- Research Article
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On Fixed Point Theorems for Multivalued Contractions
Fixed Point Theory and Applications volume 2010, Article number: 870980 (2010)
Abstract
Three concepts of multivalued contractions in complete metric spaces are introduced, and the conditions guaranteeing the existence of fixed points for the multivalued contractions are established. The results obtained in this paper extend genuinely a few fixed point theorems due to Ćirić (2009) Feng and Liu (2006) and Klim and Wardowski (2007). Five examples are given to explain our results.
1. Introduction and Preliminaries
Let be a metric space, and let
,
, and
denote the families of all nonempty closed, all nonempty closed and bounded, and all nonempty compact subsets of
, respectively. For any
and
, let
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ1_HTML.gif)
Such a mapping is called a generalized Hausdorff metric in
induced by
.
Throughout this paper, we assume that ,
, and
denote the sets of all real numbers, nonnegative real numbers, and positive integers, respectively.
The existence of fixed points for various multivalued contractive mappings had been studied by many authors under different conditions. For details, we refer the reader to [1–7] and the references therein. In 1969, Nadler Jr [7] extended the famous Banach Contraction Principle from single-valued mapping to multivalued mapping and proved the following fixed point theorem for the multivalued contraction.
Theorem 1.1 (see [7]).
Let be a complete metric space, and let
be a mapping from
into
. Assume that there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ2_HTML.gif)
Then, has a fixed point.
In 1989, Mizoguchi and Takahashi [6] generalized the Nadler fixed point theorem and got a fixed point theorem for the multivalued contraction as follows.
Theorem 1.2 (see [6]).
Let be a complete metric space, and let
be a mapping from
into
. Assume that there exists a map
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ3_HTML.gif)
Then, has a fixed point.
In 2006, Feng and Liu [3] obtained a new extension of the Nadler fixed point theorem and proved the following fixed point theorem.
Theorem 1.3 (see [3]).
Let be a complete metric space, and let
be a multivalued mapping from
into
. If there exist constants
, such that for any
there is
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ4_HTML.gif)
then has a fixed point in
provided a function
,
is lower semicontinuous.
In 2007, Klim and Wardowski [5] improved the result of Feng and Liu and proved the following results.
Theorem 1.4 (see [5]).
Let be a complete metric space, and let
be a multivalued mapping from
into
. Assume that
(a)the mapping , defined by
,
, is lower semi-continuous;
(b)there exist and
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ5_HTML.gif)
and for any there is
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ6_HTML.gif)
Then, has a fixed point in
.
Theorem 1.5 (see [5]).
Let be a complete metric space, and let
be a mapping from
into
. Assume that
(a)the mapping , defined by
,
, is lower semi-continuous;
(b)there exists a function satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ7_HTML.gif)
and for any there is
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ8_HTML.gif)
Then, has a fixed point in
.
In 2008 and 2009, Ćirić [1, 2] introduced new multivalued nonlinear contractions and established a few nice fixed point theorems for the multivalued nonlinear contractions, one of which is as follows.
Theorem 1.6 (see [2]).
Let be a complete metric space, and let
be a mapping from
into
. Assume that
(a)the mapping , defined by
,
, is lower semi-continuous;
(b)there exists a function ,
, satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ9_HTML.gif)
and for any there is
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ10_HTML.gif)
Then has a fixed point in
.
The aim of this paper is both to introduce three new multivalued contractions in complete metric spaces and to prove the existence of fixed points for the multivalued contractions under weaker conditions than the ones in [2, 3, 5]. Five nontrivial examples are given to show that the results presented in this paper generalize substantially and unify the corresponding fixed point theorems of Ćirić [2], Feng and Liu [3], and Klim and Wardowski [5] and are different from those results of Mizoguchi and Takahashi [6] and Nadler Jr [7].
Next we recall and introduce the following result in [4] and some notions and terminologies.
Lemma 1.7 (see [4]).
