1. Introduction
Modular function spaces (MFS) find their roots in the study of the classical function spaces
and their extensions by Orlicz and others. For more details on MFS, we recommend the book by Kozlowski [
1]. Another interesting use of the modular structure, for whoever is looking for more applications, is the excellent book by Diening et al. [
2] about Lebesgue and Sobolev spaces with variable exponents. Fixed point theory in MFS was initiated in 1990 in the original paper [
3]. Since then this theory has become prevalent, culminating in the publication of the recent book by Khamsi and Kozlowski [
4]. In this work, we continue investigating the fixed point problem in MFS. To be precise, we investigate the case of monotone mappings. This area of metric fixed point theory is new and attracted some attention after the publication of Ran and Reuring’s paper [
5]. An interesting reference with many applications of the fixed point theory of monotone mappings is the excellent book by Carl and Heikkilä [
6]. We also suggest the work of Marin [
7] for some applications associated to the iteration problem.
Since this work deals with the metric fixed point theory, we recommend the book by Khamsi and Kirk [
8].
2. Preliminaries
For the basic definitions and properties of MFS, we refer the readers to the books [
1,
4]. Throughout this work, we assume the following:
- (i)
is a nonempty set;
- (ii)
is a nontrivial -algebra of subsets of ;
- (iii)
is a -ring of subsets of such that for any and ;
- (iv)
there exists an increasing sequence in such that .
Consider the set of extended measurable functions such that there exists a sequence of simple functions whose supports are in with and , for any .
Definition 1. ([1,4]) A convex and even function is called regular modular if - (i)
implies ;
- (ii)
for all implies , where (we will say that ρ is monotone);
- (iii)
for all implies , where (ρ has the Fatou property).
Recall that is said to be -null if , for any simple function f whose support is in , where denotes the characteristic function of C. A property holds -almost everywhere (-a.e.) if the subset where it does not hold is -null.
Remark 1. Let ρ be convex regular modular. Let be such that ρ-a.e. Then and ρ-a.e., which imply The MFS
is defined as
The following theorem is essential throughout this work.
Theorem 1 ([
1,
4])
. Let ρ be convex regular modular.- (1)
If , for some , then there exists a subsequence such that
- (2)
If , then .
The following definition mimics the metric properties using the modular.
Definition 2 ([
1,
4])
. Let ρ be convex regular modular.- (1)
is said to ρ-converge to g if
- (2)
A sequence is called ρ-Cauchy if .
- (3)
A subset C of is said to be ρ-closed if for any sequence in Cρ-convergent to g implies that
- (4)
A subset A of is called ρ-bounded if its ρ-diameteris finite.
Note that, despite the fact that
does not satisfy the triangle inequality in general, the
limit is unique. However, the
-convergence may not imply
-Cauchy behavior. Despite this setback, we can prove that any
-Cauchy sequence in
is
-convergent, i.e.,
is
-complete [
1,
4]. Moreover, the Fatou property will show that the
-balls
are
-closed.
The following result will be used throughout:
Theorem 2 ([
1,
4])
. Let ρ be convex regular modular. Let be a sequence which ρ-converges to g. The following hold:- (i)
if is monotone increasing, i.e., ρ-a.e., for any , then ρ-a.e., for any .;
- (ii)
if is monotone decreasing, i.e., ρ-a.e., for any , then ρ-a.e., for any .
Next we discuss a property called uniform convexity, which plays an important part in metric fixed point theory.
Definition 3 ([
4])
. Let ρ be convex regular modular. Let and . Consider the following set:Then define- (i)
ρ is said to be uniformly convex if for every and , we have .
- (ii)
ρ is said to be if for every there exists such that , for .
Example 1. As an example of modular function spaces, we consider the Orlicz spaces. These spaces were introduced by Orlicz and Birnbaum [9]. The function space is defined as follows:where is assumed to be a convex function which is increasing to infinity. In other words, φ behaves similarly to the power functions , . The functional defined byis convex regular modular. In this case, the uniform convexity of the modular was investigated in [10,11]. For example, the Orlicz functions that will generate a uniformly convex modular, one may take and [12,13]. Modular functions which are uniformly convex enjoy a property similar to reflexivity in Banach spaces.
Theorem 3 ([
4,
10])
. Let ρ be convex regular modular. Then has the property , i.e., every sequence of nonempty, ρ-bounded, ρ-closed, convex subsets of such that , for any , has a nonempty intersection, i.e., . Remark 2. Let ρ be convex regular modular. Let K be a ρ-bounded convex ρ-closed nonempty subset of . Let be a monotone increasing sequence. Since order intervals in are convex and ρ-closed, then the property impliesIn other words, there exists such that ρ-a.e., for any . A similar conclusion holds for decreasing sequences. The following lemma is useful throughout this work.
Lemma 1 ([
14])
. Let ρ be convex regular modular. Let and with . Let and be in . Assume thatThen holds. The concept of -type function is fundamental in investigating the existence of fixed points. First let us recall the definition of a modular type function.
