# One-Local Retract and Common Fixed Point in Modular Metric Spaces

Authors: Afrah A. N. Abdou
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

The notion of a modular metric on an arbitrary set and the corresponding
modular spaces, generalizing classical modulars over linear spaces
like Orlicz spaces, were recently introduced. In this paper we introduced and
study the concept of one-local retract in modular metric space. In particular,
we investigate the existence of common fixed points of modular nonexpansive
mappings defined on nonemptyω-closed ω-bounded subset of modular metric
space.


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## Body

## 1. Introduction

The purpose of this paper is to give an outline of a common fixed-point theory for nonexpansive mappings (i.e., mappings with the modular Lipschitz constant 1) on some subsets of modular metric spaces which are natural generalization of classical modulars over linear spaces like Lebesgue, Orlicz, Musielak-Orlicz, Lorentz, Orlicz-Lorentz, Calderon-Lozanovskii, and many other spaces. Modular metric spaces were introduced in [1, 2]. The main idea behind this new concept 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” λ>0 (the absolute value of) an average velocity ωλ(x,y) is associated in such a way that in order to cover the “distance” between points x,y∈X it takes time λ to move from x to y with velocity ωλ(x,y). But the way we approached the concept of modular metric spaces is different. Indeed we look at these spaces as the nonlinear version of the classical modular spaces introduced by Nakano [3] on vector spaces and Musielak-Orlicz spaces introduced by Musielak [4] and Orlicz [5].In recent years, there was an uptake interest in the study of electrorheological fluids, sometimes referred to as “smart fluids” (for instance, lithium polymethacrylate). For these fluids, modeling with sufficient accuracy using classical Lebesgue and Sobolev spaces,Lp and W1,p, where p is a fixed constant is not adequate, but rather the exponent p should be able to vary [6, 7]. One of the most interesting problems in this setting is the famous Dirichlet energy problem [8, 9]. The classical technique used so far in studying this problem is to convert the energy function, naturally defined by a modular, to a convoluted and complicated problem which involves a norm (the Luxemburg norm). The modular metric approach is more natural and has not been used extensively.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 recent years, there was a strong interest to study the fixed point property in modular function spaces after the first paper [10] was published in 1990. More recently, the authors presented a fixed point result for pointwise nonexpansive and asymptotic pointwise nonexpansive acting in modular functions spaces [11]. The theory of nonexpansive mappings defined on convex subsets of Banach spaces has been well developed since the 1960s (see, e.g., Belluce and Kirk [12], Browder [13], Bruck [14], and Lim [15]), and generalized to other metric spaces (see e.g., [16–18]), and modular function spaces (see e.g., [10]). The corresponding fixed-point results were then extended to larger classes of mappings like pointwise contractions, asymptotic pointwise contractions [18–22], and asymptotic pointwise nonexpansive mappings [11]. In [23], Penot presented an abstract version of Kirk’s fixed point theorem [24] for nonexpansive mappings. Many results of fixed point in metric spaces were developed after Penot’s formulation. Using Penot’s work, the author in [25] proved some results in metric spaces with uniform normal structure similar to the ones known in Banach spaces. In [26], Khamsi introduced the concept of one-local retract in metric spaces and proved that any commutative family of nonexpansive mappings defined on a metric space with a compact and normal convexity structure has a common fixed point. Recently in [27], the authors introduced the concept of one-local retract in modular function spaces and proved the existence of common fixed points for commutative mappings.In this paper, we study the concept of one-local retract in more general setting in modular metric space; therefore, we prove the existence of common fixed points for a family of modular nonexpansive mappings defined on nonemptyω-closed ω-bounded subsets in modular metric space.For more on metric fixed point theory, the reader may consult the book [28] and for modular function spaces the book [29].


## 2. Basic Definitions and Properties

LetX be a nonempty set. Throughout this paper for a function ω:(0,∞)×X×X→[0,∞], we will write
(1)ωλ(x,y)=ω(λ,x,y),
for all λ>0 and x,y∈X.Definition 1 (see [1, 2]).
