- Open Access
Asymptotic pointwise contractive type in modular function spaces
Fixed Point Theory and Applications volume 2013, Article number: 101 (2013)
In this paper, we introduce asymptotic pointwise contractive type conditions in modular function spaces and present fixed point results for mappings under such conditions.
MSC:47H09, 47H10, 54H25.
The notion of asymptotic pointwise contraction was introduced by Kirk : Let be a metric space. A mapping is called an asymptotic pointwise contraction if there exists a function such that for each integer ,
where pointwise on M. Moreover, Kirk and Xu  proved that if C is a weakly compact convex subset of a Banach space E and an asymptotic pointwise contraction, then T has a unique fixed point , and for each , the sequence of Picard iterates converges in norm to v.
Very recently, Saeidi  introduced the concept of (weak) asymptotic pointwise contraction type: Let be a metric space. A mapping is said to be of asymptotic pointwise contraction type (resp. of weak asymptotic pointwise contraction type) if is continuous for some integer and there exists a function such that for each x in M,
where pointwise on M.
Theorem 1.1 Let C be a nonempty weakly compact subset of a Banach space E, and let be a mapping of weak asymptotic pointwise contraction type. Then T has a unique fixed point and, for each , the sequence of Picard iterates converges in norm to v.
On the other hand, Khamsi and Kozlowski  studied the concept of asymptotic pointwise contractions in modular function spaces.
In this paper, motivated by Khamsi and Kozlowski [4, 5] and Saeidi , we study the notion of asymptotic pointwise contraction type in a modular function space. Moreover, we present fixed results which extend the earlier results in [3, 4].
Let Ω be a nonempty set, and let Σ be a nontrivial σ-algebra of subsets of Ω. Let be a δ-ring of subsets of Ω such that for any and . Let us assume that there exists an increasing sequence of sets such that . By ξ we denote the linear space of all simple functions with supports from . By we denote the space of all extended measurable function, i.e., all function such that there exists a sequence , and for all .
By we denote the characteristic function of the set A.
Definition 2.1 
Let be a nontrivial, convex and even function. We say that ρ is a regular convex function pseudomodular if:
ρ is monotone, i.e., for all implies , where ;
ρ is orthogonally subadditive, i.e., for any such that , ;
ρ has the Fatou property, i.e., for all implies , where ;
ρ is order continuous in ξ, i.e., and implies .
Similarly as in the case of measure spaces, we say that a set is ρ-null if for every . We say that a property holds ρ-almost everywhere if the exceptional set is ρ-null. As usual we identify any pair of measurable sets whose symmetric difference is ρ-null as well as any pair of measurable functions differing only on a ρ-null set. With this in mind, we define
where each is actually an equivalence class of functions equal ρ-a.e. rather than an individual function. Where no confusion exists, we write ℳ instead of .
Let ρ be a regular function pseudomodular;
we say that ρ is a regular convex function semimodular if for every implies ρ-a.e.;
we say that ρ is a regular convex function modular if implies ρ-a.e.
The class of all nonzero regular convex function modulars on Ω is denoted by ℜ.
Let ρ be a convex function modular.
A modular function space is the vector space , or briefly , defined by
The following formula defines a norm in (frequently called Luxemburg norm):
In the following theorem, we recall some of the properties of modular function spaces that will be used later on in this paper.
Let . Defining and , we have
has the Lebesgue property, i.e., , for , and ;
is the closure of ξ (in the sense of ).
We say that is ρ-convergent to f and write if and only if .
A sequence where is called ρ-Cauchy if as .
A set is called ρ-closed if for any sequence in C, the convergence implies that f belongs to C.
A set is called ρ-bounded if .
For a set , the mapping is called ρ-continuous if , then .
A set is called ρ-a.e. closed if for any sequence in C which ρ-a.e. converges to some f, then we must have .
A set is called ρ-a.e. compact if for any sequence in C, there exists a subsequence which ρ-a.e. converges to some .
Let and . The ρ-distance between f and C is defined as
Let us recall that ρ-convergence does not necessarily imply ρ-Cauchy condition. Also, does not imply in general , .
Definition 2.6 
We say that has the property if and only if every nonincreasing sequence of nonempty, ρ-bounded, ρ-closed, convex subsets of has nonempty intersection.
Definition 2.7 
We say that the function modular ρ is uniformly continuous if for every and , there exists such that
Definition 2.8 
A function , where is nonempty and ρ-closed, is called ρ-lower semicontinuous if for any , the set is ρ-closed.
It can be proved that ρ-lower semicontinuity is equivalent to the condition
The following result plays an important role in the proof of the main results.
Lemma 2.9 
Assume that has the property . Let be nonempty, convex, ρ-closed and ρ-bounded. If is a ρ-lower semicontinuous convex function, then there exists such that
Let us recall the notion of ρ-type.
Definition 2.10 
Let be convex and ρ-bounded. A function is called a -type (or shortly a type) if there exists a sequence of elements of C such that for any , the following holds:
Lemma 2.11 
Let be uniformly continuous. Let be nonempty, convex, ρ-closed and ρ-bounded. Then any ρ-type is ρ-lower semicontinuous in C.
3 Asymptotic pointwise contractive type conditions in modular function spaces
Definition 3.1 
Let and be non-empty and ρ-closed. A mapping is called an asymptotic pointwise mapping if there exists a sequence of mappings such that
If converges pointwise to , then T is called asymptotic pointwise ρ-contraction.
If for any , then T is called asymptotic pointwise nonexpansive.
If for any with , then T is called strongly asymptotic pointwise ρ-contraction.
Khamsi and Kozlowski proved the following results in modular function spaces.
