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Some New Weakly Contractive Type Multimaps and Fixed Point Results in Metric Spaces
Fixed Point Theory and Applications volume 2009, Article number: 412898 (2010)
Abstract
Some new weakly contractive type multimaps in the setting of metric spaces are introduced, and we prove some results on the existence of fixed points for such maps under certain conditions. Our results extend and improve several known results including the corresponding recent fixed point results of Pathak and Shahzad (2009), Latif and Abdou (2009), Latif and Albar (2008), Cirić (2008), Feng and Liu (2006), and Klim and Wardowski (2007).
1. Introduction
Let be a metric space. Let
denote a collection of nonempty subsets of
,
a collection of nonempty closed subsets of
and
a collection of nonempty closed bounded subsets of
Let
be the Hausdorff metric with respect to
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F412898/MediaObjects/13663_2009_Article_1140_Equ1_HTML.gif)
for every where
An element is called a fixed point of a multivalued map (multimap)
if
. We denote
A sequence in
is called an orbit of
at
if
for all
. A map
is called lower semicontinuous if for any sequence
such that
we have
.
Using the concept of Hausdorff metric, Nadler [1] established the following multivalued version of the Banach contraction principle.
Theorem 1.1.
Let be a complete metric space and let
be a map such that for a fixed constant
and for each
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F412898/MediaObjects/13663_2009_Article_1140_Equ2_HTML.gif)
Then
This result has been generalized in many directions. For instance, Mizoguchi and Takahashi [2] have obtained the following general form of the Nadler's theorem.
Theorem 1.2.
Let be a complete metric space and let
. Assume that there exists a function
such that for every
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F412898/MediaObjects/13663_2009_Article_1140_Equ3_HTML.gif)
and for all
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F412898/MediaObjects/13663_2009_Article_1140_Equ4_HTML.gif)
Then
Many authors have been using the Hausdorff metric to obtain fixed point results for multivalued maps. 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 [3] extended Nadler's fixed point theorem without using the concept of the Hausdorff metric. They proved the following result.
Theorem 1.3.
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%2F412898/MediaObjects/13663_2009_Article_1140_Equ5_HTML.gif)
Then provided that a real-valued function
on
,  
is lower semicontinuous.
Recently, Klim and Wardowski [4] generalized Theorem 1.3 as follows.
Theorem 1.4.
Let be a complete metric space and let
. Assume that the following conditions hold:
(I)if there exist a number and a function
such that for each
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F412898/MediaObjects/13663_2009_Article_1140_Equ6_HTML.gif)
(II)for any there is
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F412898/MediaObjects/13663_2009_Article_1140_Equ7_HTML.gif)
Then provided that a real-valued function
on
,  
is lower semicontinuous.
The above results have been generalized in many directions; see for instance [5–9] and references therein.
In [10], 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
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F412898/MediaObjects/13663_2009_Article_1140_IEq64_HTML.gif)
a map is lower semicontinuous; that is, if there is a sequence
in
with
, then
;
for any there exists
such that
and
imply
Note that, in general for ,  
and neither 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 [10]. Many other examples and properties of the
-distance can be found in [10, 11].
The following lemmas concerning -distance are crucial for the proofs of our results.
Lemma 1.5 (see [10]).
Let and
be sequences in
and let
and
be sequences in
converging to
Then, for the
-distance
on
the following conditions hold for every
:
(a)if and
for any
then
in particular, if
and
then
;
(b)if and
for any
then
converges to
;
(c)if for any
with
then
is a Cauchy sequence;
(d)if for any
then
is a Cauchy sequence.
Lemma 1.6 (see [12]).
Let be a closed subset of
and
be a
-distance on
Suppose that there exists
such that
. Then
where
Using the concept of -distance, the authors of this paper most recently extended and generalized Theorem 1.4 and [8, Theorem
] as follows.
Theorem 1.7 (see [13]).
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%2F412898/MediaObjects/13663_2009_Article_1140_Equ8_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
.
Let Let
satisfy that
(i) and
for each
(ii) is nondecreasing on
(iii) is subadditive; that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F412898/MediaObjects/13663_2009_Article_1140_Equ9_HTML.gif)
We define
Remark 1.8.
