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Fixed Points of Multivalued Maps in Modular Function Spaces
Fixed Point Theory and Applications volume 2009, Article number: 786357 (2009)
Abstract
The purpose of this paper is to study the existence of fixed points for contractive-type and nonexpansive-type multivalued maps in the setting of modular function spaces. We also discuss the concept of -modular function and prove fixed point results for weakly-modular contractive maps in modular function spaces. These results extend several similar results proved in metric and Banach spaces settings.
1. Introduction and Preliminaries
The well-known Banach fixed point theorem on complete metric spaces (specifically, each contraction self-map of a complete metric space has a unique fixed point) has been extended and generalized in different directions. For example, see Edelstein [1, 2], Kasahara [3], Rhoades [4], Siddiq and Ansari [5], and others. One of its generalizations is for nonexpansive single-valued maps on certain subsets of a Banach space. Indeed, these fixed points are not necessarily unique. See, for example, Browder [6–8] and Kirk [9]. Fixed point theorems for contractive and nonexpansive multivalued maps have also been established by several authors. Let denote the Hausdorff metric on the space of all bounded nonempty subsets of a metric space
. A multivalued map
(where
denotes the collection of all nonempty subsets of
) with bounded subsets as values is called contractive [10] if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F786357/MediaObjects/13663_2009_Article_1176_Equ1_HTML.gif)
for all and for a fixed number
. If the Lipschitz constant
, then
is called a multivalued nonexpansive mapping [11]. Nadler [10], Markin [11], Lami-Dozo [12], and others proved fixed point theorems for these maps under certain conditions in the setting of metric and Banach spaces. Note that an element
is called a fixed point of a multivalued map
if
. Among others, without using the concept of the Hausdorff metric, Husain and Tarafdar [13] introduced the notion of a nonexpansive-type multivalued map and proved a fixed point theorem on compact intervals of the real line. Using such type of notions Husain and Latif [14] extended their result to general Banach space setting.
The fixed point results in modular function spaces were given by Khamsi et al. [15]. Even though a metric is not defined, many problems in metric fixed point theory can be reformulated in modular spaces. For instance, fixed point theorems are proved in [15, 16] for nonexpansive maps.
In this paper, we define nonexpansive-type and contractive-type multivalued maps in modular function spaces, investigate the existence of fixed points of such mappings, and prove similar results found in [17].
Now, we recall some basic notions and facts about modular spaces as formulated by Kozlowski [18]. For more details the reader may consult [15, 16].
Let be a nonempty set and let
be a nontrivial
-algebra of subsets of
. Let
be a
-ring of subsets of
, such that
for any
and
Let us assume that there exists an increasing sequence of sets such that
. By
we denote the linear space of all simple functions with supports from
. By
we will denote the space of all measurable functions, that is, all functions
such that there exists a sequence
,
and
for all
. By
we denote the characteristic function of the set
.
Definition 1.1.
A functional is called a function modular if
() for any
,
() whenever
for any
,
and
,
() is a
-subadditive measure for every
,
() as
decreases to
for every
, where
,
()if there exists such that
, then
for every
, and
()for any is order continuous on
, that is,
if
and decreases to
.
The definition of is then extended to
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F786357/MediaObjects/13663_2009_Article_1176_Equ2_HTML.gif)
For the sake of simplicity we write instead of
.
Definition 1.2.
A set is said to be
-null if
for every
A property
is said to hold
-almost everywhere (
-a.e.) if the set
is
-null.
Definition 1.3.
A modular function is called
-finite if there exists an increasing sequence of sets
such that
and
It is easy to see that the functional
is a modular and satisfies the following properties:
(i) if and only if
-a.e.,
(ii) for every scalar
with
and
, and
(iii) if
,
and
.
In addition, if the following property is satisfied,
(iii)' if
,
and,
,
we say that is a convex modular.
The modular defines a corresponding modular space, that is, the vector space
given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F786357/MediaObjects/13663_2009_Article_1176_Equ3_HTML.gif)
When is convex, the formula
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F786357/MediaObjects/13663_2009_Article_1176_Equ4_HTML.gif)
defines a norm in the modular space which is frequently called the Luxemburg norm. We can also consider the space
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F786357/MediaObjects/13663_2009_Article_1176_Equ5_HTML.gif)
Definition 1.4.