A functionω:(0,∞)×X×X→[0,∞] is said to be modular metric on X if it satisfies the following axioms:  (i)
x=y if and only if ωλ(x,y)=0, for all λ>0;(ii)
ωλ(x,y)=ωλ(y,x), for all λ>0, and x,y∈X;(iii)
ωλ+μ(x,y)≤ωλ(x,z)+ωμ(z,y), for all λ,μ>0 and x,y,z∈X.If, instead of (i), we have only the condition (i′)
(2)ωλ(x,x)=0,∀λ>0,x∈X,
then ω is said to be a pseudomodular (metric) on X. A modular metric ω on X is said to be regular if the following weaker version of (i) is satisfied:
(3)x=yiffωλ(x,y)=0,forsomeλ>0.
Finally, ω is said to be convex if, for λ,μ>0 and x,y,z∈X, it satisfies the inequality
(4)ωλ+μ(x,y)≤λλ+μωλ(x,z)+μλ+μωμ(z,y).Note that, for a metric pseudomodularω on a set X, and any x,y∈X, the function λ→ωλ(x,y) is nonincreasing on (0,∞). Indeed, if 0<μ<λ, then
(5)ωλ(x,y)≤ωλ-μ(x,x)+ωμ(x,y)=ωμ(x,y).Definition 2 (see [1, 2]).
Letω be a pseudomodular on X. Fix x0∈X. The two sets:
(6)Xω=Xω(x0)={x∈X:ωλ(x,x0)⟶0asλ⟶∞},Xω*=Xω*(x0)={suchthatωλ(x,x0)<∞x∈X:∃λ=λ(x)>0suchthatωλ(x,x0)<∞}
are said to be modular spaces (around x0).It is clear thatXω⊂Xω* but this inclusion may be proper in general. It follows from [1, 2] that if ω is a modular on X, then the modular space Xω can be equipped with a (nontrivial) metric, generated by ω and given by
(7)dω(x,y)=inf{λ>0:ωλ(x,y)≤λ},
for any x,y∈Xω. If ω is a convex modular on X, according to [1, 2] the two modular spaces coincide, that is Xω*=Xω, and this common set can be endowed with the metric dω* given by
(8)dω*(x,y)=inf{λ>0:ωλ(x,y)≤1},
for any x,y∈Xω. These distances will be called Luxemburg distances (see example below for the justification).Definition 3.
LetXω be a modular metric space.  (1)
The sequence(xn)n∈ℕ in Xω is said to be ω-convergent to x∈Xω if and only if ω1(xn,x)→0, as n→∞. x will be called the ω-limit of (xn).(2)
The sequence(xn)n∈N in Xω is said to be ω-Cauchy if ω1(xm,xn)→0, as m,n→∞.(3)
A subsetC of Xω is said to be ω-closed if the ω-limit of a ω-convergent sequence of C always belongs to C.(4)
A subsetC of Xω is said to be ω-complete if any ω-Cauchy sequence in C is a ω-convergent sequence and its ω-limit is in C.(5)
Letx∈Xω and C⊂Xω. The ω-distance between x and C is defined as
(9)dω(x,C)=inf{ω1(x,y);y∈C}.(6)
A subsetC of Xω is said to be ω-bounded if we have
(10)δω(C)=sup{ω1(x,y);x,y∈C}<∞.In general iflimn→∞ωλ(xn,x)=0, for some λ>0, then we may not have limn→∞ωλ(xn,x)=0, for all λ>0. Therefore, as it is done in modular function spaces, we will say that ω satisfies Δ2 condition if this is the case; that is limn→∞ωλ(xn,x)=0, for some λ>0 implies limn→∞ωλ(xn,x)=0, for all λ>0. In [1, 2], one will find a discussion about the connection between ω-convergence and metric convergence with respect to the Luxemburg distances. In particular, we have
(11)limn→∞dω(xn,x)=0ifflimn→∞ωλ(xn,x)=0,∀λ>0,
for any {xn}∈Xω and x∈Xω. And in particular we have that ω-convergence and dω-convergence are equivalent if and only if the modular ω satisfies the Δ2-condition. Moreover, if the modular ω is convex, then we know that dω* and dω are equivalent which implies that
(12)limn→∞dω*(xn,x)=0ifflimn→∞ωλ(xn,x)=0,∀λ>0,
for any {xn}∈Xω and x∈Xω [1, 2]. Another question that arises in this setting is the uniqueness of the ω-limit. Assume ω is regular, and let {xn}∈Xω be a sequence such that {xn}  ω-converges to a∈Xω and b∈Xω. Then we have
(13)ω2(a,b)≤ω1(a,xn)+ω1(xn,b),
for any n≥1. Our assumptions will imply ω2(a,b)=0. Since ω is regular, we get a=b; that is, the ω-limit of a sequence is unique.Let(X,ω) be a modular metric space. Throughout the rest of this work, we will assume that ω satisfies the Fatou property; that is, if {xn}  ω-converges to x and {yn}  ω-converges to y, then we must have
(14)ω1(x,y)≤liminfn→∞ω1(xn,yn).
