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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 501631, 7 pages
On Asymptotic Pointwise Contractions in Modular Metric Spaces
Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
Received 20 April 2013; Accepted 17 May 2013
Academic Editor: Geraldo Botelho
Copyright © 2013 Afrah A. N. Abdou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
In this paper we study and prove some new fixed points theorems for pointwise and asymptotic pointwise contraction mappings in modular metric spaces.
The notion of modulars on linear spaces and the corresponding theory of modular linear spaces were founded by Nakano  and were extensively developed by his mathematical school. 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 present time the theory of modulars and modular spaces is extensively investigated. Chistyakov in [2, 3] introduced the notion of a Modular metric on arbitrary set and their corresponding modular spaces. The main idea behind this new concept according to Chistyakov 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 to move from to with velocity . But our approach to 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 Musielak  and Orlicz .
In recent years, there was a strong interest to study the fixed point property in modular function spaces, which are natural generalization of both function and sequence spaces, after the first paper  was published by Khamsi et al. in 1990. More recently, the authors presented some fixed point results for pointwise contractions and asymptotic pointwise contractions acting in modular functions spaces [7, 8]. 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 [9, 10], Browder , Bruck , DeMarr , and Lim ) and generalized to other metric spaces (see, e.g., [15–17]) and modular function spaces (see, e.g., ). The corresponding fixed point results were then extended to larger classes of mappings like pointwise contractions [18, 19] and asymptotic pointwise contractions and nonexpansive mappings [20, 21].
In this paper we prove the existence of fixed point theorems for pointwise mappings without the use of ultrapower technique. Our results extend and improve several known results including the corresponding recent fixed point results of [7, 8, 20].
For more on metric fixed point theory, the reader may consult the book .
2. Basic Definitions and Properties
Let be a nonempty set. Throughout this paper for a function , we will write for all and .
Definition 1 (see [2, 3]). A function is said to be modular metric on if it satisfies the following axioms: (i) if and only if , for all ; (ii), for all , and ; (iii), for all and .If instead of (i), we have only the condition (i′) then is said to be a pseudomodular (metric) on . A modular metric on is said to be regular if the following weaker version of (i) is satisfied: Finally is said to be convex if for and , it satisfies the inequality
Note that for a metric pseudomodular on a set , and any , the function is nonincreasing on . Indeed if , then
It is clear that , but this inclusion may be proper in general. It follows from [2, 3] that if is a modular on , then the modular space can be equipped with a (nontrivial) metric, generated by and given by for any . If is convex modular on , according to [2, 3] the two modular spaces coincide, , and this common set can be endowed with the metric given by for any . These distances will be called Luxemburg distances (see example below for the justification).
Next we give an example of a modular metric space.
Example 3. Let be a nonempty set and be a nontrivial -algebra of subsets of . Let be a -ring of subsets of , such that for any and . Let us assume that there exists an increasing sequence of sets such that . By we denote the linear space of all simple functions with supports from . By we will denote the space of all extended measurable functions, that is all functions such that there exists a sequence , and for all . By we denote the characteristic function of the set . Let be a nontrivial, convex, and even function. We say that is a regular convex function pseudomodular if(i); (ii) is monotone; that is, for all implies , where ; (iii) is orthogonally subadditive; that is, for any such that ; (iv) has the Fatou property; that is, for all implies , where ; (v) is order continuous in ; that is, and implies . Similarly as in the case of measure spaces, we say that a set is -null if for every . We say that a property holds -almost everywhere if the exceptional set is -null. As usual we identify any pair of measurable sets whose symmetric difference is -null as well as any pair of measurable functions differing only on a -null set. With this in mind we define where each is actually an equivalence class of functions equal -a.e. rather than an individual function. Where no confusion exists we will write instead of . Let be a regular function pseudomodular. (a)We say that is a regular function semimodular if for every implies .(b)We say that is a regular function modular if implies .The class of all nonzero regular convex function modulars defined on will be denoted by . Let us denote for . It is easy to prove that is a function pseudomodular in the sense of Definition 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 spaces as per the framework defined by Kozlowski in [23–25], and see also Musielak  for the basics of the general modular theory. Let be a convex function modular. (a)The associated modular function space is the vector space , or briefly , defined by (b)The following formula defines a norm in (frequently called Luxemburg norm): Modular function space furnishes a wonderful example of a modular metric space. Indeed let be modular function space. Define the function by for all , and ; then is a modular metric on . Note that is convex if and only if is convex. Moreover we have for any .