A function modular is said to satisfy the -condition if
as
whenever
decreases to
and
as
We know from [18] that when
satisfies the
-condition.
Definition 1.5.
A function modular is said to satisfy the -type condition if there exists
such that for any
we have
In general, -type condition and
-condition are not equivalent, even though it is obvious that
-type condition implies
-condition on the modular space
Definition 1.6.
Let be a modular space.
(1)The sequence is said to be
-convergent to
if
as
.
(2)The sequence is said to be
-a.e. convergent to
if the set
is
-null.
(3)The sequence is said to be
-Cauchy if
as
and
go to
.
(4)A subset of
is called
-closed if the
-limit of a
-convergent sequence of
always belongs to
.
(5)A subset of
is called
-a.e. closed if the
-a.e. limit of a
-a.e. convergent sequence of
always belongs to
.
(6)A subset of
is called
-a.e. compact if every sequence in
has a
-a.e. convergent subsequence in
.
(7)A subset of
is called
-bounded if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F786357/MediaObjects/13663_2009_Article_1176_Equ6_HTML.gif)
We recall two basic results (see [15]) in the theory of modular spaces.
(i)If there exists a number such that
then there exists a subsequence
of
such that
-a.e.
(ii)(Lebesgue's Theorem) If ,
-a.e. and there exists a function
such that
-a.e. for all
then
We know, by [15, 16] that under -condition the norm convergence and modular convergence are equivalent, which implies that the norm and modular convergence are also the same when we deal with the
-type condition. In the sequel we will assume that the modular function
is convex and satisfies the
-type condition.
Definition 1.7.
Let be as aforementioned. We define a growth function
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F786357/MediaObjects/13663_2009_Article_1176_Equ7_HTML.gif)
We have the following:
Lemma 1.8 (see [19]).
Let be as aforementioned. Then the growth function
has the following properties:
(1) ,
,
(2) is a convex, strictly increasing function. So, it is continuous,
(3),
(4);
where
is the function inverse of
.
The following lemma shows that the growth function can be used to give an upper bound for the norm of a function.
Lemma 1.9 (see [19]).
Let be a convex function modular satisfying the
-type condition. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F786357/MediaObjects/13663_2009_Article_1176_Equ8_HTML.gif)
The next lemma will be of major interest throughout this work.
Lemma 1.10 (see [16]).
Let be a function modular satisfying the
-condition and let
be a sequence in
such that
, and there exists
such that
. Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F786357/MediaObjects/13663_2009_Article_1176_Equ9_HTML.gif)
Moreover, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F786357/MediaObjects/13663_2009_Article_1176_Equ10_HTML.gif)
2. Fixed Points of Contractive-Type and Nonexpansive-Type Maps
In the sequel we assume that is a convex,
-finite modular function satisfying the
-type condition, and
is a nonempty
-bounded subset of the modular function space
. We denote that
is a collection of all nonempty
-closed subsets of
, and
is a collection of all nonempty
-compact subsets of
.
We say that a multivalued map is
-contractive-type if there exists
such that for any
and for any
, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F786357/MediaObjects/13663_2009_Article_1176_Equ11_HTML.gif)
and -nonexpansive-type if for any
and for any
, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F786357/MediaObjects/13663_2009_Article_1176_Equ12_HTML.gif)
We have the following fixed point theorem (for which a similar result may be found in [17]).
Theorem 2.1.
Let be a nonempty
-closed subset of the modular function space
. Then any
-contractive-type map has a fixed point, that is, there exists
such that
.
Proof.