For any x∈Xω and r≥0, we define the modular ball
(15)Bω(x,r)={y∈Xω;ω1(x,y)≤r}.
Note that if ω satisfies the Fatou property, then modular balls (ω-balls) are ω-closed. An admissible subset of Xω is defined as an intersection of modular balls. We say A is an admissible subset of C if
(16)A=⋂i∈I‍Bω(bi,ri)∩C,
where bi∈C, ri≥0, and I is an arbitrary index set. Denote by 𝒜ω(Xω) the family of admissible subsets of Xω. Note that 𝒜ω(Xω) is stable by intersection. At this point we will need to define the concept of Chebyshev center and radius in modular metric spaces. Let A⊂X be a nonempty ω-bounded subset. For any x∈A, define
(17)rx(A)=sup{ω1(x,y);y∈A}.
The Chebyshev radius of A is defined by
(18)Rω(A)=inf{rx(A);x∈A}.
Obviously we have Rω(A)≤rx(A)≤δω(A), for any x∈A. The Chebyshev center of A is defined as
(19)𝒞ω(A)={x∈A;rx(A)=Rω(A)}.Definition 4.
Let(X,ω) be a modular metric space. Let C be a nonempty subset of Xω.  (i)
We will say that𝒜ω(C) is compact if any family (Aα)α∈Γ of elements of 𝒜ω(C) has a nonempty intersection provided ∩α∈FAα≠∅, for any finite subset F⊂Γ.(ii)
We will say that𝒜ω(C) is normal if  for any A∈𝒜ω(C), not reduced to one point, ω-bounded, we have Rω(A)<δω(A).Remark 5.
Note that if𝒜ω(Xω) is compact, then Xω is ω-complete.Definition 6.
Let(X,ω) be a modular metric space. Let C be a nonempty subset of Xω. A mapping T:C→C is said to be ω-nonexpansive if
(20)ω1(T(x),T(y))≤ω1(x,y)foranyx,y∈C.
For such mapping we will denote by Fix(T) the set of its fixed points; that is, Fix(T)={x∈C;T(x)=x}.In [1, 2] the author defined Lipschitzian mappings in modular metric spaces and proved some fixed point theorems. Our definition is more general. Indeed, in the case of modular function spaces, it is proved in [10] that
(21)ωλ(T(x),T(y))≤ωλ(x,y),foranyλ>0
if and only if dω(T(x),T(y))≤dω(x,y), for any x,y∈C. Next we give an example, which first appeared in [10], of a mapping which is ω-nonexpansive in our sense but fails to be nonexpansive with respect to dω.Example 7.
LetX=(0,∞). Define the Musielak-Orlicz function modular on the space of all Lebesgue measurable functions by
(22)ρ(f)=1e2∫0∞‍|f(x)|x+1dm(x).
Let B be the set of all measurable functions f:(0,∞)→ℝ such that 0≤f(x)≤1/2. Consider the map
(23)T(f)(x)={f(x-1),forx≥10,forx∈[0,1].
Clearly, T(B)⊂B. In [10], it was proved that, for every λ≤1 and for all f,g∈B, we have
(24)ρ(λ(T(f)-T(g)))≤λρ(λ(f-g)).
This inequality clearly implies that T is ω-nonexpansive. On the other hand, if we take f=1[0,1], then
(25)∥T(f)∥ρ>e≥∥f∥ρ,
which clearly implies that T is not dω-nonexpansive.Next we present the analog of Kirk’s fixed point theorem [24] in modular metric spaces.Theorem 8 (see [30]).
Let(X,ω) be a modular metric space and C be a nonempty ω-closed ω-bounded subset of Xω. Assume that the family 𝒜ω(C) is normal and compact. Let T:C→C be ω-nonexpansive. Then T has a fixed point.


