is metrically inward on . Then has a fixed point in .

Example 2.6.

Let  be a differentiable function with   such that     is  positive and decreasing in     and   .  Now we show that     satisfies all the conditions of Theorem 2.3. Obviously,   is continuous and increasing. Since   ,  then by L'Hopital's rule   . Thus  Now we prove that for each   ,   . To show this let   for . Then   . Since     and is increasing, we get     for each     and we are done. Finally, we show that for each   ,  we have   .  Let     for   .  Then   .  If   ,  then   .  Since   ,  we obtain     for each     and  we  are finished. In the case,   ,   if and only  if   .  Since     for     and     for   ,  then   ,  and we  are finished (note that we proved above that   ).

Declarations

Acknowledgments

The authors would like to thank the referees for careful reading and giving valuable comments. This work was supported in part by the Shahrekord University. The first author would like to thank this support.

Authors’ Affiliations

(1)
Department of Mathematics, University of Shahrekord
(2)
Department of Mathematics, National University of Ireland

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Copyright

© A. Amini-Harandi and D. O’Regan. 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.