Fixed Points of Asymptotic Pointwise Nonexpansive Mappings in Modular Spaces

Abstract

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.

1. 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 Δ₂-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 𝑇^𝑛(𝑥)−𝑇^𝑛𝑦≤𝛼_𝑛(𝑥)𝑥−𝑦,∀𝑥,𝑦∈𝐶, (1.1) 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].

2. Preliminaries

Definition 2.1. Let 𝑋 be an arbitrary vector space over 𝐾=(𝑅or𝐶).(a)A functional 𝜌∶𝑋→[0,∞] is called modular if(i)𝜌(𝑥)=0 if and only if 𝑥=0,(ii)𝜌(𝛼𝑥)=𝜌(𝑥) for 𝛼∈𝐾 with |𝛼|=1, for all 𝑥∈𝑋,(iii)𝜌(𝛼𝑥+𝛽𝑦)≤𝜌(𝑥)+𝜌(𝑦) if 𝛼,𝛽≥0, for all 𝑥,𝑦∈𝑋. If (iii) is replaced by(iii′)𝜌(𝛼𝑥+𝛽𝑦)≤𝛼𝜌(𝑥)+𝛽𝜌(𝑦), for 𝛼,𝛽≥0,𝛼+𝛽=1, for all 𝑥,𝑦∈𝑋, then the modular 𝜌 is called convex modular.(b)A modular 𝜌 defines a corresponding modular space, that is, the space 𝑋_𝜌 given by 𝑋_𝜌=𝑥∈𝑋∣𝜌(𝛼𝑥)⟶0as𝛼⟶0. (2.1)

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 ‖𝑥‖_𝜌𝑥=inf𝛼>0;𝜌𝛼≤𝛼. (2.2) Namely, if 𝜌 is convex, then the functional ‖𝑥‖_𝜌=inf{𝛼>0;𝜌(𝑥/𝛼)≤1} is a norm in 𝑋_𝜌 which is equivalent to the 𝐹-norm ‖⋅‖_𝜌.

Definition 2.4. Let 𝑋_𝜌 be a modular space.(a)A sequence {𝑥_𝑛}⊂𝑋_𝜌 is said to be 𝜌-convergent to 𝑥∈𝑋_𝜌 and write 𝑥−→_𝑛^𝜌𝑥 if 𝜌(𝑥_𝑛−𝑥)→0 as 𝑛→∞.(b)A sequence {𝑥_𝑛} is called 𝜌-Cauchy whenever 𝜌(𝑥_𝑛−𝑥_𝑚)→0 as 𝑛,𝑚→∞.(c)The modular 𝜌 is called 𝜌-complete if any 𝜌-Cauchy sequence is 𝜌-convergent.(d)A subset 𝐵⊂𝑋_𝜌 is called 𝜌-closed if for any sequence {𝑥_𝑛}⊂𝐵𝜌-convergent to 𝑥∈𝑋_𝜌, we have 𝑥∈𝐵.(e)A 𝜌-closed subset 𝐵⊂𝑋_𝜌 is called 𝜌-compact if any sequence {𝑥_𝑛}⊂𝐵 has a 𝜌-convergent subsequence. (f)𝜌 is said to satisfy the Δ₂-condition if 𝜌(2𝑥_𝑛)→0 whenever 𝜌(𝑥_𝑛)→0 as 𝑛→∞.(g)We say that 𝜌 has the Fâtou property if 𝑥−𝑦≤liminf_𝑛→∞𝜌𝑥_𝑛−𝑦_𝑛 (2.3) whenever 𝑥−→_𝑛^𝜌𝑥 and 𝑦−→_𝑛^𝜌𝑦 as 𝑛→∞.(h)A subset 𝐵⊂𝑋_𝜌 is said to be 𝜌-bounded if diam_𝜌(𝐵)<∞, (2.4) 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 asdis_𝜌(𝑥,𝐵)=inf𝜌𝑥−𝑦;𝑦∈𝐵. (2.5)(j)Define the 𝜌-Ball, 𝐵_𝜌(𝑥,𝑟), centered at 𝑥∈𝑋_𝜌 with radius 𝑟 as𝐵_𝜌(𝑥,𝑟)=𝑦∈𝑋_𝜌;𝜌𝑥−𝑦≤𝑟. (2.6)