## 3. One-Local Retract Subsets in Modular Metric Spaces

LetC be a nonempty subset of Xω. A nonempty subset D of C is said to be a one-local retract of C if, for every family {Bi;i∈I} of ω-balls centered in D such that C∩(∩i∈IBi)≠∅, it is the case that D∩(∩i∈IBi)≠∅. It is immediate that each ω-nonexpansive retract of Xω is a one-local retract (but not conversely). Recall that D⊂C is a ω-nonexpansive retract of C if there exists a ω-nonexpansive map R:C→D such that R(x)=x, for every x∈D.The result in [26] may be stated in modular metric spaces as follows.Theorem 9.
Let(X,ω) be a modular metric space and C be a nonempty ω-closed ω-bounded subset of Xω. Assume that 𝒜ω(C) is normal and compact. Then, for any ω-nonexpansive mapping T:C→C, the fixed point set Fix(T) is nonempty one-local retract of C.Proof.
Theorem8 shows that Fix(T) is nonempty. Let us complete the proof by showing that it is a one-local retract of C. Let {Bω(xi,ri)}i∈I be any family of ω-closed balls such that xi∈Fix(T), for  any i∈I, and
(26)C0=C∩(⋂i∈I‍Bω(xi,ri))≠∅.
Let us prove that Fix(T)∩(∩i∈IBω(xi,ri))≠∅. Since {xi}i∈I⊂Fix(T) and T is ω-nonexpansive, then T(C0)⊂C0. Clearly, C0∈𝒜ω(C) and is nonempty. Then we have 𝒜ω(C0)⊂𝒜ω(C). Therefore, 𝒜ω(C0) is compact and normal. Theorem 8 will imply that T has a fixed point in C0 which will imply
(27)Fix(T)∩(⋂i∈I‍Bω(xi,ri))≠∅.Now, we discuss some properties of one-local retract subsets.Theorem 10.
Let(X,ω) be a modular metric space. Let C be a nonempty ω-closed ω-bounded subset of Xω. Let D be a nonempty subset of C. The following are equivalent.  (i)
Dis a one-local retract ofC.(ii)
Dis aω-nonexpansive retract ofD∪{x}→D, for everyx∈C.Proof.
Let us prove(i)⇒(ii). Let x∈C. We may assume that x does not belong to D. In order to construct a ω-nonexpansive retract R:D∪{x}→D, we only need to find R(x)∈D such that
(28)ω1(R(x),y)≤ω1(x,y),foreveryy∈D.
Since x∈∩y∈DBω(y,ω1(x,y)) and x∈C, then
(29)C∩(⋂y∈D‍Bω(y,ω1(x,y)))≠∅.
Since D is one-local retract of C, we get
(30)D0=D∩(⋂y∈D‍Bω(y,ω1(x,y)))≠∅.
Any point in D0 will work as R(x).Next, we prove that(ii)⇒(i). In order to prove that D is a one-local retract of C, let {Bω(xi,ri)}i∈I be any family of ω-closed balls such that xi∈D, for any i∈I, and
(31)C0=C∩(⋂i∈I‍Bω(xi,ri))≠∅.
Let us prove that D∩(∩i∈IBω(xi,ri))≠∅. Let x∈C0. If x∈D, we have nothing to prove. Assume otherwise that x does not belong to D. Property (ii) implies the existence of a ω-nonexpansive retract R:D∪{x}→C. It is easy to check that R(x)∈D∩(∩i∈IBω(xi,ri))=∅, which completes the proof of our theorem.For the rest of this work, we will need the following technical result.Lemma 11.
Let(X,ω) be a modular metric space and C be a nonempty ω-closed ω-bounded subset of Xω. Let D be a nonempty one-local retract of C. Set coC(D)=C∩(∩{A;A∈𝒜ω(C)andD⊂A}). Then  (i)
rx(D)=rx(coC(D)), for anyx∈C;(ii)
Rω(coC(D))=Rω(D);(iii)
δω(coC(D))=δω(D).Proof.