Definition 4 ([
4])
. Let ρ be convex regular modular. Let K be a nonempty subset of . The function is said to be a ρ-type if there exists a sequence in such thatfor any . A sequence in K is called a minimizing sequence of φ if . We have the following amazing result about -type functions in MFS.
Lemma 2 ([
14])
. Let ρ be convex regular modular. Then any minimizing sequence of any ρ-type defined on a ρ-bounded ρ-closed convex nonempty subset C of is ρ-convergent. Its ρ-limit is independent of the minimizing sequence. Before we finish this section, we give the modular definitions of monotone Lipschitzian mappings.
Definition 5. Let ρ be convex regular modular. Let C be nonempty subset of . Let be a mapping.
- (i)
T is said to be monotone if ρ-a.e. implies ρ-a.e., for any .
- (ii)
T is called monotone asymptotically nonexpansive (in short M-A-N) if T is monotone and there exists , with for any such that andfor any g and h in C such that ρ-a.e., and . - (iii)
g is a fixed point of T whenever .
The fixed point theory for asymptotically nonexpansive mappings finds its root in the work of Goebel and Kirk [
15]. Following the success of the fixed point theory of monotone mappings, an existence fixed point theorem for M-A-N mappings was proved in [
16] and its modular version in [
17]. Before we state the main result of [
17], recall that a map
T is said to be
-continuous if
-convergent to
g implies
-convergent to
.
Theorem 4 ([
17])
. Let ρ be convex regular modular. Let C be a ρ-bounded ρ-closed convex nonempty subset of . Let be a ρ-continuous M-A-N mapping. Assume there exists such that (resp. ) ρ-a.e. Then T has a fixed point f such that (resp. ) ρ-a.e. The original proof of the existence of a fixed point of asymptotically nonexpansive mappings was not constructive. It was Shu [
18] who considered a modified Mann iteration to generate an approximate fixed point sequence for such mappings. While studying asymptotically nonexpansive mappings, Schu modified the Mann iteration by
for
and
. Schu used the iterate
because it is becoming almost nonexpansive. In the investigation of monotone mappings, it is unknown whether Schu’s iteration sequence generates a sequence which is monotone provided
and
are comparable. A very important fact when investigating the existence of fixed points of such mappings. This problem forced the authors in [
16] to modify Schu’s iteration sequence by using the Fibonacci sequence
defined by
for any
. The Fibonacci–Mann iteration [
16] ((FMI) in short) is defined by
for
and
. This new iteration scheme allowed the authors of [
16] to prove the main results of Schu [
18] for M-A-N mappings defined in uniformly convex Banach spaces. This is surprising since this class of mappings may fail to be continuous.
Next we discuss the behavior of the iteration (FMI) which will generate an approximate fixed point of M-A-N mapping in MFS.
The proof of the following lemma uses solely the partial order and is similar to the original proof done in [
16] in the context of Banach spaces.
Lemma 3. [16] Let ρ be convex regular modular. Let C be a convex nonempty subset of . Let be a monotone mapping. Let be such that (resp. ) ρ-a.e. Let . Consider the (FMI) sequence generated by and . Let be a fixed point of T such that (resp. ) ρ-a.e. Then - (i)
(resp. ) ρ-a.e.;
- (ii)
(resp. ) ρ-a.e.;
for any .
The next lemma is crucial in the proof of the main results of this work.
Lemma 4. Let ρ be convex regular modular. Let C be a ρ-bounded and convex nonempty subset of . Assume is an M-A-N mapping with the Lipschitz constants satisfying . Let be such that (resp. ) ρ-a.e. Let . Consider the (FMI) sequence generated by and . Let be a fixed point of T such that (resp. ) ρ-a.e. Then exists.
Proof. Without loss of generality, assume that
-a.e. Note that, since
C is
-bounded, we must have
. From the definition of
, we have
for any
, where we used the fact that
f is a fixed point of
T, the definition of the Lipschitz constants
and
. Hence
for any
, which implies
for any
. Let us rewrite this inequality as
for any
. Next, we let
to obtain
for any
. Finally if we let
, we have
since the series
is convergent. Therefore, we have
which implies the desired conclusion. ☐
3. Main Results
The next result shows that the sequence generated by (FMI) has an approximate fixed point behavior which is crucial throughout.
Proposition 1. Let ρ be convex regular modular. Let be a ρ-bounded ρ-closed convex nonempty subset. Let be an M-A-N mapping with the associated constants satisfying . Let be such that (resp. ) ρ-a.e. Let be a fixed point of T such that (resp. ) ρ-a.e. Let , with . Consider the (FMI) sequence generated by and . Then Proof. Without loss of generality, we assume
-a.e. From Lemma 3, we know that
-a.e. Using Remark 1, we have
, for any
, i.e.,
is a decreasing sequence of positive numbers. Hence
exists. Assume that
, i.e.,
-converges to
f. From Lemma 3, we obtain
-a.e., which implies
for any
. Hence, we have
. Next, we assume
. We have
since
and
f is a fixed point of
T. On the other hand, we have
, for any
. Let
be a non-trivial ultrafilter over
. We have
with
. Since
, we have
. Since
was an arbitrary ultrafilter, we have
Therefore,
. Since
and
is
, then, by using Lemma 1, we conclude that
which completes the proof of our claim. ☐
Recall that the map
is said to be
-compact if
has a
-convergent subsequence for any sequence
in
C. The following result is the monotone version of Theorem 2.2 of [
18].
