On the fixed points of nonexpansive mappings in modular metric spaces
© Abdou and Khamsi; licensee Springer. 2013
Received: 2 June 2013
Accepted: 12 August 2013
Published: 28 August 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, have been recently introduced. In this paper we investigate the existence of fixed points of modular nonexpansive mappings. We also discuss some compactness properties of the family of admissible sets in modular metric spaces with uniform normal structure property.
MSC:47H09, 46B20, 47H10, 47E10.
The purpose of this paper is to give an outline of fixed point theory for nonexpansive mappings (i.e., mappings with the modular Lipschitz constant 1) on subsets of modular metric spaces which are natural generalization of classical modulars over linear 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 approach the concept of modular metric spaces is different. Indeed, we look at these spaces as a nonlinear version of the classical modular spaces, introduced by Nakano , on vector spaces and modular function spaces, introduced by Musielak  and Orlicz .
In recent years, there was an increasing 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 converting 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.
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. In recent years, there has been a great interest in the study of the fixed point property in modular function spaces after the first paper  was published in 1990. More recently, the authors presented a fixed point result for pointwise nonexpansive and asymptotic pointwise nonexpansive acting in modular functions spaces . The theory of nonexpansive mappings defined on convex subsets of Banach spaces has been well developed since the 1960s (see, e.g., Belluce and Kirk , Browder , Bruck , and Lim ) and generalized to other metric spaces (see, e.g., [16–18]) and modular function spaces (see, e.g., ). The corresponding fixed point results were then extended to larger classes of mappings like pointwise contractions and asymptotic pointwise contractions [18–22], and asymptotic pointwise nonexpansive mappings . In , Penot presented an abstract version of Kirk’s fixed point theorem  for nonexpansive mappings. Many results of a fixed point in metric spaces have been developed after Penot’s formulation. Using Penot’s work, the author in  proved some results in metric spaces with uniform normal structure similar to the ones known in Banach spaces.
In this paper we investigate the existence of fixed points of modular nonexpansive mappings defined in modular metric spaces. We also discuss some compactness properties of the family of admissible sets in modular metric spaces with uniform normal structure and prove similar results to the ones obtained in .
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 are 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.
- (a)The associated modular function space is the vector space , or briefly defined by
- (b)The following formula defines a norm in (frequently called the Luxemburg norm):
for any .
The sequence in is said to be ω-convergent to if and only if as . x is called the ω-limit of .
The sequence in is said to be ω-Cauchy if as .
A subset M of is said to be ω-closed if the ω-limit of an ω-convergent sequence of M always belongs to M.
A subset M of is said to be ω-complete if any ω-Cauchy sequence in M is an ω-convergent sequence and its ω-limit is in M.
- (5)A subset M of is said to be ω-bounded if we have
for any . Our assumptions imply . Since ω is regular, we get , i.e., the ω-limit of a sequence is unique.
Recall that A is ω-bounded if .
We say that is compact if any family of elements of has a nonempty intersection provided for any finite subset .
We say that is countably compact or satisfies the property if any sequence of elements of , which are nonempty and decreasing, has a nonempty intersection.
We say that is normal if for any , not reduced to one point, ω-bounded, we have .
We say that is uniformly normal if there exists such that for any , not reduced to one point, ω-bounded, we have .
If , then we have for any . Hence ω-converges to z, which completes the proof of our statement.
3 Main results
Let us first start this section with the definition of nonexpansive mappings in the modular metric sense.
For such a mapping, we denote by the set of its fixed points, i.e., .
if and only if for any . Next we give an example, which first appeared in , of a mapping which is ω-nonexpansive in our sense but fails to be nonexpansive with respect to .
which clearly implies that T is not -nonexpansive.
Next we discuss the analog of Kirk’s fixed point theorem  in modular metric spaces.
Theorem 3.1 Let be a modular metric space. Let C be a nonempty ω-closed ω-bounded subset of . Assume that the family is normal and compact. Let be ω-nonexpansive. Then T has a fixed point.
And since , for any , we get . The minimality behavior of A implies . In particular we have for any . Hence for any . Since is normal, we get , which is a contradiction. Thus we must have , i.e., A is reduced to one point which is fixed by T. □
Next we give a constructive result discovered by Kirk  which relaxes the compactness assumption in the above theorem. The main ingredient in Kirk’s constructive proof is a technical lemma due to Gillespie and Williams . The next lemma is a modular version of the Gillespie and Williams result.
Next we give the analogue of the main fixed point result in .
Theorem 3.2 Let be an ω-complete modular metric space, and let C be a nonempty ω-closed ω-bounded subset of . Assume that the family is uniformly normal and is ω-nonexpansive. Then T has a fixed point.
Since , we get . Since ω is regular, we get , i.e., . Since for any , we get . □
The following technical proposition is needed to show an analogue to the main result in .
is an Ω-complete modular metric space.
is Ω-bounded with .
- (iii)For any and , we have
is uniformly normal.
So, . Since ε was arbitrary, we get . This completes the proof of (iv). □
The following theorem shows that although we do not need compactness of the family of admissible sets in Theorem 3.2, its assumptions imply a weaker form of compactness, mainly countable compactness.
Theorem 3.3 Let be an ω-complete modular metric space, and let C be a nonempty ω-closed ω-bounded subset of . Assume that the family is uniformly normal. Then has the property .
Obviously, T is ω-nonexpansive. Theorem 3.2 implies that T has a fixed point, i.e., there exists such that . The definition of T forces to be a constant sequence, i.e., , for any . Obviously, we have for any , which implies . □
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