Let . Without loss of generality, assume that
is not a fixed point of
. Then there exists
such that
. Hence
. Since
is
-contractive-type, then there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F786357/MediaObjects/13663_2009_Article_1176_Equ13_HTML.gif)
By induction, one can easily construct a sequence such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F786357/MediaObjects/13663_2009_Article_1176_Equ14_HTML.gif)
for any . In particular we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F786357/MediaObjects/13663_2009_Article_1176_Equ15_HTML.gif)
Without loss of generality, we may assume , otherwise
is a fixed point of
. Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F786357/MediaObjects/13663_2009_Article_1176_Equ16_HTML.gif)
Using Lemma 1.9, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F786357/MediaObjects/13663_2009_Article_1176_Equ17_HTML.gif)
Using the properties of , we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F786357/MediaObjects/13663_2009_Article_1176_Equ18_HTML.gif)
So
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F786357/MediaObjects/13663_2009_Article_1176_Equ19_HTML.gif)
which implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F786357/MediaObjects/13663_2009_Article_1176_Equ20_HTML.gif)
Since and
, then
. This forces
to be
-Cauchy. Hence the sequence
-converges to some
. Since
satisfies the
-condition, then
-converges to
. Since
is
-closed, then
. Let us prove that
is indeed a fixed point of
. Since
is a
-contractive-type mapping, then for any
, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F786357/MediaObjects/13663_2009_Article_1176_Equ21_HTML.gif)
Hence converges to 0. Since
satisfies the
-condition, we have
converges to 0. Since
-converges to
, then
-converges to
. Hence
-converges to
. Since
is
-closed and
, we get
.
Remark 2.2.
Consider the multivalued map , where
is a nonempty
-closed subset of
. Then it is easy to show that
is a
-contractive-type map. The set of all fixed point of
is exactly the set
. In particular,
-contractive-type maps may not have a unique fixed point.
As an application of the above theorem, we have the following result.
Proposition 2.3.
Let be a
-closed convex subset of the modular function space
. Let
be
-nonexpansive-type map. Then there exists an approximate fixed points sequence
in
, that is, for any
there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F786357/MediaObjects/13663_2009_Article_1176_Equ22_HTML.gif)
In particular one has , where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F786357/MediaObjects/13663_2009_Article_1176_Equ23_HTML.gif)
Proof.
Let and let
be a fixed point in
. For each
, define a map
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F786357/MediaObjects/13663_2009_Article_1176_Equ24_HTML.gif)
Note that is nonempty and
-closed subset of
because
is
-closed and
is convex. Since
is a
-nonexpansive-type map, for each
and for any
, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F786357/MediaObjects/13663_2009_Article_1176_Equ25_HTML.gif)
Since is convex we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F786357/MediaObjects/13663_2009_Article_1176_Equ26_HTML.gif)
which implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F786357/MediaObjects/13663_2009_Article_1176_Equ27_HTML.gif)
In other words, the map is a
-contractive-type. Theorem 2.1 implies the existence of a fixed point
of
, thus there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F786357/MediaObjects/13663_2009_Article_1176_Equ28_HTML.gif)
In particular, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F786357/MediaObjects/13663_2009_Article_1176_Equ29_HTML.gif)
where is the
-diameter of
. Note that since
is
-bounded, then
. If we choose
, for
and write
and
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F786357/MediaObjects/13663_2009_Article_1176_Equ30_HTML.gif)
for any , which implies
.
Using the above result, we are now ready to prove the main fixed point result for -nonexpansive-type multivalued maps.
Theorem 2.4.
Let be a nonempty
-closed convex subset of the modular function space
. Assume that
is
-a.e. compact. Then each
-nonexpansive-type map
has a fixed point.
Proof.
Proposition 2.3 ensures the existence of a sequence in
and a sequence
such that
and
. Without loss of generality we may assume that
-a.e. converges to
and
-a.e. converges to
. Lemma 1.10 implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F786357/MediaObjects/13663_2009_Article_1176_Equ31_HTML.gif)
Hence . Since
is a
-nonexpansive-type map, then there exists a sequence
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F786357/MediaObjects/13663_2009_Article_1176_Equ32_HTML.gif)
for all . Since
is
-compact, we may assume that
is
-convergent to some
. Lemma 1.10 implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F786357/MediaObjects/13663_2009_Article_1176_Equ33_HTML.gif)
Since satisfies the
-condition, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F786357/MediaObjects/13663_2009_Article_1176_Equ34_HTML.gif)
(see, [20]). Since , we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F786357/MediaObjects/13663_2009_Article_1176_Equ35_HTML.gif)
which implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F786357/MediaObjects/13663_2009_Article_1176_Equ36_HTML.gif)
Hence or
. Hence
; that is,
is a fixed point of
.
