Fixed Points of Asymptotic Pointwise Nonexpansive Mappings in Modular Spaces

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.


Introduction
The theory of modular spaces was initiated by Nakano 1 in 1950 in connection with the theory of order spaces and redefined and generalized by Musielak and Orlicz 2 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. 3 studied some fixed points of nonlinear and asymptotic contractions in the modular spaces.In addition, quasi-contraction mappings in modular spaces without Δ 2 -condition were considered by Khamsi 4 in 2008.Recently, Kuaket and Kumam 5 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 6 in 2008, that is, mappings T : C → C, such that where limsup n → ∞ α n x ≤ 1.Their main result states that every asymptotic pointwise nonexpansive self-mapping of a nonempty, closed, bounded, and convex subset C of a uniformly convex Banach space X has a fixed point.
The above-mentioned result of Kirk and Xu is a generalization of the 1972 Geobel-Kirk fixed point theorem 7 for a narrower class of mappings-the class of asymptotic nonexpansive mappings, where-using our notation-every function α n 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 8 .In 2009, the results of 6 were extended to the case of metric spaces by Hussain and Khamsi 9 .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 6 .

Preliminaries
Definition 2.1.Let X be an arbitrary vector space over b A modular ρ defines a corresponding modular space, that is, the space X ρ given by Remark 2.2.Note that ρ is an increasing function.Suppose 0 < a < b.Then, property iii with y 0 shows that ρ ax ρ a/b bx ≤ ρ bx .
Remark 2.3.In general, the modular ρ is not subadditive and therefore does not behave as a norm or a distance.But one can associate to a modular with a F-norm see 10 .
The modular space X ρ can be equipped with a F-norm defined by f ρ is said to satisfy the Δ 2 -condition if ρ 2x n → 0 whenever ρ x n → 0 as n → ∞.
g We say that ρ has the Fâtou property if where diam ρ B sup{ρ x − y ; x, y ∈ B} is called the ρ-diameter of B.
i Define the ρ-distance between x ∈ X ρ and B ⊂ X ρ as 2.5 j Define the ρ-Ball, B ρ x, r , centered at x ∈ X ρ with radius r as B ρ x, r y ∈ X ρ ; ρ x − y ≤ r .
a A function τ : C → 0, ∞ is called a ρ -type or shortly a type if there exists a sequence {y m } of elements of C such that for any z ∈ C there holds The following definitions are straightforward generalizations of their norm and metric equivalents.
Definition 2.7.Let C ⊂ X ρ be nonempty and ρ-closed.A mapping T : C → C is called an asymptotic pointwise mapping if there exists a sequence of mapping 2.12 i If {α n } converges pointwise to α : C → 0, 1 , then T is called an asymptotic pointwise contraction.
ii If limsup n → ∞ α n x ≤ 1 for any x ∈ X ρ , then T is called an asymptotic pointwise nonexpansive mapping.
iii If limsup n → ∞ α n x ≤ k for any x ∈ X ρ with 0 < k < 1, then T is called a strongly asymptotic pointwise contraction.

Main Results
Lemma 3.1 see 11 .Let C be a ρ-closed ρ-bounded convex nonempty subset of X ρ , let ρ satisfy (UUC), and let τ be a type defined on C. Then any minimizing sequence of τ is ρ-convergent and its limit is independent of the minimizing sequence.
Theorem 3.2.Let X ρ be a modular space, let C be a ρ-bounded ρ-closed convex nonempty subset of X ρ , and let ρ satisfy (UUC), and T : C → C is asymptotic pointwise nonexpansive.Then T has a fixed point.Moreover, the set of all fixed points Fix T is ρ-closed.
Proof.For a fixed x ∈ C, define the type Let τ 0 inf{τ h ; h ∈ C}.Let {f n } ⊂ C be a minimizing sequence of τ and f n → f ∈ C, which exists in view of Lemma 3.1.
Let us prove that f is a fixed point of T .First notice that τ T m h ≤ α m h τ h , for any h ∈ C and m ≥ 1.Indeed, for fixed x ∈ C, τ T m h lim sup

3.2
In particular, we have By induction, we build an increasing sequence {m k } such that Indeed, since T is asymptotic pointwise nonexpansive mapping, we have limsup m → ∞ α m f 1 ≤ 1, so there exists m 1 ≥ 1, such that for any m ≥ m 1 , we have α m f 1 ≤ 1 1/1.Since limsup m → ∞ α m f 2 ≤ 1, there exists m 2 > m 1 such that for any m ≥ m 2 , we have α m f 2 ≤ 1 1/2.Assume m k is built, then since limsup m → ∞ α m f k 1 ≤ 1, there exists m k 1 > m k such that for any m ≥ m k 1 , we have α m f k 1 ≤ 1 1/ k 1 , which completes our induction claim.By 3.3 and Definition 2.6 b , it is easy to observe that {T m k p f k } is a minimizing sequence of τ, for any p ≥ 0. Lemma 3.1 implies {T m k p f k } is ρ-convergent to f, for any p ≥ 0.
In particular, we have we conclude that {T m k 1 f k } is also ρ-convergent to T f .Since the ρ-limit of any ρ-convergent sequence is unique by Lemma 3.1, we must have T f f.
In the following, we will prove that Fix T is ρ-closed.
To prove that Fix T is ρ-closed, let x n ∈ Fix T and ρ x n − x → 0. Observe that

3.6
Hence, x ∈ Fix T , so Fix T is ρ-closed.This completes the proof.
Remark 3.3.It is not hard to see that if C is ρ-bounded, then an asymptotic pointwise nonexpansive mapping T is of asymptotic nonexpansive type 12 ; that is, there exists a sequence {k n } of positive numbers with the property k n → 1 as n → ∞ and such that ρ T n x − T y y ≤ k n ρ x − y , 3.7 for all n and x, y ∈ C.