Theorem 5. Let ρ be convex regular modular. Let be a ρ-bounded and ρ-closed convex nonempty subset of . Let be an M-A-N mapping with the Lipschitz constants . Assume that is ρ-compact for some . Let be such that (resp. ) ρ-a.e. Let with . Consider the (FMI) sequence generated by and . Then ρ-converges to a fixed point f of T such that (resp. ) ρ-a.e.
Proof. Without loss of generality, we assume
-a.e. Since
T is monotone, the sequence
is monotone increasing. Since
is
-compact, there exists a subsequence
which
-converges to
. Let us show that
-converges to
f and
f is a fixed point of
T. Using the properties of the
-a.e. partial order, we have
-a.e., for any
. In particular, we have
for any
. Using Remark 1, we have
for any
. This will imply
-converges to
f. But
for any
, which implies that
-converges to
as well, which implies
from the uniqueness of the
-limit. It is clear from the properties of the modular
, that
is a decreasing sequence of positive real numbers. Hence,
i.e.,
-converges to
f. Let us finish the proof of Theorem 5 by showing that
-converges to
f. Since
f is a fixed point of
T, which satisfies
, then Lemma 3 implies
-a.e., which implies
for any
. Hence
-converges to
f. Since
is monotone increasing and bounded above by
f, we know that
is a decreasing sequence of positive real numbers. Hence,
exists. Let us prove that
. Let
be a non-trivial ultrafilter over
. Using the definition of
, we have
for any
. If we set
, we get
Since , and , we get . Since , we conclude that , i.e., -converges to f. ☐
Before we investigate a weaker convergence of the (FMI) sequence, we will need the following result, which may be seen as similar to the classical Opial condition [
19]. First, we recall that a subset
C of
is
-a.e.-compact if any sequence
in
C has a
-a.e.-convergent subsequence and its
-a.e.-limit is in
C.
Proposition 2. Let be a ρ-a.e.-compact and ρ-bounded convex nonempty subset of . Let be a monotone increasing (resp. decreasing) bounded sequence in C. Set for any . Consider the ρ-type function defined by Then is ρ-a.e. convergent to and Moreover, if ρ is , then any minimizing sequence of φ in ρ-converges to f. In particular, φ has a unique minimum point in .
Proof. Without loss of generality, assume that
is monotone increasing. Since
C is
-a.e.-compact, there exists a subsequence
, which is
-a.e. convergent to some
. Using Theorem 2, we conclude that
-a.e., for any
. Hence,
, which implies that
is nonempty. Let
. Then the sequence
is a decreasing sequence of finite positive numbers since
C is
-bounded. Hence
exists. As we saw before, there exists a subsequence
of
, which
-a.e.-converges to
. Let us prove that
-a.e.-converges to
f. Indeed, for any
, there exists a unique
such that
. Clearly, we have
when
. Moreover, we have
, for any
. Since
-a.e. converges to
f, we conclude that
also
-a.e. converges to
f. Next let
. Then we must have
-a.e., which implies
for any
. Hence,
, i.e.,
The last part of Proposition 2 is a classical result which may be found in [
14]. ☐
Now we are ready to state a modular monotone version of Theorem 2.1 of [
18].
Theorem 6. Let ρ be convex regular modular. Let be a ρ-a.e.-compact and ρ-bounded convex nonempty subset of . Let be an M-A-N mapping with the Lipschitz constants . Assume that . Let be such that and are ρ-a.e.-comparable. Let with . Consider the (FMI) sequence generated by and . Then is ρ-a.e.-convergent. Its ρ-a.e.-limit is a fixed point of Tρ-a.e.-comparable to .
Proof. Without loss of generality, assume that
-a.e. In this case, we know that
is monotone increasing. Proposition 2 implies that
is
-a.e.-convergent to
with
Since
is
,
f is the unique minimum point of the
-type
defined by
By definition of
, we get
for any
. Hence
is a minimizing sequence of
since
. Using Proposition 2, we conclude that
-converges to
f. Note that since
-a.e., we get
-a.e., for any
, which implies
-a.e., for any
. Hence
is monotone increasing and
-converges to
f, which implies
-a.e. Hence
holds, i.e.,
f is a fixed point of
T. Using Lemma 3, we have
for any
, which implies that
also
-a.e.-converges to
f. Proposition 1 implies
Using the properties of
-convergence and
-a.e.-convergence [
4], there exists a sequence of increasing integers
such that
-a.e.-converges to 0. Therefore, we must have
-a.e.-converges to
f. Since
is monotone increasing and
-a.e., we conclude that
-a.e.-converges to
f. This completes the proof of Theorem 6 by noting that
f is a fixed point of
T and
-a.e. ☐