- Open Access
Fixed point results of pointwise contractions in modular metric spaces
© Abdou and Khamsi; licensee Springer. 2013
Received: 28 March 2013
Accepted: 3 June 2013
Published: 25 June 2013
The notion of a modular metric on an arbitrary set and the corresponding modular spaces, generalizing classical modulars over linear spaces like Orlicz spaces, were recently introduced. In this paper we investigate the existence of fixed points of modular contractive mappings in modular metric spaces. These are related to the successive approximations of fixed points (via orbits) which converge to the fixed points in the modular sense, which is weaker than the metric convergence.
MSC: 47H09, 46B20, 47H10, 47E10.
The purpose of this paper is to give an outline of a fixed point theory for Lipschitzian mappings defined on some subsets of modular metric spaces which are natural generalizations of both function and sequence variants of many important, from applications perspective, spaces like Lebesgue, Orlicz, Musielak-Orlicz, Lorentz, Orlicz-Lorentz, Calderon-Lozanovskii spaces and many others. Modular metric spaces were introduced in [1, 2]. The main idea behind this new concept is the physical interpretation of the modular. Informally speaking, whereas a metric on a set represents nonnegative finite distances between any two points of the set, a modular on a set attributes a nonnegative (possibly, infinite valued) ‘field of (generalized) velocities’: to each ‘time’ (the absolute value of) an average velocity is associated in such a way that in order to cover the - ‘distance’ between points , it takes time λ to move from x to y with velocity . But the way we approached the concept of modular metric spaces is different. Indeed, we look at these spaces as the nonlinear version of the classical modular spaces as introduced by Nakano  on vector spaces and modular function spaces introduced by Musielack  and Orlicz .
In recent years, there was an uptake of interest in the study of electrorheological fluids, sometimes referred to as ‘smart fluids’ (for instance, lithium polymetachrylate). For these fluids, modeling with sufficient accuracy using classical Lebesgue and Sobolev spaces, and , where p is a fixed constant, is not adequate, but rather the exponent p should be able to vary [6, 7]. One of the most interesting problems in this setting is the famous Dirichlet energy problem [8, 9]. The classical technique used so far in studying this problem is to convert the energy functional, naturally defined by a modular, to a convoluted and complicated problem which involves a norm (the Luxemburg norm). The modular metric approach is more natural and has not been used extensively.
The fixed point property in modular function spaces was initiated after the publication of the paper  in 1990. More recently, the authors presented a series of fixed point results for pointwise contractions and asymptotic pointwise contractions acting in modular functions spaces [11, 12]. In this paper, we define and investigate the fixed point property in the framework of modular metric spaces. It seems that a timid start has been going on for a few years now but it still lacks some generalities. The importance of applications of fixed points of mappings in modular metric spaces follows the success of such applications in modular function spaces because of the richness of structure of modular function spaces that - besides being Banach spaces (or F-spaces in a more general settings) - are equipped with modular equivalents of norm or metric notions, and also are equipped with almost everywhere convergence and convergence in submeasure. In many cases, particularly in applications to integral operators, approximation and fixed point results, modular type conditions are much more natural as modular-type assumptions can be more easily verified than their metric or norm counterparts. From this perspective, the fixed point theory in modular metric spaces should be considered as complementary to the fixed point theory in modular function spaces and in metric spaces. The theory of contractions and nonexpansive mappings defined on convex subsets of Banach spaces has been well developed since the 1960s (see, e.g., Belluce and Kirk [13, 14], Browder , Bruck , DeMarr , and Lim ) and generalized to other metric spaces (see, e.g., [19–21]) and modular function spaces (see, e.g., ). The corresponding fixed point results were then extended to larger classes of mappings like pointwise contractions [22, 23] and asymptotic pointwise contractions and nonexpansive mappings [24, 25].
2 Basic definitions and properties
for all and .
if and only if for all ;
for all and ;
for all and .
are said to be modular spaces (around ).
for any . These distances will be called Luxemburg distances (see example below for the justification).
Next we give the main example that motivated this paper.