Let be a complete metric space and
. Then, for each
and
there exists an element
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ11_HTML.gif)
In the rest of this paper, for a multivalued mapping , we put
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ12_HTML.gif)
where and
, for all
. The function
is said to be
-orbitally lower semicontinuous at
if
is an arbitrary orbit of
with
impling that
.
2. Main Results
In this section, we establish three fixed point theorems for three new multivalued contractions in complete metric spaces.
Theorem 2.1.
Let be a multivalued mapping from a complete metric space
into
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ13_HTML.gif)
where and
satisfy that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ14_HTML.gif)
Then,
(a1) for each there exists an orbit
of
and
such that
;
(a2) is a fixed point of
in
if and only if the function
is
-orbitally lower semi-continuous at
.
Proof.
Put
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ15_HTML.gif)
It follows from (2.1) that for each there exists
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ16_HTML.gif)
which together with (2.3) yield that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ17_HTML.gif)
Continuing this process, we choose easily an orbit of
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ18_HTML.gif)
which imply that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ19_HTML.gif)
Now, we claim that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ20_HTML.gif)
Notice that the ranges of , (2.2), and (2.3) ensure that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ21_HTML.gif)
Using (2.7) and (2.9), we conclude that is a nonnegative and nonincreasing sequence, which means that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ22_HTML.gif)
for some . Suppose that
. Taking limits superior as
in (2.7) and using (2.2), (2.3), (2.9), and (2.10), we get that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ23_HTML.gif)
which is a contradiction. Thus, ; that is, (2.8) holds.
Next, we show that is a Cauchy sequence. Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ24_HTML.gif)
It follows from (2.2), (2.3), (2.9), and (2.12) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ25_HTML.gif)
Let and
. In light of (2.12) and (2.13), we deduce that there exists some
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ26_HTML.gif)
which together with (2.6) and (2.7) yield that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ27_HTML.gif)
which imply that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ28_HTML.gif)
which give that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ29_HTML.gif)
which implies that is a Cauchy sequence because
. It follows from completeness of
that there is some
such that
.
Suppose that is
-orbitally lower semi-continuous in
. It follows from (2.8) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ30_HTML.gif)
which means that because
is closed.
Suppose that is a fixed point of
in
. For any orbit
of
with
and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ31_HTML.gif)
that is, is
-orbitally lower semi-continuous in
. This completes the proof.
Theorem 2.2.
Let be a multivalued mapping from a complete metric space
into
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ32_HTML.gif)
where satisfies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ33_HTML.gif)
Then,
(a1) for each there exists an orbit
of
and
such that
;
(a2) is a fixed point of
in
if and only if the function
is
-orbitally lower semi-continuous at
.
Proof.
It follows from Lemma 1.7 that for each there exists
with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ34_HTML.gif)
which together with (2.20) and (2.21) ensures that (2.1) and (2.2) hold with and
. Thus, Theorem 2.2 follows from Theorem 2.1. This completes the proof.
Theorem 2.3.
Let be a multivalued mapping from a complete metric space
into
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ35_HTML.gif)
where and
satisfy that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ36_HTML.gif)
and one of and
is nondecreasing. Then,
(a1) for each there exists an orbit
of
and
such that
;
(a2) is a fixed point of
in
if and only if the function
is
-orbitally lower semi-continuous at
.
Proof.
