- Research Article
- Open access
- Published:
Fixed Point Results for Generalized Contractive Multimaps in Metric Spaces
Fixed Point Theory and Applications volume 2009, Article number: 432130 (2009)
Abstract
The concept of generalized contractive multimaps in the setting of metric spaces is introduced, and the existence of fixed points for such maps is guaranteed under certain conditions. Consequently, our results either generalize or improve a number of fixed point results including the corresponding recent fixed point results of Ciric (2008), Latif-Albar (2008), Klim-Wardowski (2007), and Feng-Liu (2006). Examples are also given.
1. Introduction and Preliminaries
Let be a metric space,
a collection of nonempty subsets of
,
a collection of nonempty closed bounded subsets of
,
a collection of nonempty closed subsets of
a collection of nonempty compact subsets of
and
the Hausdorff metric induced by
Then for any
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ1_HTML.gif)
where
An element is called a fixed point of a multivalued map
if
. We denote
A sequence
in
is called an
of
at
if
for all
.
A map is called lower semicontinuous if for any sequence
with
it implies that
.
Using the concept of Hausdorff metric, Nadler [1] established the following fixed point result for multivalued contraction maps, known as Nadler's contraction principle which in turn is a generalization of the well-known Banach contraction principle.
Theorem 1.1 (see [1]).
Let be a complete metric space and let
be a contraction map. Then
Using the concept of the Hausdorff metric, many authors have generalized Nadler's contraction principle in many directions. But, in fact for most cases the existence part of the results can be proved without using the concept of Hausdorff metric. Recently, Feng and Liu [2] extended Nadler's fixed point theorem without using the concept of Hausdorff metric. They proved the following result.
Theorem 1.2.
Let be a complete metric space and let
be a map such that for any fixed constants
and for each
there is
satisfying the following conditions:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ2_HTML.gif)
Then provided a real-valued function
on
,
is lower semicontinuous.
Recently, Klim and Wardowski [3] generalized Theorem 1.2 and proved the following two results.
Theorem 1.3.
Let be a complete metric space and let
. Assume that the following conditions hold:
(i)there exist a number and a function
such that for each
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ3_HTML.gif)
(ii)for any there is
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ4_HTML.gif)
Then provided a real-valued function
on
,
is lower semicontinuous.
Theorem 1.4.
Let be a complete metric space and let
. Assume that the following conditions hold:
(i)there exists a function such that for each
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ5_HTML.gif)
(ii)for any there is
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ6_HTML.gif)
Then provided a real-valued function
on
,
is lower semicontinuous.
Note that Theorem 1.3 generalizes Nadler's contraction principle and Theorem 1.2. Most recently, Ciric [4] obtained some interesting fixed point results which extend and generalize the cited results. Namely, [4, Theorem  5] generalizes [5, Theorem  5], [4, Theorem  6] generalizes [4, Theorems  1.2,  1.3], and [3, theorem  7] generalizes Theorem 1.4.
In [6], Kada et al. introduced the concept of -distance on a metric space as follows:
A function is called
-
on
if it satisfies the following for each
:
()
()a map is lower semicontinuous; that is, if a sequence
in
with
, then
;
()for any there exists
such that
and
imply
Note that, in general for ,
and not either of the implications
necessarily hold. Clearly, the metric
is a
-distance on
. Let
be a normed space. Then the functions
defined by
and
for all
are
-distances [6]. Many other examples and properties of the
-distance can be found in [6, 7].
The following lemma is crucial for the proofs of our results.
Lemma 1.5 (see [8]).
Let be a closed subset of
and
be a w-distance on
Suppose that there exists
such that
. Then
where
Most recently, the authors of this paper generalized Latif and Albar [9, Theorem  1.3] as follows.
Theorem 1.6 (see [10]).
Let be a complete metric space with a
-distance
. Let
be a multivalued map satisfying that for any constant
and for each
there is
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ7_HTML.gif)
where and
is a function from
to
with
for every
Suppose that a real-valued function
on
defined by
is lower semicontinuous. Then there exists
such that
Further, if
then
.
The aim of this paper is to present some more general results on the existence of fixed points for multivalued maps satisfying certain conditions. Our results unify and generalize the corresponding results of Mizoguchi and Takahashi [5], Klim and Wardowski [3], Latif and Abdou [10], Ciric [4], Feng and Liu [2], Latif and Albar [9] and several others.
2. The Results
First we prove a theorem which is a generalization of Ciric [4, Theorem  5] and due to Klim and Wardowski [3, Theorem  1.4].
Theorem 2.1.
Let be a complete metric space with a
-distance
Let
be a multivalued map. Assume that the following conditions hold:
(i)there exists a function such that for each
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ8_HTML.gif)
(ii)for any there exists
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ9_HTML.gif)
(iii)the map defined by
is lower semicontinuous.