Definition 4. Let be a modular metric space. (1)The sequence in is said to be -convergent to if and only if , as . will be called the -limit of . (2)The sequence in is said to be -Cauchy if , as . (3)A subset of is said to be -closed if the -limit of a -convergent sequence of always belong to . (4)A subset of is said to be -complete if any -Cauchy sequence in is a -convergent sequence and its -limit is in .(5)A subset of is said to be -bounded one has
Note that if is regular, then the -limit of a sequence is unique. In general if , for some , then we may not have , for all . Therefore, as it is done in modular function spaces, we will say that satisfies -condition if this is the case; that is, , for some implies , for all . In [2, 3], one will find a discussion about the connection between -convergence and metric convergence with respect to the Luxemburg distances. In particular, we have
for any and . And in particular we have -convergence and convergence are equivalent if and only if the modular satisfies the -condition. Moreover if the modular is convex, then we know that and are equivalent which implies
for any and [2, 3].
Let be a modular metric space. Throughout the rest of this work, we will assume that satisfies the Fatou property; that is, if then we must have For any and , we define the modular ball Note that if satisfies the Fatou property, then modular balls are -closed. An admissible subset of is defined as an intersection of modular balls. Denote by the family of admissible subsets of . Note that is stable by intersection. At this point we introduce some notation which will be used throughout the remainder of this work. For a subset of a modular metric space set Recall that is -bounded if .
Definition 5. Let be a modular metric space. One will say that is compact if any family of elements of has a nonempty intersection provided for any finite subset .
Remark 6. Note that if is compact, then is -complete.
Definition 7. Let be a modular metric space. A function is said to be (i)-convex if
(ii)a type if it is defined as
where is a -bounded sequence in .
Types are very useful in the study of the geometry of Banach spaces and the existence of fixed point of mappings. We will say that is type-stable if types are -convex. We have the following lemma.
Lemma 8. Let be a modular metric space such that is compact on which is type-stable. Then for any type , there exists such that The proof is easy and will be omitted.
3. Pointwise Contraction in Modular Metric Spaces
In [2, 3] the author defined Lipschitzian mappings in modular metric spaces and proved some fixed point theorems. In this paper, we propose a more general definition. Indeed, in the case of modular function spaces, it is proved in  that if and only if Moreover an example is given such that but is not Lipschitzian with respect to with constant 1.
Definition 9. Let be a modular metric space. Let be a nonempty subset of . A map is called (i)generalized contraction if there is an increasing and upper semicontinuous function satisfying , for , such that for any (ii)generalized pointwise contraction if there is an increasing and upper semicontinuous function with respect to the second variable satisfying , for and , such that for any .
Now, we are ready to prove an analogue to Boyd and Wong fixed point theorem  in modular metric spaces.
Theorem 10. Let be a modular metric space. Let be a nonempty -closed -bounded subset of . Assume that the family is compact and is a generalized contraction. Then has a unique fixed point . Moreover the orbit converges to , for each .
Proof. Let , we define the -type Since is compact, then for any , we have Since is a generalized contraction, for any , we have which shows that is decreasing and bounded below. Therefore converges to . Thus we get since is upper semicontinuous. Our assumptions on force , which means that Now let be a fixed point of ; that is, , and then we have for any Since , then ; that is, if has a fixed point , then any orbit -converges to . Note that if is another fixed point of , then , so ; that is, . Therefore has at most one fixed point. Let us finish the proof of the theorem by showing that has a fixed point. Fix . Let . We have Let . There exists , such that for every , Now, for , , and since is increasing, we get In particular, we have , for any . So Since , we get Since is upper semicontinuous, if we let go to 0, we get Note that . Indeed we have ; that is, is decreasing. Since is a positive function, the sequence converges to some . Since is upper semicontinuous, we get . Our assumptions on will imply . Therefore, we have Similarly, we have , thus Since is regular, the uniqueness of -limit implies which prove that is a fixed point by . In particular is reduced to one point.
