# Fixed Points of Asymptotic Pointwise Nonexpansive Mappings in Modular Spaces

Authors: Xue Wang; Ying Chen; Rudong Chen
Journal: Journal of Applied Mathematics
 (2012)

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## 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.


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## Body

## 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 (1.1)‖Tn(x)-Tn(y)‖≤αn(x)‖x-y‖,∀x,y∈C,
where limsupn→∞αn(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 αn 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.
LetX be an arbitrary vector space over K=(RorC).(a)
A functionalρ: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 modularρ defines a corresponding modular space, that is, the space Xρ given by
(2.1)Xρ={x∈X∣ρ(αx)⟶0asα⟶0}.Remark 2.2.
Note thatρ 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 modularρ 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 spaceXρ can be equipped with a F-norm defined by
(2.2)‖x‖ρ=inf⁡⁡{α>0;ρ(xα)≤α}.
Namely, if ρ is convex, then the functional ∥x∥ρ=inf{α>0;ρ(x/α)≤1} is a norm in Xρ which is equivalent to the F-norm ∥·∥ρ.Definition 2.4.
LetXρ be a modular space.(a)
A sequence{xn}⊂Xρ is said to be ρ-convergent to x∈Xρ and write xn→ρx if ρ(xn-x)→0 as n→∞.(b)
A sequence{xn} is called ρ-Cauchy whenever ρ(xn-xm)→0 as n,m→∞.(c)
The modularρ is called ρ-complete if any ρ-Cauchy sequence is ρ-convergent.(d)
A subsetB⊂Xρ is called ρ-closed if for any sequence {xn}⊂Bρ-convergent to x∈Xρ, we have x∈B.(e)
Aρ-closed subset B⊂Xρ is called ρ-compact if any sequence {xn}⊂B has a ρ-convergent subsequence.(f)
ρ is said to satisfy the Δ2-condition if ρ(2xn)→0 whenever ρ(xn)→0 as n→∞.(g)
We say thatρ has the Fâtou property if
(2.3)(x-y)≤liminfn→∞ρ(xn-yn)
whenever xn→ρx and yn→ρy as n→∞.(h)
A subsetB⊂Xρ is said to be ρ-bounded if
(2.4)diamρ(B)<∞,
where diamρ(B)=sup{ρ(x-y);x,y∈B} is called the ρ-diameter of B.(i)
Define theρ-distance between x∈Xρ and B⊂Xρ as(2.5)disρ(x,B)=inf⁡⁡{ρ(x-y);y∈B}.(j)
Define theρ-Ball, Bρ(x,r), centered at x∈Xρ with radius r as(2.6)Bρ(x,r)={y∈Xρ;ρ(x-y)≤r}.Definition 2.5.
LetXρ be a modular space. For any ε>0 and r>0, the modular of ρr-uniform convexity of Xρ is defined by
(2.7)δρ(r,ε)=inf⁡⁡{1-1rρ(x+y2):ρ(x)≤r,ρ(y)≤r,ρ(x-y)≥rε}.(i)
We sayρ satisfies uniform convexity (UC) if for every ε>0, r>0, δρ(r,ε)>0.(ii)
We say thatρ satisfies unique uniform convexity (UUC) if for every s≥0, ε>0, there exists
(2.8)η(s,ε)>0,
depending on s and ε such that
(2.9)δρ(r,ε)>η(s,ε)>0,forr>s.Definition 2.6.
LetC⊂Xρ be convex and ρ-bounded.(a)
A functionτ:C→[0,∞] is called a (ρ)-type (or shortly a type) if there exists a sequence {ym} of elements of C such that for any z∈C there holds
(2.10)τ(z)=limsup⁡m→∞⁡ρ(ym-z).(b)
Letτ be a type. A sequence {gn} is called a minimizing sequence of τ if
(2.11)lim⁡n→∞⁡τ(gn)=inf⁡⁡{τ(h);h∈C}.The following definitions are straightforward generalizations of their norm and metric equivalents.Definition 2.7.
LetC⊂Xρ be nonempty and ρ-closed. A mapping T:C→C is called an asymptotic pointwise mapping if there exists a sequence of mapping αn:C→[0,∞) such that
(2.12)ρ(Tn(x)-Tn(y))≤αn(x)ρ(x-y),∀x,y∈Xρ.(i)
If{αn} converges pointwise to α:C→[0,1), then T is called an asymptotic pointwise contraction.(ii)
Iflimsupn→∞αn(x)≤1 for any x∈Xρ, then T is called an asymptotic pointwise nonexpansive mapping.(iii)
Iflimsupn→∞αn(x)≤k for any x∈Xρ with 0<k<1, then T is called a strongly asymptotic pointwise contraction.