Then there exists such that
Further if
then
Proof.
let be any initial point. Then there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ10_HTML.gif)
From (2.3) we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ11_HTML.gif)
Define a function by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ12_HTML.gif)
Using the facts that for each and
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ13_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ14_HTML.gif)
From (2.4) and (2.5), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ15_HTML.gif)
Similarly, for , there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ16_HTML.gif)
Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ17_HTML.gif)
Continuing this process we can get an orbit of
in
satisfying the following:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ18_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ19_HTML.gif)
for each integer . Since
for each
and from (2.12), we have for all
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ20_HTML.gif)
Thus the sequence of nonnegative real numbers is decreasing and bounded below, thus convergent. Therefore, there is some
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ21_HTML.gif)
From (2.11), as for all
we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ22_HTML.gif)
Thus, we conclude that the sequence of nonnegative reals is bounded. Therefore, there is some
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ23_HTML.gif)
Note that for each
so we have
Now we will show that
Suppose that
Then we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ24_HTML.gif)
Now consider Suppose to the contrary, that
Then
and so from (2.14) and (2.16) there is a positive integer
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ25_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ26_HTML.gif)
Then from (2.19), (2.11) and (2.18), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ27_HTML.gif)
Thus for all
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ28_HTML.gif)
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ29_HTML.gif)
and we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ30_HTML.gif)
Thus for all
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ31_HTML.gif)
Thus, from (2.12) and (2.24), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ32_HTML.gif)
where Clearly
as
From (2.18) and (2.25), we have for any
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ33_HTML.gif)
Since and
there is a positive integer
such that
Now, since
for each
by (2.26) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ34_HTML.gif)
a contradiction. Hence, our assumption is wrong. Thus
Now we will show that
Since
then from (2.16) we can read as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ35_HTML.gif)
so, there exists a subsequence of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ36_HTML.gif)
Now from (2.7) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ37_HTML.gif)
and from (2.12), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ38_HTML.gif)
Taking the limit as and using (2.14), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ39_HTML.gif)
If we suppose that then from last inequality, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ40_HTML.gif)
which contradicts with (2.30). Thus Then from (2.14) and (2.15), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ41_HTML.gif)
and thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ42_HTML.gif)
Now, let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ43_HTML.gif)
Then by (2.7), Let
be such that
Then there is some
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ44_HTML.gif)
Thus it follows from (2.12),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ45_HTML.gif)
By induction we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ46_HTML.gif)
Now, using (2.15) and (2.39), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ47_HTML.gif)
Now, we show that is a Cauchy sequence, for all
we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ48_HTML.gif)
Hence we conclude, as that
is Cauchy sequence. Due to the completeness of
, there exists some
such that
. Since
is lower semicontinuous and from (2.34), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ49_HTML.gif)
and thus, Since
and
is closed, it follows from Lemma 1.5 that
We also have the following interesting result by replacing the hypothesis (iii) of Theorem 2.1 with another natural condition.
Theorem 2.2.
Suppose that all the hypotheses of Theorem 2.1 except (iii) hold. Assume that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ50_HTML.gif)
for every with
Then
Proof.
Following the proof of Theorem 2.1, there exists a Cauchy sequence with
Due to the completeness of
, there exists
such that
Since
is lower semicontinuous and
it follows for all
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ51_HTML.gif)
Assume that Then, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ52_HTML.gif)
which is impossible and hence
Now, we present an improved version of Ciric [4, Theorem  6] and which also generalizes due to Latif and Abdou [10, Theorem  1.6] and due to Klim and Wardowski [3, Theorem  1.3].
Theorem 2.3.
Let be a complete metric space with a
-distance
Let
, be a multivalued map. Assume that the following condition hold:
(i)there exist functions and
with
nondecreasing such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ53_HTML.gif)
(ii)for any there exists
satisfying the following conditions:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ54_HTML.gif)
(iii)the map defined by
is lower semicontinuous.
Then there exists such that
Further if
then
Proof.
Let be an arbitrary, then there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ55_HTML.gif)
From (2.48) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ56_HTML.gif)
Define a function by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ57_HTML.gif)
Since we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ58_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ59_HTML.gif)
Thus from (2.49)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ60_HTML.gif)
Similarly, there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ61_HTML.gif)
Then by definition of we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ62_HTML.gif)
Continuing this process, we get an orbit of
at
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ63_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ64_HTML.gif)
Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ65_HTML.gif)
Since for all
we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ66_HTML.gif)
Thus the sequence of nonnegative real numbers is decreasing and bounded below, thus convergent. Now, we want to show that the sequence
is also decreasing. Suppose to the contrary, that
then as
is nondecreasing, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ67_HTML.gif)
Now using (2.56), (2.57) and (2.60) with , we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ68_HTML.gif)
a contradiction. Thus the sequences is decreasing. Now let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ69_HTML.gif)
Thus by (2.52), Then for any
there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ70_HTML.gif)
So, from (2.58), for all we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ71_HTML.gif)
Thus by induction we get for all
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ72_HTML.gif)
Since from (2.56) and (2.65), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ73_HTML.gif)
for all Note that
Now, we show that
is a Cauchy sequence. For all
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ74_HTML.gif)
Thus we conclude that is a Cauchy sequence. Now, proceeding the proof of Theorem 2.1, we get some
such that
and
Following the proof of Theorem 2.2, we can obtain the following result.