Put
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ37_HTML.gif)
It follows from the ranges of , (2.24), and (2.25) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ38_HTML.gif)
As in the proof of Theorem 2.1, we select an orbit of
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ39_HTML.gif)
which imply that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ40_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ41_HTML.gif)
Using (2.26) and (2.28), we conclude easily that is a nonnegative and nonincreasing sequence. Consequently there exists some
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ42_HTML.gif)
Now we claim that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ43_HTML.gif)
Suppose to the contrary, that is, there exists some satisfying
. Note that one of
and
is nondecreasing. In view of (2.25), (2.26), and (2.29), we get that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ44_HTML.gif)
which is a contradiction. Thus, (2.31) holds. Therefore, there exists some such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ45_HTML.gif)
Next, we show that . Suppose that
. Taking limits superior as
in (2.28) and using (2.24), (2.25), (2.30), and (2.33), we get that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ46_HTML.gif)
which is impossible. Thus, . Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ47_HTML.gif)
It follows from (2.24), (2.25), (2.33), and (2.35) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ48_HTML.gif)
Let and
. By means of (2.35) and (2.36), we infer that there exists some
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ49_HTML.gif)
which together with (2.27) and (2.28) yield that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ50_HTML.gif)
The rest of the proof is similar to that of Theorem 2.1 and is omitted. This completes the proof.
3. Remarks and Examples
In this section, we construct five examples to illustrate the superiority and applications of the results presented in this paper.
Remark 3.1.
In case and
for all
, where
and
are constants in
with
, then Theorem 2.1 reduces to a result, which is a generalization of Theorem 1.3. The following example reveals that Theorem 2.1 extends both essentially Theorem 1.3 and is different from Theorems 1.1 and 1.2.
Example 3.2.
Let be endowed with the Euclidean metric
, and let
be defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ51_HTML.gif)
It is easy to see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ52_HTML.gif)
is -orbitally lower semi-continuous in
and
. Define
and
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ53_HTML.gif)
Obviously,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ54_HTML.gif)
For , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ55_HTML.gif)
For , we infer that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ56_HTML.gif)
For , there exists
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ57_HTML.gif)
For , there exists
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ58_HTML.gif)
That is, the conditions of Theorem 2.1 are fulfilled. It follows from Theorem 2.1 that has a fixed point in
. However, we cannot invoke each of Theorems 1.1–1.3 to show that the mapping
has a fixed point in
.
In fact, for any , we consider two possible cases as follows.
Case 1.
Let. Take
and
. Notice that
and
. It is clear that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ59_HTML.gif)
Case 2.
Let. Put
and
. It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ60_HTML.gif)
Set and
. Note that
and
. It is easy to verify that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ61_HTML.gif)
That is, the conditions of Theorem 1.3 do not hold.
Put and
. Clearly
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ62_HTML.gif)
for any with
. That is, the conditions of Theorems 1.1 and 1.2 do not hold.
Remark 3.3.
If and
for all
, then Theorem 2.1 changes into a result, which is an extension of Theorem 1.6. The following example demonstrates that Theorem 2.1 generalizes substantially Theorem 1.6.
Example 3.4.
Let be endowed with the Euclidean metric
. Let
be defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ63_HTML.gif)
It is easy to see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ64_HTML.gif)
is -orbitally lower semi-continuous in
and
. Define
and
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ65_HTML.gif)
Obviously,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ66_HTML.gif)
For , there exists
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ67_HTML.gif)
For , there exists
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ68_HTML.gif)
That is, the conditions of Theorem 2.1 are fulfilled. It follows from Theorem 2.1 that has a fixed point in
. However, Theorem 1.6 is inapplicable in ensuring the existence of fixed points for the mapping
in
because there does not exist
and
satisfying the assumptions of Theorem 1.6. In fact, for any
and
, we consider the following two possible cases.
Case 1.
Let . Put
. Note that
. If
, we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ69_HTML.gif)
If , then we infer that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ70_HTML.gif)
Case 2.
Let . Put
. Note that
. Suppose that
. Let
. It is clear that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ71_HTML.gif)
Take . Obviously,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ72_HTML.gif)
Suppose that . As in the proof of Case 1 stated first, we conclude similarly the conclusion of Case 1 stated after then. Therefore, the assumptions of Theorem 1.6 do not hold.
Remark 3.5.
The following example is an application of Theorem 2.2.
Example 3.6.