Theorem 3.2 
Let be nonempty, ρ-closed and ρ-bounded. Let be an asymptotic pointwise ρ-contraction. Then T has at most one fixed point in C. Moreover, if is a fixed point of T, then the orbit is ρ-convergent to for any .
Theorem 3.3 
Let us assume that is uniformly continuous and has the property . Let be nonempty, convex, ρ-closed and ρ-bounded. Let be an asymptotic pointwise ρ-contraction. Then T has a unique fixed point . Moreover, the orbit is ρ-convergent to for any .
Below, we introduce the notion of asymptotic pointwise ρ-contraction type in modular function spaces.
Definition 3.4 Let be nonempty, ρ-bounded and ρ-closed. A mapping is said to be of asymptotic pointwise ρ-contraction type (resp. of weak asymptotic pointwise ρ-contraction type) if is ρ-continuous for some integer and there exists a function such that, for each x in C,
where pointwise on M.
it can be easily seen from (3.1) (resp. (3.2)) that
for all , and
We will obtain fixed point results for these mappings in modular function spaces.
First, it is worth mentioning that the ρ-limit of any ρ-convergent sequence in is unique. This fact follows from the following reasoning: Assume that and . Then
which implies that .
The following theorem is our main result.
Theorem 3.5 Let be uniformly continuous and have the property (R). Let be nonempty, convex, ρ-closed and ρ-bounded. Let be a mapping of weak asymptotic pointwise ρ-contraction type. Then T has a unique fixed point and, for each , the sequence of Picard iterates is ρ-convergent to v.
Proof Fix an and define a function τ by
By Lemma 2.11, τ is ρ-lower semicontinuous in C. By Lemma 2.9, then there exists such that
Let us prove that . Indeed, for any , we have
Since T is of weak asymptotic pointwise ρ-contraction type, by (3.4) we have . Thus, for a subsequence of , we have
Now, by (3.6) and (3.7), we obtain
which forces as . Hence as . From this and the continuity of , for some , it follows that as . Since the ρ-limit of any ρ-convergent sequence is unique, we must have , namely, is a fixed point of . Now, repeating the above proof for instead of x, we deduce that is ρ-convergent to a member v of C; i.e., . But for all . Hence, and then .
We show that ; for this purpose, consider an arbitrary . Then there exists a such that for all . So, by choosing a natural number , we obtain
Since the choice of is arbitrary and , we get .
It is easy to verify that T can have only one fixed point. Indeed, if are fixed points of T, then by (3.5), we have
Taking lim inf in the above inequality, we obtain
Since and , we immediately get . □
Next, using the ρ-a.e. strong Opial property of the function modular, we prove a fixed point theorem which does not assume the uniform continuity of ρ.
We say that satisfies the ρ-a.e. strong Opial property (or shortly SO-property) if for every which is ρ-a.e. convergent to zero such that there exists a for which
the following equality holds for any :
Lemma 3.7 
Let . Assume that has the ρ-a.e. strong Opial property. Let be a nonempty, ρ-a.e. compact subset such that there exists such that . Let be a nonempty ρ-a.e. closed subset. For any , let be such that for any , there exists a sequence such that, for every , the following holds:
and for every and . Let for any . Then there exists at which λ attains infimum, i.e.,
Theorem 3.8 Let . Assume that has the ρ-a.e. strong Opial property. Let be a nonempty ρ-a.e. compact convex subset such that for some . Then any of weak asymptotic pointwise ρ-contraction type has a unique fixed point . Moreover, the orbit is ρ-convergent to for any .
Proof Fix an and define a function τ by
By Lemma 3.7 applied with , , , and with chosen for all , there exists such that
The rest of the proof is like the one used for Theorem 3.5. □
Kirk WA: Asymptotic pointwise contraction. Plenary Lecture, the 8th International Conference on Fixed Point Theory and Its Applications 2007.
Kirk WA, Xu H-K: Asymptotic pointwise contractions. Nonlinear Anal. 2008, 69: 4706–4712. 10.1016/j.na.2007.11.023
Saeidi, S: Mapping under asymptotic pointwise contractive type conditions. J. Nonlinear Convex Anal. (in press)
Khamsi MA, Kozlowski WM: On asymptotic pointwise contraction in modular function spaces. Nonlinear Anal. 2010, 73: 2957–2967. 10.1016/j.na.2010.06.061
Khamsi MA, Kozlowski WM: On asymptotic pointwise nonexpansive mappings in modular function spaces. J. Math. Anal. Appl. 2011, 380: 697–708. 10.1016/j.jmaa.2011.03.031
Khamsi MA: Fixed point theory in modular function spaces. 48. In Recent Advances on Metric Fixed Point Theorem. Universidad de Silva, Silva; 1996:31–58.
Kozlowski WM: Notes on modular function spaces I. Comment. Math. 1988, 28: 91–104.
Kozlowski WM: Notes on modular function spaces II. Comment. Math. 1988, 28: 105–120.
Kozlowski WM Ser. Monogr. Textbooks Pure Appl. Math. 122. In Modular Function Spaces. Dekker, New York; 1988.
Khamsi MA: A convexity property in modular function spaces. Math. Jpn. 1996, 44: 269–279.
The authors are grateful to the referees for their careful reading of the paper and several helpful comments.
The authors declare that they have no competing interests.
All the authors contributed equally. All authors read and approved the final manuscript.
About this article
Cite this article
Golkarmanesh, F., Saeidi, S. Asymptotic pointwise contractive type in modular function spaces. Fixed Point Theory Appl 2013, 101 (2013). https://doi.org/10.1186/1687-1812-2013-101
- asymptotic pointwise ρ-contraction type
- modular function space