Proposition 2.3 and Theorem 2.4 are also hold if we assume that is starshaped instead of Convex. (A set
is called starshaped if there exists
such that
provided
and
)
3. Fixed Points of
-Contractive-Type Maps
In [21] the authors introduced the concept of -distance in metric spaces which they connected to the existence of fixed point of single and multivalued maps (see also [22]). Similarly we extend their definition and results to modular spaces. Indeed let
be a convex,
-finite modular function. A function
is called
-modular on the modular function space
if the following are satisfied:
(1) for any
;
(2)for any ,
is lower semicontinuous; that is, if
-converges to
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F786357/MediaObjects/13663_2009_Article_1176_Equ37_HTML.gif)
(3)for any , there exists
such that
and
imply
.
As it was done in [21], we need the following technical lemma.
Lemma 3.1.
Let be
-modular on the modular function space
. Let
and
be sequences in
, and let
and
be sequences in
converging to 0, and
. Then the following hold:
(1)if and
, for all
, then
; in particular if
and
, then
;
(2)if and
, for any
, then
-converges to
;
(3)if for any
with
, then
is a
-Cauchy sequence;
(4)if for any
, then
is a
-Cauchy sequence.
The proof is easy and similar to the one given in [21]. Now we are ready to give the first fixed point result in this setting. Let be a nonempty
-closed subset of the modular function space
. We say that a multivalued map
is weakly
-contractive-type map if there exists
-modular
on
and
such that for any
and any
, there exists
such that
.
Theorem 3.2.
Let be a nonempty
-closed subset of the modular function space
. Then each weakly
-contractive-type map
has a fixed point
, and
.
Proof.
Let be a
-modular and
associated to
, that is, for any
and any
, there exists
such that
. Fix
and
. By induction one can construct a sequence
such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F786357/MediaObjects/13663_2009_Article_1176_Equ38_HTML.gif)
for every . In particular we have
, for every
. Using the properties of
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F786357/MediaObjects/13663_2009_Article_1176_Equ39_HTML.gif)
for any . Lemma 3.1 implies that the sequence
is
-Cauchy. Hence
-converges to some
. Using the lower semicontinuity of
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F786357/MediaObjects/13663_2009_Article_1176_Equ40_HTML.gif)
for any . Since
and
is weakly
-contractive-type map, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F786357/MediaObjects/13663_2009_Article_1176_Equ41_HTML.gif)
for any . Lemma 3.1 implies that
- converges to
as well. Since
is
-closed, then
, that is,
is a fixed point of
. Let us complete the proof by showing that
. Since
, there exists
such that
. By induction we can construct a sequence
in
such that
and
, for any
. So we have
, for any
. Lemma 3.1 implies that
is
-Cauchy. Hence
- converges to some
. Using the lower semicontinuity of
we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F786357/MediaObjects/13663_2009_Article_1176_Equ42_HTML.gif)
Hence . Then for any
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F786357/MediaObjects/13663_2009_Article_1176_Equ43_HTML.gif)
Lemma 3.1 implies , or
.
Note that in the proof above we did not use the -condition. The reason behind is that
satisfies the triangle inequality. If
is single valued, then we have little more information about the fixed point. Indeed, let
be a nonempty
-closed subset of the modular function space
. The map
is called a weakly
-contractive type map if there exists
-modular
on
and
such that for any
.
Theorem 3.3.
Let be a nonempty
-closed subset of the modular function space
. Then each weakly
-contractive type map
has a unique fixed point
, and
.
Proof.
Theorem 3.2 ensures the existence of a fixed point , that is,
and
. Let us show that
is the only fixed point of
. Assume that
is another fixed point of
. Then we must have
. Combining this with
, Lemma 3.1 implies
.
Similar extensions of the results as found in [21–23] may be proved in our setting.
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Acknowledgments
The authors thank the referees for their valuable comments and suggestions. The authors would also like to thank Professor M.A. Khamsi for productive discussion and cooperation regarding this work.
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Kutbi, M.A., Latif, A. Fixed Points of Multivalued Maps in Modular Function Spaces. Fixed Point Theory Appl 2009, 786357 (2009). https://doi.org/10.1155/2009/786357
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DOI: https://doi.org/10.1155/2009/786357