- Open Access
Convergence of Ishikawa iterates of two mappings in modular function spaces
© Abdou et al.; licensee Springer. 2014
Received: 27 December 2013
Accepted: 7 March 2014
Published: 25 March 2014
We establish convergence in the modular sense of an iteration scheme associated with a pair of mappings on a nonlinear domain in modular function spaces. In particular, we prove that such a scheme converges to a common fixed point of the mappings. Our results are generalization of known similar results in the non-modular setting. In particular, we avoid smoothness of the norm in the case of Banach spaces and that of the triangle inequality of the distance in metric spaces.
MSC:47H09, 46B20, 47H10, 47E10.
1 Introduction and basic definitions
The possibility of introducing the structure of a linear metric in as well as the interesting properties of these spaces and many applications to differential and integral equations with kernels of nonpower types were among the reasons for the development of the theory of Orlicz spaces, their applications, and generalizations for more than half a century.
We may observe two principal directions of further development. The first is the theory of Banach function spaces initiated in 1955 by Luxemburg  and developed further in a series of joint papers with Zaanen . The second one is inspired by the theory of Orlicz spaces, based on replacing the particular integral form of the nonlinear functional, which controls the growth of members of the space, by an abstract given functional with some good properties. This idea was the basis of the theory of modular spaces initiated by Nakano  in connection with the theory of order spaces and redefined and generalized by Luxemburg and Orlicz in 1959. Such spaces have been studied for almost 40 years and a number of applications of such spaces in various parts of analysis, probability, and mathematical statistics are known.
In this paper, we will consider modular spaces which lie somewhere in between the abstract modular theory and Musielak-Orlicz theory, i.e. the class of modular spaces given by modulars not of any particular form but, nevertheless, having much more convenient properties than the abstract modulars can possess. In other words, we present a useful tool for applications whenever there is a need to introduce a function space by means of functionals which have some reasonable properties but are far from being norms or F-norms.
Let us introduce basic notions in modular function spaces and related notations which will be used in this chapter. For further details we refer the reader to preliminary sections of the recent articles [4–6] or to the survey article ; see also [8–10] for the standard framework of modular function spaces.
Let Ω be a nonempty set and 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 ℰ we denote the linear space of all simple functions with support from . By we will denote the space of all extended measurable functions, i.e. all functions such that there exists a sequence , and for all . By we denote the characteristic function of the set A.
Definition 1.1 
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 ℰ, 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 element is actually an equivalence class of functions equal ρ-a.e. rather than an individual function.
Definition 1.2 
We say that a regular function pseudomodular ρ is a regular convex function modular if implies ρ-a.e. The class of all nonzero regular convex function modulars defined on Ω will be denoted by ℜ.
In this paper, we only consider convex function modulars.
defines a norm, frequently called the Luxemburg norm.
Note that the monographic exposition of the theory of Orlicz spaces may be found in the book of Krasnosel’skii and Rutickii . For a current review of the theory of Musielak-Orlicz spaces and modular spaces, the reader is referred to Musielak  and Kozlowski .
The following definitions will be needed in this paper.
Definition 1.4 
The sequence is said to be ρ-convergent to if as .
The sequence is said to be ρ-Cauchy if as n and .
We say that is ρ-complete if and only if any ρ-Cauchy sequence in is ρ-convergent.
A subset C of is called ρ-closed if the ρ-limit of a ρ-convergent sequence of C always belongs to C.
A subset C of is called ρ-compact if every sequence in C has a ρ-convergent subsequence in C.
- (f)A subset C of is called ρ-bounded if
- (g)Let and . Define the ρ-distance between f and C as:
The above terminology is used because of its similarity to the metric case. Since ρ does not behave in general as a distance, one should be very careful when dealing with these notions. In particular, ρ-convergence does not imply ρ-Cauchy since ρ does not satisfy the triangle inequality.
The following proposition brings together a few facts that will be often used.
Proposition 1.1 
ρ-balls are ρ-closed.
If for an then there exists a subsequence of such that ρ-a.e.
whenever ρ-a.e. (Note: this property is equivalent to the Fatou property.)
