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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 319394, 6 pages
Fixed Points of Asymptotic Pointwise Nonexpansive Mappings in Modular Spaces
1Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China
2Tianjin Higher Occupation Technique Education Department, Tianjin University of Technology and Education, Tianjin 300222, China
Received 9 February 2012; Accepted 1 March 2012
Academic Editor: Yonghong Yao
Copyright © 2012 Xue Wang et al. 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.
Kirk and Xu studied the existence of fixed points of asymptotic pointwise nonexpansive mappings in the Banach space. In this paper, we investigate these kinds of mappings in modular spaces. Moreover, we prove the existence of fixed points of asymptotic pointwise nonexpansive mappings in modular spaces. The results improve and extend the corresponding results of Kirk and Xu (2008) to modular spaces.
The theory of modular spaces was initiated by Nakano  in 1950 in connection with the theory of order spaces and redefined and generalized by Musielak and Orlicz  in 1959. These spaces were developed following the successful theory of Orlicz spaces, which replaces the particular, integral form of the nonlinear functional, which controls the growth of members of the space, by an abstractly given functional with some good properties. In 2007, Razani et al.  studied some fixed points of nonlinear and asymptotic contractions in the modular spaces. In addition, quasi-contraction mappings in modular spaces without -condition were considered by Khamsi  in 2008. Recently, Kuaket and Kumam  proved the existence of fixed points of asymptotic pointwise contractions in modular spaces. Even though a metric is not defined, many problems in fixed point theory for nonexpansive mappings can be reformulated in modular spaces.
The existence of fixed points of asymptotic pointwise nonexpansive mappings was studied by Kirk and Xu  in 2008, that is, mappings , such that where . Their main result states that every asymptotic pointwise nonexpansive self-mapping of a nonempty, closed, bounded, and convex subset of a uniformly convex Banach space has a fixed point.
The above-mentioned result of Kirk and Xu is a generalization of the 1972 Geobel-Kirk fixed point theorem  for a narrower class of mappings—the class of asymptotic nonexpansive mappings, where-using our notation-every function is a constant function. The latter result was in its own a generalization of the classical Browder-Gohde-Kirk fixed point theorem for nonexpansive mappings . In 2009, the results of  were extended to the case of metric spaces by Hussain and Khamsi .
In this paper, we investigate asymptotic pointwise nonexpansive mappings in modular spaces. Moreover, we obtain similar results in the sense of modular spaces. The results presented in this paper extend and improve the corresponding results of Kirk and Xu .
Definition 2.1. Let be an arbitrary vector space over .(a)A functional is called modular if(i) if and only if ,(ii) for with , for all ,(iii) if , for all . If (iii) is replaced by(iii′), for , for all , then the modular is called convex modular.(b)A modular defines a corresponding modular space, that is, the space given by
Remark 2.2. Note that is an increasing function. Suppose . Then, property (iii′) with shows that .
Remark 2.3. In general, the modular is not subadditive and therefore does not behave as a norm or a distance. But one can associate to a modular with a -norm (see ).
The modular space can be equipped with a -norm defined by Namely, if is convex, then the functional is a norm in which is equivalent to the -norm .
Definition 2.4. Let be a modular space.(a)A sequence is said to be convergent to and write if as .(b)A sequence is called Cauchy whenever as .(c)The modular is called complete if any Cauchy sequence is convergent.(d)A subset is called closed if for any sequence convergent to , we have .(e)A closed subset is called -compact if any sequence has a -convergent subsequence. (f) is said to satisfy the condition if whenever as .(g)We say that has the Fâtou property if whenever and as .(h)A subset is said to be bounded if where is called the diameter of .(i)Define the distance between and as(j)Define the Ball, , centered at with radius as
Definition 2.5. Let be a modular space. For any and , the modular of -uniform convexity of is defined by (i)We say satisfies uniform convexity (UC) if for every , , .(ii)We say that satisfies unique uniform convexity (UUC) if for every , , there exists depending on and such that
Definition 2.6. Let be convex and bounded.(a)A function is called a -type (or shortly a type) if there exists a sequence of elements of such that for any there holds (b)Let be a type. A sequence is called a minimizing sequence of if
The following definitions are straightforward generalizations of their norm and metric equivalents.
Definition 2.7. Let be nonempty and closed. A mapping is called an asymptotic pointwise mapping if there exists a sequence of mapping such that (i)If converges pointwise to , then is called an asymptotic pointwise contraction.(ii)If for any , then is called an asymptotic pointwise nonexpansive mapping.(iii)If for any with , then is called a strongly asymptotic pointwise contraction.
3. Main Results
Lemma 3.1 (see ). Let be a closed bounded convex nonempty subset of , let satisfy (UUC), and let be a type defined on . Then any minimizing sequence of is convergent and its limit is independent of the minimizing sequence.
Theorem 3.2. Let be a modular space, let be a bounded closed convex nonempty subset of , and let satisfy (UUC), and is asymptotic pointwise nonexpansive. Then has a fixed point. Moreover, the set of all fixed points is closed.
Proof. For a fixed , define the type
Let Let be a minimizing sequence of and , which exists in view of Lemma 3.1.
Let us prove that is a fixed point of .
First notice that , for any and .
Indeed, for fixed ,
In particular, we have
By induction, we build an increasing sequence such that
Indeed, since is asymptotic pointwise nonexpansive mapping, we have , so there exists , such that for any , we have . Since , there exists such that for any , we have . Assume is built, then since , there exists such that for any , we have , which completes our induction claim.
By (3.3) and Definition 2.6 (b), it is easy to observe that is a minimizing sequence of , for any . Lemma 3.1 implies is convergent to , for any .
In particular, we have is convergent to . Since we conclude that is also convergent to .
Since the limit of any convergent sequence is unique by Lemma 3.1, we must have .
In the following, we will prove that is closed.
To prove that is closed, let and . Observe that Hence, , so is -closed.
This completes the proof.
Remark 3.3. It is not hard to see that if is -bounded, then an asymptotic pointwise nonexpansive mapping is of asymptotic nonexpansive type ; that is, there exists a sequence of positive numbers with the property as and such that for all and .
R. Chen was supported in part by NSFC 11071279. Y. Chen was supported in part by the Tianjin University of Technology and Education Science Research Development Foundation (2011-28) of Tianjin.
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