1. Introduction
Variable exponent spaces first appeared in a work of Orlicz in 1931 [
1] (see also [
2]), where he defined the following space:
They became very important because of their use in the mathematical modeling of non-Newtonian fluids [
3,
4]. The typical example of such fluids are electrorheological fluids, the viscosity of which exhibits dramatic and sudden changes when exposed to an electric or magnetic field. The necessity of a clear understanding of the spaces with variable integrability is reinforced by their potential applications.
The properties of this vector space have been extensively studied in [
5,
6,
7]. The norm that was commonly used to investigate the geometrical properties of
X is the Minkowski functional associated to the modular unit ball and it is known as the Luxembourg norm. Whereas in the case of classical
spaces, the natural norm is suitable for making calculations, the Luxembourg norm on
X is very difficult to manipulate.
In 1950, Nakano [
8] introduced for the first time the notion of modular vector space (see also [
9,
10]). This abstract point of view has been crucial to the development of the research on geometrical and topological properties of the variable exponent spaces
.
In this work, we will introduce a class of subsets of
that have some interesting geometrical properties. This will allow us to prove a new fixed-point theorem concerning
spaces. For the study of metric fixed-point theory, we recommend the book [
9].
2. Basic Notations and Terminology
For a function
, define the vector space
Nakano [
8,
11] introduced the concept of modular vector space.
Proposition 1 ([
6,
9]).
Consider the function defined bythen ρ satisfies the following properties
- (1)
if and only if ,
- (2)
, if ,
- (3)
.
for any . The function ρ is called a convex modular.
For any subset
I of
, we consider the functional
If , we set . We define on modular spaces a modular topology which is similar to the topology induced by a metric.
Definition 1. Consider the vector space .
- (a)
We say that a sequence is ρ-convergent to if and only if . The ρ-limit is unique if it exists.
- (b)
A sequence is called ρ-Cauchy if as .
- (c)
A nonempty subset is called ρ-closed if for any sequence which ρ-converges to x implies that .
- (d)
A nonempty subset is called ρ-bounded if and only if
Note that
satisfies the Fatou property, i.e.,
holds whenever
-converges to
y, for any
. Throughout, we will use the notation
to denote the
-ball with radius
centered at
and defined as
Note that Fatou property holds if and only if the -balls are -closed. That is, all -balls are -closed in .
Definition 2. Let be a nonempty subset. A mapping is called ρ-Lipschitzian if there exists a constant such thatIf , T is called ρ-nonexpansive. A point is called a fixed point of T if . The concept of modular uniform convexity was first introduced by Nakano [
11], but a weaker definition of modular uniform convexity called
was introduced in [
9] and seems to be more suitable to hold in
when weaker assumptions on the exponent function
hold. The following definition is given in terms of subsets because of the subsequent results discovered in this work.
Definition 3 ([
9])
. Consider the vector space . Let C be a nonempty subset of .- 1.
Let and . Define If , let If , we set . We say that ρ satisfies on C if for every and , we have . When , we remark that for every , , for small enough. In this case, we will use the notation .
- 2.
We say that ρ satisfies on C if for every and , there exists depending on s and ε such that - 3.
We say that ρ is strictly convex on C (in short ), if for every such that
In the study of the properties of
(see [
12]), the following values are very important:
In [
5], the authors proved that for
, with
, the modular is
. This modular geometrical property allows to prove the following fixed-point result:
Theorem 1. Consider the vector space . Assume . Let C be a nonempty ρ-closed convex ρ-bounded subset of . Let be a ρ-nonexpansive mapping. Then T has a fixed point.
In [
13], the authors proved a similar fixed-point theorem in the case where
has at most one element which is an improvement from
.
Before we close this section, we recall the following lemma, of a rather technical nature, which plays a crucial role when dealing with spaces.
Lemma 1. The following inequalities hold:
- (i)
for any .
- (ii)
[15]. If , then for any such that .
In this work, using a different approach, we obtain some fixed-point results when without the known conditions on the function .
3. Uniform Decrease Condition
First, we introduce an interesting class of subsets of
, which will play an important part in our work. In particular, they enjoy similar modular geometric properties as
when
. Before, let us introduce the following notations:
where
.
Definition 4. Consider the vector space . A nonempty subset C of is said to satisfy the uniform decrease condition (in short ) if for any , there exists such that Obviously the condition
passes from a set to its subsets. Moreover, if
is identically equal to 1, then the only
subset is
. Since this case is not interesting, we will assume throughout that
is not identically equal to 1. Moreover, if
, then any nonempty subset of
satisfies the condition
. Indeed, let
C be a nonempty subset of
and
. Let
. Then
which implies
Therefore, the condition is interesting to study only when and is not identically equal to 1, which will be the case throughout.