Consider the sets , and . Then we have .
We already pointed out that ρ may not satisfy the triangle inequality, so the modular convergence and norm convergence may not be the same. This will only happen if ρ satisfies the so-called -condition.
Definition 1.5 
The modular function ρ is said to satisfy the -condition if as , whenever as .
We have the following proposition.
Proposition 1.2 
ρ satisfies the -condition;
if and only if , for all if and only if .
Let us mention that uniform continuity holds for a large class of function modulars. For instance, it can be proved that in Orlicz spaces over a finite atomless measure  or in sequence Orlicz spaces , the uniform continuity of the Orlicz modular is equivalent to the -type condition.
Let us recall the definition of different mappings acting in a modular function space. We start with the concept of Lipschitzian mappings.
When , T is called ρ-contraction mapping. Moreover, if , then T is called ρ-nonexpansive mapping. A point is called a fixed point of T if . The set of fixed points of T will be denoted by .
As mentioned before, one of the reasons of our interest in ρ-behavior of mappings is that the Luxemburg norm associated with the function modular is defined in an indirect way and consequently harder to handle than the function modular. Therefore, one may ask what the relationship is, if there is any, between the F-norm nonexpansiveness and the ρ-nonexpansiveness. The following example gives a partial answer.
Example 1.1 
In this paper, we will use some geometrical properties of the modular functional.
Definition 1.8 Let . We say that ρ is strictly convex (SC), if for every such that and , for some , we have .
For the discussion of some geometrical properties of Calderon-Lozanovskii and Orlicz-Lorentz spaces, the reader may consult .
Recently, special attention was given to the use of the geometric properties in modular function spaces. This is due to recent interest in the Dirichlet energy problem which we discuss in the next example.
2 Ishikawa iterates for two mappings
Our first result discusses the convergence behavior of the sequence generated by (2.1).
If and , with , then for any subsequence of which ρ-converges to f, we have .
If and , with , then for any subsequence of which ρ-converges to f, we have .
If , with , then for any subsequence of which ρ-converges to f, we have . In this case, we have that ρ-converges to f.
- (1)If and , then
The strict convexity of ρ will imply .
- (2)If and , then
- (3)If and , then
If , then and . Hence .
Let us finish the proof of Theorem 2.1. Note that (i) implies and . If , then the conclusion (2) above implies . Otherwise the conclusion (4) will imply . This proves (i).
For (ii), notice that and . Hence the conclusion (3) will imply , which proves (ii).
we find that ρ-converges to f, which completes the proof of (iii). □
If we assume compactness of the domain, Theorem 2.1 will imply the following result.
Theorem 2.2 Assume is strictly convex and uniformly continuous. Let C be a nonempty ρ-bounded, ρ-compact and convex subset of . Let be two nonexpansive mappings. Assume that . Fix . Define as in (2.1), where , with , and is the initial element of the sequence. Then ρ-converges to a common fixed point of S and T.
Proof Since C is ρ-compact, has a ρ-convergent subsequence , i.e., ρ-converges to z. By Theorem 2.1, we have and ρ-converges to z. □
Hence . As C is ρ-compact, the fixed point set of T is nonempty. As ρ is strictly convex, is convex. Clearly is ρ-closed subset of C. Hence is ρ-compact. If we assume that T and S commute, i.e., , then . Consequently, S and T have a common fixed point. In general, it is not the case that S and T have a common fixed point.
If we take in Theorem 2.2, we get the following result.
Theorem 2.3 
Assume is strictly convex and uniformly continuous. Let C be a nonempty ρ-bounded, ρ-compact and convex subset of . Let be two nonexpansive mapping. Fix . Define as in (2.2), where , with , and is the initial element of the sequence. Then ρ-converges to a fixed point of T.
The authors MA Khamsi and AR Khan acknowledge gratefully KACST, Riyad, Saudi Arabia, for supporting Research Project ARP-32-34. The author AAN Abdou gratefully acknowledges the Deanship of Scientific Research at King Abdulaziz University, Jeddah, Saudi Arabia, for supporting this research.
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