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 .
We say that ρ is a regular function semimodular if for every implies ρ-a.e.;
We say that ρ is a regular function modular if implies ρ-a.e.
The class of all nonzero regular convex function modulars defined on Ω is denoted by ℜ. Let us denote for , . It is easy to prove that is a function pseudomodular in the sense of Def. 2.1.1 in  (more precisely, it is a function pseudomodular with the Fatou property). Therefore, we can use all results of the standard theory of modular function space as per the framework defined by Kozlowski in [27–29]; see also Musielak  for the basics of the general modular theory. Let ρ be a convex function modular.
for any .
The sequence in is said to be w-convergent to if and only if as for some .
The sequence in is said to be w-Cauchy if as for some .
A subset C of is said to be w-closed if the limit of a w-convergent sequence of C always belongs to C.
A subset C of is said to be w-complete if any w-Cauchy sequence in C is a convergent sequence and its limit is in C.
- (5)A subset C of is said to be w-bounded if for some , we have
for any . Our assumptions imply . Since w is regular, we must have as claimed.
Let us finish this section with the modular metric definitions of Lipschitzian mappings. The definitions are straightforward generalizations of their norm and metric equivalents .
The smallest such constant k is known as . A point is called a fixed point of T whenever . The set of fixed points of T is denoted by .
In the next definition, we introduce the concept of pointwise contraction mappings in this setting.
- (i)a contraction mapping if there exists a constant such that
- (ii)a pointwise contraction mapping if there exists a function such that
but T is not Lipschitzian with respect to with constant 1.
3 Banach contraction principle in modular metric spaces
The statement of Banach contraction principle in modular metric spaces is as follows.
Theorem 3.1 Let be a modular metric space. Assume that w is regular. Let C be a nonempty subset of . Assume that C is w-complete and w-bounded, i.e., . Let be a contraction. Then T has a unique fixed point . Moreover, the orbit w-converges to for each .
for any . Since w-converges to , therefore we get , i.e., . Since T has at most one fixed point, we conclude that any orbit of T w-converges to the only fixed point of T. □
for any . Hence the orbit w-converges to as well. In this remark, we show how one has to be careful when dealing with modulars. Indeed, a modular may take infinite value. This is the problem that the authors of  did not pay attention to. This was also pointed out in the short note . In fact, the authors of  did try to fix this problem in another short note  but they used the triangle inequality in their proof knowing that w does not in general satisfy the triangle inequality. Therefore our Theorem 3.1 properly establishes the classical Banach contraction principle in the best possible way in modular metric spaces and improves a number of earlier known results in this setting.
4 Pointwise contraction mappings in modular metric spaces
Pointwise contractive behavior was introduced in  to extend the contractive behavior in the Banach contraction principle. The central fixed point result in the metric setting for such mappings is the following theorem.
Let K be a weakly compact convex subset of a Banach space and suppose that is a pointwise contraction. Then T has a unique fixed point . Moreover, the orbit converges to for each .
for any . Since , we conclude that w-converges to a. Moreover, if b is another fixed point of T, then from , we must have . But it is not clear how to prove the existence of the fixed point from the convergence of the orbits, which is the case in the classical proof of the Banach contraction principle. In this case, we have to use a different technique than the one used in Theorem 3.1.
Recall that A is w-bounded if .
Definition 4.1 Let be a modular metric space. We say that is compact if any family of elements of has a nonempty intersection provided for any finite subset .
If , then we have for any . Hence w-converges to z, which completes the proof of our statement.
Theorem 4.2 Let be a modular metric space. Let C be a nonempty w-closed w-bounded subset of . Assume that the family is compact. Let be a pointwise contraction. Then T has a unique fixed point . Moreover, the orbit w-converges to for each .
And since for any , we get . The minimality behavior of K implies . In particular, we have for any . Hence for any . Hence . And since , we get , i.e., K is reduced to one point which is fixed by T. Hence the fixed point set of T is not empty. The remaining conclusion of the theorem follows from the general properties of pointwise contractions. □
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