- Research Article
- Open Access

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

- A Amini-Harandi
^{1}Email author and - D O'Regan
^{2}

**2010**:390183

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

© A. Amini-Harandi and D. O’Regan. 2010

**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.

## Keywords

- Real Number
- Point Theorem
- Differentiable Function
- Fixed Point Theorem
- Triangle Inequality

## 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,

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

Theorem 1.1 (Nadler [1]).

for all . Then has a fixed point.

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

Corollary (Mizoguchi and Takahashi [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]).

hold. Then has a fixed point in .

Definition.

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

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
,

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.

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

Proof.

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

Corollary.

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.

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

Proof.

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.

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

## References

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**Mizoguchi-Takahashi's fixed point theorem is a real generalization of Nadler's.***Journal of Mathematical Analysis and Applications*2008,**340**(1):752–755. 10.1016/j.jmaa.2007.08.022MathSciNetView ArticleMATHGoogle Scholar

## Copyright

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.