Example 1. Consider the function defined byConsider the subsetC is nonempty, convex and ρ-closed. Let us show that it satisfies the condition . Indeed, fix . Let be such that . Set . We have for all , which proves our claim that C is .
Before we give a characterization of subsets which satisfy the condition , we need to introduce a new class of subsets of .
Definition 5. Consider the vector space such that and not identically equal to 1. Let be a nondecreasing function. Define the set to be Note that is never empty since . Some of the basic properties of are given in the following lemma.
Lemma 2. Consider the vector space such that and not identically equal to 1. Let be a non-decreasing function. Then the following properties hold:
- 1.
is convex.
- 2.
is symmetrical, i.e., whenever .
- 3.
The Fatou property implies easily that is ρ-closed as a subset of which in turn implies that is ρ-complete.
Proposition 2. Consider the vector space such that and not identically equal to 1. A subset C of satisfies the condition if and only if there exists non-decreasing such that .
Proof. First, we prove that
satisfies the condition
. Fix
. If we take
, we obtain
which proves our claim. Clearly, any subset
C of
will also satisfy the condition
. Conversely, let
C be a nonempty subset of
which satisfies the condition
. For any
, there exists
such that
. Set
Clearly,
f is well defined and
, for all
. Let
and
be such that
. We claim that
. Indeed, it is easy to see that
. If
, then we have
which easily implies
. Otherwise, assume
. Let
. We have
and
. By definition of the sets
J, we have
. Since
, for all
, we obtain
i.e.,
. This fact, will force
. In all cases, we have
. In other words, the function
is non-decreasing. Finally, let us show that
, where
, for all
. Since
, then we have
, for all
. If
, pick
. Then
which implies
. Hence
which implies
. Otherwise, assume
, then
. Since
, there exists
such that
. Similar argument will show that
In both cases, we showed that
, for all
, i.e.,
as claimed. □
Proposition 2 allows us to focus on the subsets instead of subsets which satisfy the condition . The next result is amazing and surprising since it tells us that the subsets enjoy nice modular geometric properties despite the fact that .
Theorem 2. Consider the vector space such that and not identically equal to 1. Let be a non-decreasing function. Then, ρ is on .
Proof. Let
and
. Let
such that
,
and
. Since
is convex, we have
which implies
. Set
. The properties of
imply
. So
which implies
Since
, we obtain
, for all
. From our assumptions, we have
Using Lemma 1, we obtain
which implies
Using the convexity of the modular, we have
which implies
For the second case, assume
Since
, we obtain
Hence
Our assumption on
implies
For any
, we have
Using Lemma 1, we obtain
for any
. Hence
which implies
Both cases imply that
is
on
with
since
, for any
. Since
is nondecreasing, we may set
to see that in fact
is
on
which completes the proof of Theorem 2. □
The following lemma will be useful:
Lemma 3. Consider the vector space such that and not identically equal to 1. Let be a non-decreasing function. Set , for . We have Proof. Let
. For any
, we have
which implies
Hence
Therefore
, that is
, which completes the proof of Lemma 3. □
In the next section, we will prove a fixed-point theorem for modular nonexpansive mappings.
4. Application
As an application to Theorem 2, we will prove a fixed-point result for modular nonexpansive mappings. The classical ingredients will be needed. First, we prove the proximinality of -closed convex subsets which satisfies the condition .
Proposition 3. Consider the vector space such that and not identically equal to 1. Let non-decreasing. Any nonempty ρ-closed convex subset C of is proximinal, i.e., for any such thatthere exists a unique such that Proof. Without loss of generality, we assume that
. Since
C is
-closed we have,
. For any
, there exists
such that
. We claim that
is
-Cauchy. Assume not. Then there exists a subsequence
of
and
such that
for any
. According to Lemma 3,
is in
, where
, for any
. Fix
. We have
Since
with
, and using Theorem 2, we obtain
where
Since
and
are in
C and
C is convex, we obtain
If we let
, we obtain
This contradiction implies that
is
-Cauchy. Since
is
-complete, there exists
such that
-converges to
y. Since
C is convex and
-closed, we conclude that
. Using the Fatou property, we have
If we set
, we obtain
. The uniqueness of the point
c comes from the fact that
is strictly convex on
since it is
. □
The next result discusses an intersection property known as the property
[
9]. Recall that a nonempty
-closed convex subset
C of
is said to satisfy the property
if for any decreasing sequence of nonempty
-closed
-bounded convex subsets of
C have a nonempty intersection.