Before we state our next result, we will need to define the concept of Chebyshev center and radius in modular metric spaces. Let be a modular metric space and let be a nonempty -bounded subset. For any , define The Chebyshev radius of is defined by Obviously we have , for any . The Chebyshev center of is defined as The following result is a generalization of Kirk's fixed point theorem  on pointwise contraction mappings.
Theorem 11. Let be a modular metric space. Let be a nonempty -closed -bounded subset of . Assume that the family is compact and is a generalized pointwise contraction. Then has a unique fixed point . Moreover the orbit converges to , for each .
Proof. Since is compact, there exists a minimal nonempty such that . It is easy to check that . Let us prove that ; that is, is reduced to one point. Indeed since is -bounded, then ; that is, is also -bounded. Let ; then we have . Since is a generalized pointwise contraction, there exists an increasing upper semicontinuous mapping with respect to the second variable such that In particular, we have , for any , which implies So, , which implies This will force . Indeed fix and define Clearly is not empty since . Moreover we have Since for any , we get . The minimality behavior of implies . In particular we have for any . Hence , for any . Since , for any , we get Assume , then , which gives a contradiction. Thus ; that is, is reduced to one point which is a fixed point by since is -invariant. Hence has a fixed point. Next we prove that has a unique fixed point. Let and be two fixed points of . We have Our assumptions on will then imply , that is, . Next we finish the proof of our theorem by showing that, for any , the orbit -converges to a fixed point of . Indeed for any , we have that is, is decreasing. Let . Suppose that . Since is upper semicontinuous with respect to the second variable, we get which is a contradiction. Thus ; that is, the orbit -converges to the fixed point .
4. Asymptotically Pointwise Contraction in Modular Metric Spaces
Definition 12. Let be a modular metric space. Let be a nonempty subset of . A mapping is called an asymptotic pointwise Lipschitzian mapping if there exists a sequence of mappings such that
(i)If converges pointwise to , then is called an asymptotic pointwise contraction.(ii)If , with , then is called strongly asymptotic pointwise contraction.
Note that if is asymptotic pointwise contraction, then has at most one fixed point. Indeed let and be two fixed points of . Then Since , then we must have ; that is, . A similar conclusion holds if is a strongly asymptotic pointwise contraction.
Theorem 13. Let be a modular metric space. Let be a nonempty -closed -bounded subset of . Assume that the family is compact and is a strongly asymptotic pointwise contraction. Then has a unique fixed point . Moreover the orbit converges to , for each .
Proof. Let and define the type for any . Since is compact, then For any , we have If we let go to infinity, we get Next we let go to infinity to get for some , which easily implies that . Fix and notice that Indeed let ; then for any , there exists such that for any , we have In particular we have , for any . So which implies . This is true for any . Hence for any we have . Therefore Therefore we have ; that is, -converges to . This will force to be a fixed point of . Indeed we have which implies for any . Since -converges to , we get ; that is, , since is regular. Since has at most one fixed point, we conclude that any orbit of -converges to the only fixed point of .
Next we relax the strong behavior of but assume that types are convex to obtain the following result.
Theorem 14. Let be a modular metric space. Let be a nonempty -closed -bounded subset of . Assume that the family is compact. Assume that is compact and type-stable. Let be an asymptotic pointwise contraction. Then has a unique fixed point . Moreover the orbit converges to , for each .
Proof. Similarly one can easily show that has at most one fixed point. As we did in the proof of the previous result, let and define the type Since is compact and type-stable, then there exists such that Let us show that . Indeed we have for any . If we let go to infinity, we get which implies If we let go to infinity, we get . Since , we get , which implies that -converges to . This will force to be a fixed point of . Since we already noticed that has at most one fixed point, then has a fixed point and any orbit -converges to .
The author would like to thank Professor M. A. Khamsi with whom the author had many fruitful discussions regarding this work.
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