Fixed Point Theorems for Set-Valued Contraction Type Maps in Metric Spaces
© A. Amini-Harandi and D. O’Regan. 2010
Received: 13 August 2009
Accepted: 13 January 2010
Published: 14 February 2010
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,
Theorem 1.1 (Nadler ).
for all . Then has a fixed point.
Mizoguchi and Takahashi  proved the following generalization of Theorem 1.1.
Corollary (Mizoguchi and Takahashi ).
where satisfies for each . Then has a fixed point.
Also, Reich  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.
Throughout the paper, let be the family of all functions satisfying the following conditions:
(b) is lower semicontinuous and nondecreasing;
Theorem 1.4 (Bae ).
hold. Then has a fixed point in .
Let be a complete metric space and be a nonempty closed subset of .
Then is called the metrically inward set of at (see );
(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  to extend Corollary 1.2.
where satisfies for each and . Then has a fixed point.
and this with closed and (a) of Definition 1.3 implies .
where and satisfying for each . Then has a fixed point.
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.
Assume that is a lower semicontinuous function satisfying and for . Suppose that is metrically inward on . Then has a fixed point in .
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.
Note that Theorem 2.3 does not follow from Theorem 3.3 of Bae  by replacing the metric by . In Theorem 2.3, we assume is metrically inward with respect to but to apply Theorem 3.3 of  with rather than , we need to be metrically inward with respect to .
Letting for each , we get the following corollary due to Bae .
for which is a lower semicontinuous function satisfying . Suppose that is metrically inward on . Then has a fixed point in .
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 ).
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.
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