-
(a)
It follows from (ii) property of
that for each
(1.10)
-
(b)
If
and
is continuous at
, then due to the following two facts
must be continuous at each point of
. First, every sub-additive and continuous function
at 0 such that
is right upper and left lower semicontinuous [14]. Second, each nondecreasing function is left upper and right lower semicontinuous.
-
(c)
For any
and for each sequence
in
satisfying
we have
For a metric space , we denote
In the sequel, we consider
if
and
if
Assuming that the function is continuous and satisfies the conditions (i) and (ii) above, Zhang [15] proved some fixed point results for single-valued maps which satisfy some contractive type condition involving such function
. Recently, using
Pathak and Shahzad [9] generalized Theorem 1.4.
In this paper, we prove some results on the existence of fixed points for contractive type multimaps involving the function where
and the function
is a
-distance on a metric space
Our results either generalize or improve several known fixed point results in the setting of metric spaces, (see Remarks 2.3 and 2.6).
2. The Results
Theorem 2.1.
Let be a complete metric space with a
-distance
Let
be a multimap. 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%2F412898/MediaObjects/13663_2009_Article_1140_Equ11_HTML.gif)
(II)there exists a function such that for any
there exists
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F412898/MediaObjects/13663_2009_Article_1140_Equ12_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%2F412898/MediaObjects/13663_2009_Article_1140_Equ13_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F412898/MediaObjects/13663_2009_Article_1140_Equ14_HTML.gif)
From (2.3) and (2.4), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F412898/MediaObjects/13663_2009_Article_1140_Equ15_HTML.gif)
Similarly, there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F412898/MediaObjects/13663_2009_Article_1140_Equ16_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F412898/MediaObjects/13663_2009_Article_1140_Equ17_HTML.gif)
From (2.6) and (2.7), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F412898/MediaObjects/13663_2009_Article_1140_Equ18_HTML.gif)
From (2.4) and (2.6), it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F412898/MediaObjects/13663_2009_Article_1140_Equ19_HTML.gif)
Continuing this process, we get an orbit of
at
satisfying the following:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F412898/MediaObjects/13663_2009_Article_1140_Equ20_HTML.gif)
From (2.10), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F412898/MediaObjects/13663_2009_Article_1140_Equ21_HTML.gif)
Note that for each ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F412898/MediaObjects/13663_2009_Article_1140_Equ22_HTML.gif)
Thus the sequences of non-negative real numbers and
are decreasing. Now, since
is nondecreasing, it follows that
and
are decreasing sequences and are bounded from below, thus convergent. Now, by the definition of the function
there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F412898/MediaObjects/13663_2009_Article_1140_Equ23_HTML.gif)
Thus, for any there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F412898/MediaObjects/13663_2009_Article_1140_Equ24_HTML.gif)
and thus for all we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F412898/MediaObjects/13663_2009_Article_1140_Equ25_HTML.gif)
Also, it follows from (2.11) that for all
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F412898/MediaObjects/13663_2009_Article_1140_Equ26_HTML.gif)
where Note that for all
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F412898/MediaObjects/13663_2009_Article_1140_Equ27_HTML.gif)
Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F412898/MediaObjects/13663_2009_Article_1140_Equ28_HTML.gif)
where Now, since
we have
and we get the decreasing sequence
converging to
. Thus we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F412898/MediaObjects/13663_2009_Article_1140_Equ29_HTML.gif)
Note that for all
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F412898/MediaObjects/13663_2009_Article_1140_Equ30_HTML.gif)
where Now, for any
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F412898/MediaObjects/13663_2009_Article_1140_Equ31_HTML.gif)
Clearly, and thus we get that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F412898/MediaObjects/13663_2009_Article_1140_Equ32_HTML.gif)
that is, is Cauchy sequence in
. Due to the completeness of
, there exists some
such that
. Due to the fact that the function
is lower semicontinuous and (2.19), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F412898/MediaObjects/13663_2009_Article_1140_Equ33_HTML.gif)
thus, Since
and
is closed, it follows from Lemma 1.6 that
If we consider a constant map in Theorem 2.1, then we obtain the following result.
Corollary 2.2.
Let be a complete metric space with a
-distance
. Let
be a multimap satisfying that for any constants
and for each
there is
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F412898/MediaObjects/13663_2009_Article_1140_Equ34_HTML.gif)
where Suppose that a real-valued function
on
defined by
is lower semicontinuous. Then there exists
such that
Further, if
then
.