Proposition 4. Consider the vector space such that and not identically equal to 1. Let be a non-decreasing function. Then satisfies the property .
Proof. Let
be a decreasing sequence of nonempty
-closed
-bounded convex subsets of
. Let
. We have
Since
is decreasing, the sequence
is increasing bounded above by
. Set
. If
, then
for any
, which will imply
. Otherwise, assume
. Using Proposition 3, there exists
such that
, for any
. Similar argument as the one used in the proof of Proposition 3 will show that
is
-Cauchy and converges to
. Since
is a decreasing sequence of
-closed subsets, we conclude that
. Again this will show that
which completes the proof of Proposition 4. Moreover, using Fatou property, we note that
which will imply
□
Remark 1. Let us note that under the assumptions of Proposition 4, the conclusion still holds when we consider any family of nonempty, convex, ρ-closed subsets of C, where is upward directed, such that there exists which satisfies . Indeed, set . Without loss of generality, we may assume . For any , there exists such that Since is upward directed, we may assume which implies . Proposition 4 implies . Clearly is ρ-closed and using the last noted point in the proof of Proposition 4, we obtain Let such that . We claim that , for any . Indeed, fix . If for some we have , then obviously we have . Therefore let us assume that , for any . Since is upward directed, there exists such that and , for any . We can also assume that for any . Again we have . Since , for any , we obtain . Moreover we haveHence, which implies the existence of a unique point such that . Since ρ is on , we obtain . In particular, we have , for any . Since , we conclude that , for any , which implies . Since α was taking arbitrary in , we obtain , which implies as claimed. The next result is necessary to obtain the fixed-point theorem sought for -nonexpansive mappings.
Proposition 5. Consider the vector space such that and are not identically equal to 1. Let be a nondecreasing function. Then has the ρ-normal structure property, i.e., for any nonempty ρ-closed convex ρ-bounded subset C of not reduced to one point, there exists such that Proof. Let
C be a
-closed convex
-bounded subset
C of
not reduced to one point. Since
C is not reduced to one point, we have
. Let
such that
. Set
Fix
. Using Lemma 3, we have
and
are in
, where
, for any
. So far we have
Since
c was taken arbitrary in
C, we conclude that
Therefore the proof of Proposition 5 is complete. □
Putting all this together, we are ready to prove the main fixed-point result of our work.
Theorem 3. Consider the vector space such that and are not identically equal to 1. Let C be a nonempty ρ-closed convex ρ-bounded subset of , which satisfies the condition . Any ρ-nonexpansive mapping has a fixed point.
Proof. Since
C satisfies the condition
, Proposition 2 secures the existence of a nondecreasing function
such that
C is a subset of
. The conclusion is trivial if
C is reduced to one point. Therefore, we will assume that
C is not reduced to one point, i.e.,
. Consider the family
The family
is not empty since
. Since
C is bounded, we use Remark 1 to be able to use Zorn’s lemma and conclude that
contains a minimal element
. Let us show that
is reduced to one point. Assume not, i.e.,
contains more than one point. Set
to be the intersection of all
-closed convex subset of
C containing
. Hence
since
. Moreover, we have
which implies that
.
being a minimal element of
we deduce that
. Using Proposition 5, we deduce the existence of
such that
Define the subset
.
K is not empty since
. Note that we have
. Using the properties of modular balls,
K is a
-closed and convex subset of
. Next, we prove that
. Indeed, let
. Since
T is
-nonexpansive, we have
for all
. So we have
, which implies
. Since
, we conclude that
, which implies
for all
. Hence
. Since
x was taken as arbitrary in
K, we obtain
. The minimality of
will force
. Hence
This is a contradiction. Therefore, is reduced to one point and it is a fixed point of T because . □
Remark 2. In Theorem 3, the condition can be replaced by the following condition which is slightly more general:
Author Contributions
A.E.A. and M.A.K. contributed equally on the development of the theory and their respective analysis. All authors have read and agreed to the published version of the manuscript.
Funding
Khalifa University research project No. 8474000357.
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Data Availability Statement
Not applicable.
Acknowledgments
The second author was funded by Khalifa University, UAE, under grant No. 8474000357. The authors, therefore, gratefully acknowledge, with thanks, Khalifa University’s technical and financial support.
Conflicts of Interest
The authors declare no conflict of interest.
Abbreviations
The following abbreviations are used in this manuscript:
MDPI | Multidisciplinary Digital Publishing Institute |
DOAJ | Directory of open access journals |
TLA | Three letter acronym |
LD | linear dichroism |
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