Let be endowed with the Euclidean metric
. Let
be defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ73_HTML.gif)
It is easy to see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ74_HTML.gif)
is -orbitally lower semi-continuous in
and
. Define
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ75_HTML.gif)
Clearly,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ76_HTML.gif)
For , we infer that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ77_HTML.gif)
For , we conclude that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ78_HTML.gif)
For , we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ79_HTML.gif)
For , we get that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ80_HTML.gif)
Therefore, all assumptions of Theorem 2.2 are satisfied. It follows from Theorem 2.2 that has a fixed point in
.
Remark 3.7.
If and
for all
, then Theorem 2.3 comes down to a result, which extends Theorem 1.5. The following example shows that Theorem 2.3 is a genuine generalization of Theorem 1.5.
Example 3.8.
Let be endowed with the Euclidean metric
. Define
,
, and
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ81_HTML.gif)
respectively. Clearly, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ82_HTML.gif)
is continuous in . Note that for each
there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ83_HTML.gif)
It is easy to verify that the assumptions of Theorem 2.3 are satisfied. Consequently, Theorem 2.3 guarantees that has a fixed point in
. But
does not satisfy the conditions of Theorem 1.5 because
is not compact for all
.
Remark 3.9.
In case and
for all
, then Theorem 2.3 reduces to a result, which extends Theorem 1.4. The following example reveals that Theorem 2.3 generalizes properly Theorem 1.4.
Example 3.10.
Let and
be as in example 3.1. Clearly,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ84_HTML.gif)
is -orbitally lower semi-continuous in
and
. Define
and
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ85_HTML.gif)
It is easy to verify that is nondecreasing and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ86_HTML.gif)
For , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ87_HTML.gif)
For , we infer that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ88_HTML.gif)
For , there exists
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ89_HTML.gif)
For , there exists
, satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ90_HTML.gif)
That is, the conditions of Theorem 2.3 are fulfilled. It follows from Theorem 2.3 that has a fixed point in
. However, we cannot use Theorem 1.4 to show that the mapping
has a fixed point in
since there does not exist
and
satisfying the assumptions in Theorem 1.4. In fact, for any
and
, we consider two possible cases as follows.
Case 1.
Let. Take
and
. Note that
. It is clear that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ91_HTML.gif)
Case 2.
Let. Put
and
. It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ92_HTML.gif)
Let and
. Note that
. It is easy to verify that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F870980/MediaObjects/13663_2010_Article_1357_Equ93_HTML.gif)
Therefore, the assumptions of Theorem 1.4 are not satisfied.
References
Ćirić L: Fixed point theorems for multi-valued contractions in complete metric spaces. Journal of Mathematical Analysis and Applications 2008,348(1):499–507. 10.1016/j.jmaa.2008.07.062
Ćirić L: Multi-valued nonlinear contraction mappings. Nonlinear Analysis: Theory, Methods & Applications 2009,71(7–8):2716–2723. 10.1016/j.na.2009.01.116
Feng Y, Liu S: Fixed point theorems for multi-valued contractive mappings and multi-valued Caristi type mappings. Journal of Mathematical Analysis and Applications 2006,317(1):103–112. 10.1016/j.jmaa.2005.12.004
Kamran T: Mizoguchi-Takahashi's type fixed point theorem. Computers & Mathematics with Applications 2009,57(3):507–511. 10.1016/j.camwa.2008.10.075
Klim D, Wardowski D: Fixed point theorems for set-valued contractions in complete metric spaces. Journal of Mathematical Analysis and Applications 2007,334(1):132–139. 10.1016/j.jmaa.2006.12.012
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
Nadler SB Jr.: Multi-valued contraction mappings. Pacific Journal of Mathematics 1969, 30: 475–488.
Acknowledgment
This work was supported by the Korea Research Foundation (KRF) Grant funded by the Korea government (MEST) (2009-0073655).
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Liu, Z., Sun, W., Kang, S. et al. On Fixed Point Theorems for Multivalued Contractions. Fixed Point Theory Appl 2010, 870980 (2010). https://doi.org/10.1155/2010/870980
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DOI: https://doi.org/10.1155/2010/870980