Definition 2.5. Let 𝑋_𝜌 be a modular space. For any 𝜀>0 and 𝑟>0, the modular of 𝜌_𝑟-uniform convexity of 𝑋_𝜌 is defined by 𝛿_𝜌()1𝑟,𝜀=inf1−𝑟𝜌𝑥+𝑦2∶𝜌(𝑥)≤𝑟,𝜌𝑦≤𝑟,𝜌𝑥−𝑦≥𝑟𝜀. (2.7)(i)We say 𝜌 satisfies uniform convexity (UC) if for every 𝜀>0, 𝑟>0, 𝛿_𝜌(𝑟,𝜀)>0.(ii)We say that 𝜌 satisfies unique uniform convexity (UUC) if for every 𝑠≥0, 𝜀>0, there exists 𝜂(𝑠,𝜀)>0, (2.8) depending on 𝑠 and 𝜀 such that 𝛿_𝜌(𝑟,𝜀)>𝜂(𝑠,𝜀)>0,for𝑟>𝑠. (2.9)

Definition 2.6. Let 𝐶⊂𝑋_𝜌 be convex and 𝜌-bounded.(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 𝜏(𝑧)=limsup_𝑚→∞𝜌𝑦_𝑚−𝑧. (2.10)(b)Let 𝜏 be a type. A sequence {𝑔_𝑛} is called a minimizing sequence of 𝜏 if lim_𝑛→∞𝜏𝑔_𝑛=inf{𝜏(ℎ);ℎ∈𝐶}. (2.11)

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 𝛼_𝑛∶𝐶→[0,∞) such that 𝜌𝑇^𝑛(𝑥)−𝑇^𝑛𝑦≤𝛼_𝑛(𝑥)𝜌𝑥−𝑦,∀𝑥,𝑦∈𝑋_𝜌. (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.

3. 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 𝜏(ℎ)=limsup_𝑛→∞𝜌𝑇^𝑛(𝑥)−ℎ,ℎ∈𝐶. (3.1)
Let 𝜏₀=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, 𝜏𝑇^𝑚(ℎ)=limsup_𝑛→∞𝜌𝑇^𝑛(𝑥)−𝑇^𝑚(ℎ)=limsup_𝑛→∞𝜌𝑇^𝑛+𝑚(𝑥)−𝑇^𝑚(ℎ)=limsup_𝑛→∞𝜌𝑇^𝑚𝑇^𝑛(𝑥)−𝑇^𝑚(ℎ)≤limsup_𝑛→∞𝛼_𝑚(ℎ)𝜌𝑇^𝑛(𝑥)−ℎ=𝛼_𝑚(ℎ)𝜏(ℎ). (3.2)
In particular, we have 𝜏𝑇^𝑚𝑓_𝑛≤𝛼_𝑚𝑓_𝑛𝜏𝑓_𝑛,∀𝑛,𝑚≥1. (3.3)
By induction, we build an increasing sequence {𝑚_𝑘} such that 𝛼_{𝑚𝑘+𝑚}𝑓_𝑘1≤1+𝑘,∀𝑘,𝑚≥1. (3.4)
Indeed, since 𝑇 is asymptotic pointwise nonexpansive mapping, we have limsup_𝑚→∞𝛼_𝑚(𝑓₁)≤1, so there exists 𝑚₁≥1, such that for any 𝑚≥𝑚₁, we have 𝛼_𝑚(𝑓₁)≤1+1/1. Since limsup_𝑚→∞𝛼_𝑚(𝑓₂)≤1, there exists 𝑚₂>𝑚₁ such that for any 𝑚≥𝑚₂, we have 𝛼_𝑚(𝑓₂)≤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^{𝑚𝑘+𝑝}(𝑓_𝑘)} is a minimizing sequence of 𝜏, for any 𝑝≥0. Lemma 3.1 implies {𝑇^{𝑚𝑘+𝑝}(𝑓_𝑘)} is 𝜌-convergent to 𝑓, for any 𝑝≥0.
In particular, we have {𝑇^𝑚𝑘+1(𝑓_𝑘)} is 𝜌-convergent to 𝑓. Since 𝜌𝑇^𝑚𝑘+1𝑓_𝑘−𝑇𝑓≤𝛼₁𝑓𝜌𝑇^𝑚𝑘𝑓_𝑘−𝑓, (3.5) we conclude that {𝑇^𝑚𝑘+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 𝜌13(𝑇(𝑥)−𝑥)≤𝜌𝑇(𝑥)−𝑇𝑥_𝑛+𝜌𝑇𝑥_𝑛−𝑥_𝑛+𝜌𝑥_𝑛−𝑥≤𝛼₁𝜌𝑥_𝑛−𝑥+𝜌𝑥_𝑛−𝑥⟶0. (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 𝜌𝑇_𝑛(𝑥)−𝑇_𝑦𝑦≤𝑘_𝑛𝜌𝑥−𝑦, (3.7) for all 𝑛 and 𝑥,𝑦∈𝐶.

Acknowledgments

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.