## 3. Main Results

Lemma 3.1 (see [11]).
LetC be a ρ-closed ρ-bounded convex nonempty subset of Xρ, 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.
LetXρ be a modular space, let  C be a  ρ-bounded  ρ-closed convex nonempty subset of  Xρ, 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 fixedx∈C, define the type
(3.1)τ(h)=limsup⁡n→∞⁡ρ(Tn(x)-h),h∈C.
Letτ0=inf{τ(h);h∈C}. Let {fn}⊂C be a minimizing sequence of τ and fn→f∈C, which exists in view of Lemma 3.1.
Let us prove thatf is a fixed point of T.
First notice thatτ(Tm(h))≤αm(h)τ(h), for any h∈C and m≥1.
Indeed, for fixedx∈C,
(3.2)τ(Tm(h))=limsup⁡n→∞⁡ρ(Tn(x)-Tm(h))=limsup⁡n→∞⁡ρ(Tn+m(x)-Tm(h))=limsup⁡n→∞⁡ρ(TmTn(x)-Tm(h))≤limsup⁡n→∞⁡αm(h)ρ(Tn(x)-h)=αm(h)τ(h).
In particular, we have(3.3)τ(Tm(fn))≤αm(fn)τ(fn),∀n,m≥1.
By induction, we build an increasing sequence{mk} such that
(3.4)αmk+m(fk)≤1+1k,∀k,m≥1.
Indeed, sinceT is asymptotic pointwise nonexpansive mapping, we have limsupm→∞αm(f1)≤1, so there exists m1≥1, such that for any m≥m1, we have αm(f1)≤1+1/1. Since limsupm→∞αm(f2)≤1, there exists m2>m1 such that for any m≥m2, we have αm(f2)≤1+1/2. Assume mk is built, then since limsupm→∞αm(fk+1)≤1, there exists mk+1>mk such that for any m≥mk+1, we have αm(fk+1)≤1+1/(k+1), which completes our induction claim.
By (3.3) and Definition 2.6 (b), it is easy to observe that {Tmk+p(fk)} is a minimizing sequence of τ, for any p≥0. Lemma 3.1 implies {Tmk+p(fk)} is ρ-convergent to f, for any p≥0.
In particular, we have{Tmk+1(fk)} is ρ-convergent to f. Since
(3.5)ρ(Tmk+1(fk)-T(f))≤α1(f)ρ(Tmk(fk)-f),
we conclude that {Tmk+1(fk)} is also ρ-convergent to T(f).
Since theρ-limit of any ρ-convergent sequence is unique by Lemma 3.1, we must have T(f)=f.
In the following, we will prove thatFix(T) is ρ-closed.
To prove thatFix(T) is ρ-closed, let xn∈Fix(T) and ρ(xn-x)→0. Observe that
(3.6)ρ(13(T(x)-x))≤ρ(T(x)-T(xn))+ρ(T(xn)-xn)+ρ(xn-x)≤α1ρ(xn-x)+ρ(xn-x)⟶0.
Hence, x∈Fix(T), so Fix(T) is ρ-closed.
This completes the proof.Remark 3.3.
It is not hard to see that ifC is ρ-bounded, then an asymptotic pointwise nonexpansive mapping T is of asymptotic nonexpansive type [12]; that is, there exists a sequence {kn} of positive numbers with the property kn→1 as n→∞ and such that
(3.7)ρ(Tn(x)-Ty(y))≤knρ(x-y),
for all n and x,y∈C.



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*Source: 319394-2012-04-26.xml*