Theorem 2.4.
Suppose that all the hypotheses of Theorem 2.3 except (iii) hold. Assume that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ75_HTML.gif)
for every with
Then
Now, we present a result which is a generalization of Theorem 1.4 due to Klim and Wardowski [3] and Ciric [4, Theorem  7].
Theorem 2.5.
Let be a complete metric space with a
-distance
Let
be a multivalued map. Assume that the following conditions hold:
(i)there exists a function such that for each
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ76_HTML.gif)
(ii)for any there exists
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ77_HTML.gif)
(iii)the map defined by
is lower semicontinuous.
Then there exists such that
Further if
then
Proof.
Let be any initial point. Then from (ii) we can choose
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ78_HTML.gif)
Using the analogous method like in the proof of Lemma  2.1 [10], we obtain the existence of Cauchy sequence such that
and satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ79_HTML.gif)
Consequently, there exists such that
. Since
is lower semicontinuous, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ80_HTML.gif)
thus, Further by closedness of
and since
it follows from Lemma 1.5 that
3. Examples
The following example shows that Theorem 2.1 is a genuine generalization of Ciric [4, Theorem  5].
Example 3.1.
Let with the usual metric
. Define a function
, by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ81_HTML.gif)
Clearly, is a
-distance on
and
. Let
be such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ82_HTML.gif)
Define now as follows
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ83_HTML.gif)
Note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ84_HTML.gif)
and is lower semicontinuous. Moreover for each
we have
Take
then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ85_HTML.gif)
Further, note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ86_HTML.gif)
Hence, for all ,
satisfies all the conditions of Theorem 2.1. Now, if
then we have
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ87_HTML.gif)
Note that for there is
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ88_HTML.gif)
Thus, also satisfies all the conditions of Theorem 2.1 for
Hence it follows from Theorem 2.1 that
Note that
Clearly,
does not satisfy the hypotheses of Ciric [4, Theorem  5] because
is not the metric
.
Finally, we present an example which shows that Theorem 2.5 is a genuine generalization of Theorem 1.4 due to Klim-Wardowski [3].
Example 3.2.
Let with the usual metric
. Define a function
, by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ89_HTML.gif)
Then is a
-distance on
Note that
. Now, for any real number
define
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ90_HTML.gif)
and define a constant function by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ91_HTML.gif)
Note that for all
. And for each
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ92_HTML.gif)
Thus, is continuous. Now for each
there exists
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F432130/MediaObjects/13663_2009_Article_1141_Equ93_HTML.gif)
Therefore, all assumptions of Theorem 2.5 are satisfied and Note that
is not compact for all
and the
-distance
is not a metric
so
do not satisfy the hypotheses of Theorem 1.4.
References
Nadler SB Jr.: Multi-valued contraction mappings. Pacific Journal of Mathematics 1969, 30: 475–488.
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
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
Ciric 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
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
Kada O, Suzuki T, Takahashi W: Nonconvex minimization theorems and fixed point theorems in complete metric spaces. Mathematica Japonica 1996,44(2):381–391.
Takahashi W: Nonlinear Functional Analysis: Fixed Point Theory and Its Application. Yokohama Publishers, Yokohama, Japan; 2000:iv+276.
Lin L-J, Du W-S: Some equivalent formulations of the generalized Ekeland's variational principle and their applications. Nonlinear Analysis: Theory, Methods & Applications 2007,67(1):187–199. 10.1016/j.na.2006.05.006
Latif A, Albar WA: Fixed point results in complete metric spaces. Demonstratio Mathematica 2008,41(1):145–150.
Latif A, Abdou AAN: Fixed points of generalized contractive maps. Fixed Point Theory and Applications 2009, Article ID 487161, 2009:-9.
Acknowledgment
The authors thank the referees for their valuable comments.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Latif, A., Abdou, A.A.N. Fixed Point Results for Generalized Contractive Multimaps in Metric Spaces. Fixed Point Theory Appl 2009, 432130 (2009). https://doi.org/10.1155/2009/432130
Received:
Accepted:
Published:
DOI: https://doi.org/10.1155/2009/432130