Let us first prove(i). Fix x∈C. Since D⊂coC(D), we get rx(D)≤rx(coC(D)). On the other hand we have D⊂Bω(x,rx(D))∈𝒜ω(C). The definition of coC(D) implies coC(D)⊂Bω(x,rx(D)). Hence rx(coC(D))≤rx(D), which implies
(32)rx(coC(D))=rx(D).
Next, we prove (ii). Let x∈D. We have x∈coC(D). Using (i), we get
(33)rx(coC(D))=rx(D)≥Rω(coC(D)).
Hence, Rω(D)≥Rω(coC(D)). Next, let x∈coC(D). We have D⊂coC(D)⊂Bω(x,rx(coC(D))). Hence, x∈∩y∈DBω(y,rx(coC(D))). Hence
(34)C∩(⋂y∈DBω(y,rx(coC(D))))=∅.
Since D is a one-local retract of C, we get
(35)D0=D∩(⋂y∈DBω(y,rx(coC(D))))=∅.
Let y∈D0. Then it is easy to see that ry(D)≤rx(coC(D)). Hence Rω(D)≤rx(coC(D)). Since x was arbitrary taken in coC(D), we get
(36)Rω(D)≤Rω(coC(D)),
which implies
(37)Rω(D)=Rω(coC(D)).
Finally, let us prove (iii). Since D⊂coC(D), we get
(38)δω(D)≤δω(coC(D)).
Now, for any x∈D, we have
(39)D⊂Bω(x,δω(D)).
Hence
(40)coC(D)⊂Bω(x,δω(D)).
This implies
(41)x∈⋂y∈coC(D)Bω(y,δω(D)).
Since x was taken arbitrary in D, we get
(42)D⊂⋂y∈coC(D)Bω(y,δω(D)).
The definition of coC(D) implies
(43)coC(D)∈⋂y∈coC(D)Bω(y,δω(D)).
So for any x,y∈coC(D), we have
(44)ω1(x,y)≤δω(D).
Hence
(45)δω(coC(D))≤δω(D),
which implies
(46)δω(coC(D))=δω(D).As an application of this lemma we have the following result.Theorem 12.
Let(X,ω) be a modular metric space and C be a nonempty ω-closed ω-bounded subset of Xω. Assume that 𝒜ω(C) is normal and compact. If D is a nonempty one-local retract of C, then 𝒜ω(D) is compact and normal.Proof.
Using the definition of one-local retract, it is easy to see that𝒜ω(D) is compact. Let us show that 𝒜ω(D) is normal. Let A0∈𝒜ω(D) be nonempty and reduced to one point. Set
(47)coC(A0)=C∩(∩{A;A∈𝒜ω(C)andA0⊂A}).
Then from Lemma 11, we get
(48)Rω(coC(A0))=Rω(A0),δω(coC(A0))=δω(A0).
Since coC(A0)∈𝒜ω(C), then we must have
(49)Rω(coC(A0))<δω(coC(A0)),
because 𝒜ω(C) is normal. Therefore, we have
(50)Rω(A0)<δω(A0),
which completes the proof of our claim.The following result has found many application in metric spaces. Most of the ideas in its proof go back to Baillon’s work [31].Theorem 13.
Let(X,ω) be a modular metric space and C be a nonempty ω-closed ω-bounded subset of Xω. Assume that 𝒜ω(C) is normal and compact. Let (Cβ)β∈Γ be a decreasing family of one-local retracts of C, where (Γ,≺) is totally ordered. Then ∩β∈ΓCβ is not empty and is one-local retract of C.Proof.
Consider the family(51)ℱ={∏β∈Γ‍Aβ:Aβ∈𝒜ω(Cβ),(Aβ)isdecreasing}.ℱ is not empty since ∏β∈Γ‍Cβ∈ℱ. ℱ will be ordered by inclusion; that is, ∏β∈Γ‍Aβ⊂∏β∈Γ‍Bβ if and only if Aβ⊂Bβ for any β∈Γ. From Theorem 12, we know that 𝒜ω(Cβ) is compact, for every β∈Γ. Therefore, ℱ satisfies the hypothesis of Zorn’s Lemma. Hence for every D∈ℱ, there exists a minimal element A∈ℱ such that A⊂D. We claim that if A=∏β∈Γ‍Aβ is minimal, then there exists β0∈Γ such that δω(Aβ)=0, for every β>β0. Assume not, that is, δω(Aβ)>0, for every β∈Γ. Fix β∈Γ. For every K⊂C, set
(52)coβ(K)=⋂x∈CβBω(x,rx(K)).
