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Fixed Point Theorems for Set-Valued Contraction Type Maps in Metric Spaces
Fixed Point Theory and Applications volume 2010, Article number: 390183 (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,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F390183/MediaObjects/13663_2009_Article_1272_Equ1_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F390183/MediaObjects/13663_2009_Article_1272_Equ2_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F390183/MediaObjects/13663_2009_Article_1272_Equ3_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F390183/MediaObjects/13663_2009_Article_1272_Equ4_HTML.gif)
Let be a map such that for any
,
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F390183/MediaObjects/13663_2009_Article_1272_Equ5_HTML.gif)
hold. Then has a fixed point in
.
Definition.
Let be a complete metric space and
be a nonempty closed subset of
.
(i)Set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F390183/MediaObjects/13663_2009_Article_1272_Equ6_HTML.gif)
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
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F390183/MediaObjects/13663_2009_Article_1272_Equ7_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F390183/MediaObjects/13663_2009_Article_1272_Equ8_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F390183/MediaObjects/13663_2009_Article_1272_Equ9_HTML.gif)
Hence for each and
, there exists an element
such that
. Thus we can define a sequence
in
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F390183/MediaObjects/13663_2009_Article_1272_Equ10_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F390183/MediaObjects/13663_2009_Article_1272_Equ11_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F390183/MediaObjects/13663_2009_Article_1272_Equ12_HTML.gif)
Thus (note that
implies
). Also we have (note
for
)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F390183/MediaObjects/13663_2009_Article_1272_Equ13_HTML.gif)
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F390183/MediaObjects/13663_2009_Article_1272_Equ14_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F390183/MediaObjects/13663_2009_Article_1272_Equ15_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F390183/MediaObjects/13663_2009_Article_1272_Equ16_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F390183/MediaObjects/13663_2009_Article_1272_Equ17_HTML.gif)
for which is continuous and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F390183/MediaObjects/13663_2009_Article_1272_Equ18_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F390183/MediaObjects/13663_2009_Article_1272_Equ19_HTML.gif)
Since , then we get
, which contradicts our assumption on
. Let
be the graph of
. Let
be given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F390183/MediaObjects/13663_2009_Article_1272_Equ20_HTML.gif)
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,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F390183/MediaObjects/13663_2009_Article_1272_Equ21_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F390183/MediaObjects/13663_2009_Article_1272_Equ22_HTML.gif)
Since satisfies (2.10) and
is continuous, then we can choose
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F390183/MediaObjects/13663_2009_Article_1272_Equ23_HTML.gif)
Let . Then by combining (2.15) and (2.16), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F390183/MediaObjects/13663_2009_Article_1272_Equ24_HTML.gif)
From (2.11), we have (note that is nondecreasing)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F390183/MediaObjects/13663_2009_Article_1272_Equ25_HTML.gif)
Thus (2.17) and (2.18) yield
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F390183/MediaObjects/13663_2009_Article_1272_Equ26_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F390183/MediaObjects/13663_2009_Article_1272_Equ27_HTML.gif)
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  
).
References
Nadler SB Jr.: Multi-valued contraction mappings. Pacific Journal of Mathematics 1969, 30: 475–488.
Banach S: Sur les opération dans les ensembles et leur application aux équations intégrales. Fundamenta Mathematicae 1922, 3: 133–181.
Mizoguchi N, Takahashi W: Fixed point theorems for multivalued mappings on complete metric spaces. Journal of Mathematical Analysis and Applications 1989,141(1):177–188. 10.1016/0022-247X(89)90214-X
Reich S: Fixed points of contractive functions. Bollettino dell'Unione Matematica Italiana 1972,5(4):26–42.
Bae JS: Fixed point theorems for weakly contractive multivalued maps. Journal of Mathematical Analysis and Applications 2003,284(2):690–697. 10.1016/S0022-247X(03)00387-1
Suzuki T: 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.2007.08.022
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.
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Amini-Harandi, A., O'Regan, D. Fixed Point Theorems for Set-Valued Contraction Type Maps in Metric Spaces. Fixed Point Theory Appl 2010, 390183 (2010). https://doi.org/10.1155/2010/390183
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DOI: https://doi.org/10.1155/2010/390183