Remark 2.3.
-
(a)
Theorem 2.1 extends and generalizes Theorem 1.7. Indeed, if we consider
for each
in Theorem 2.1, then we can get Theorem 1.7 due to Latif and Abdou [13, Theorem
].
-
(b)
Theorem 2.1 contains Theorem
of Pathak and Shahzad [9] as a special case.
-
(c)
Corollary 2.2 extends and generalizes Theorem
of Latif and Albar [8].
We have also the following fixed point result which generalizes [13, Theorem ].
Theorem 2.4.
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%2F412898/MediaObjects/13663_2009_Article_1140_Equ35_HTML.gif)
for every with
and the function
is continuous at
. Then
Proof.
Following the proof of Theorem 2.1, there exists a Cauchy sequence and
such that
and
Since
is lower semicontinuous, it follows, from the proof of Theorem 2.1 that for all
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F412898/MediaObjects/13663_2009_Article_1140_Equ36_HTML.gif)
where Since
for all
and the function
is nondecreasing, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F412898/MediaObjects/13663_2009_Article_1140_Equ37_HTML.gif)
and thus by using (2.20), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F412898/MediaObjects/13663_2009_Article_1140_Equ38_HTML.gif)
Assume that Then, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F412898/MediaObjects/13663_2009_Article_1140_Equ39_HTML.gif)
which is impossible and hence
Theorem 2.5.
Let be a complete metric space with a
-distance
Let
be a multimap. 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%2F412898/MediaObjects/13663_2009_Article_1140_Equ40_HTML.gif)
(II)there exists a function such that for any
there exists
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F412898/MediaObjects/13663_2009_Article_1140_Equ41_HTML.gif)
(III)the map defined by
is lower semicontinuous.
Then there exists such that
Further if
then
Proof.
Let be any initial point. Following the same method as in the proof of Theorem 2.1, we obtain the existence of a Cauchy sequence
such that
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F412898/MediaObjects/13663_2009_Article_1140_Equ42_HTML.gif)
Consequently, there existssuch that
. Since
is lower semicontinuous, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F412898/MediaObjects/13663_2009_Article_1140_Equ43_HTML.gif)
thus, Further by the closedness of
and since
it follows from Lemma 1.6 that
Remark 2.6.
Theorem 2.5 extends and generalizes fixed point results of Klim and Wardowski [4, Theorem ], Cirić [5, Theorem
], and improves fixed point result of Pathak and Shahzad [9, Theorem
].
Following the same method as in the proof of Theorem 2.4, we can obtain the following fixed point result.
Theorem 2.7.
Suppose that all the hypotheses of Theorem 2.5 except (III) hold. Assume that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F412898/MediaObjects/13663_2009_Article_1140_Equ44_HTML.gif)
for every with
and the function
is continuous at
then
Now we present an example which satisfies all the conditions of the main results, namely, Theorems 2.1 and 2.5 and thus the set of fixed points of is nonempty.
Example 2.8.
Let with the usual metric
Define a
-distance function
, by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F412898/MediaObjects/13663_2009_Article_1140_Equ45_HTML.gif)
Let be defined as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F412898/MediaObjects/13663_2009_Article_1140_Equ46_HTML.gif)
Note that Let
Define a function
by
Clearly,
Define
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F412898/MediaObjects/13663_2009_Article_1140_Equ47_HTML.gif)
Note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F412898/MediaObjects/13663_2009_Article_1140_Equ48_HTML.gif)
Clearly, is lower semicontinuous. Note that for each
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F412898/MediaObjects/13663_2009_Article_1140_Equ49_HTML.gif)
Thus, for ,
satisfies all the conditions of Theorem 2.1. Now, let
then we have
Clearly, there exists
such that
Now
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F412898/MediaObjects/13663_2009_Article_1140_Equ50_HTML.gif)
Thus, all the hypotheses of Theorem 2.1 are satisfied and clearly we have Now, if we consider
, then all the hypotheses of Theorem 2.5 are also satisfied. Note that in the above example the
-distance
is not a metric
.
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Latif, A., Abdou, A.A.N. Some New Weakly Contractive Type Multimaps and Fixed Point Results in Metric Spaces. Fixed Point Theory Appl 2009, 412898 (2010). https://doi.org/10.1155/2009/412898
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DOI: https://doi.org/10.1155/2009/412898