Consider, A′=∏α∈Γ‍Aα′ where
(53)Aα′=coβ(Aβ)∩Aαifα≤β,Aα′=Aαifα≥β.
The family (Aα≥β′) is decreasing since A∈ℱ. Let α≤γ≤β. Then Aγ′⊂Aα′, since Aγ⊂Aα and Aβ=coβ(Aβ)∩Aβ. Hence the family (Aα′) is decreasing. On the other hand if α≺β, then coβ(Aβ)∩Aα∈𝒜ω(Cα) since Cβ⊂Cα. Hence Aα′∈𝒜ω(Cα). Therefore, we have A′∈ℱ. Since A is minimal, then A=A′. Hence
(54)Aα=coβ(Aβ)∩Aα,foreveryα<β.
Let x∈Cβ and α<β. Since Aβ⊂Aα, then
(55)rx(Aβ)≤rx(Aα).
Because coβ(Aβ)=∩y∈CβBω(y,ry(Aβ)), then we have
(56)coβ(Aβ)⊂Bω(y,ry(Aβ)),
which implies
(57)ry(Aβ)≤ry(Aα).
Since Aα⊂coβ(Aβ), then
(58)ry(Aβ)≤ry(Aα)≤ry(coβ(Aβ))≤ry(Aβ).
Therefore, we have
(59)ry(Aα)≤ry(Aβ),foreveryy∈Cβ.
Using the definition of Chebyshev radius Rω, we get
(60)Rω(Aα)≤Rω(Aβ).
Let x∈Aα and set s=rx(Aα). Then x∈coβ(Aβ) since Aα⊂coβ(Aβ). Hence,
(61)x∈(⋂y∈AβBω(y,s))∩coβ(Aβ).
Since Cβ is one-local retract of C, then
(62)Sβ=Cβ∩(⋂y∈AβBω(y,s))∩coβ(Aβ)≠∅.
Since Aβ=Cβ∩coβ(Aβ), then we have
(63)Sβ=Aβ∩(⋂y∈AβBω(y,s)).
Let h∈Sβ, then h∈∩y∈AβBω(y,s). Hence, rh(Aβ)≤s, which implies
(64)Rω(Aβ)≤s=rx(Aα),foreveryx∈Aα.
Hence, Rω(Aβ)≤Rω(Aα). Therefore, we have
(65)Rω(Aβ)=Rω(Aα),foreveryα,β∈Γ.
Since δω(Aβ)>0, for every β∈Γ, set Aβ′′ to the Chebyshev center of Aβ, that is, Aβ′′=Cω(Aβ), for every β∈Γ. Since Rω(Aβ)=Rω(Aα), for every α,β∈Γ, then the family (Aβ′′) is decreasing. Indeed, let α≺β and x∈Aβ′′. Then we have rx(Aβ)=Rω(Aβ). Since we proved that
(66)ry(Aβ)=ry(Aα),foreverya∈Cβ,
then
(67)rx(Aα)=rx(Aβ)=Rω(Aβ)=Rω(Aα),
which implies that x∈Aα′′. Therefore, we have A′′=∏β∈Γ‍Aβ′′∈ℱ. Since A′′⊂A and A is minimal, we get A=A′′. Therefore, we have Cω(Aβ)=Aβ for every β∈Γ. This contradicts the fact that 𝒜ω(Cβ) is normal for every β∈Γ. Hence there exists β0∈Γ such that
(68)δω(Aβ)=0,foreveryβ≻β0.
The proof of our claim is therefore complete. Then we have Aβ={x}, for every β≻β0. This clearly implies that x∈∩β∈ΓCβ≠∅. In order to complete the proof, we need to show that S=∩β∈ΓCβ is one-local retract of C. Let (Bi)i∈I be a family of ω-balls centered in S such that ∩i∈I(Bi)≠∅. Set
(69)Dβ=(⋂i∈IBi)∩Cβ,foranyβ∈Γ.
Since Cβ is a one-local retract of C and the family (Bi) is centered in Cβ, then Dβ is not empty and Dβ∈𝒜ω(Cβ). Therefore,
(70)D=∏β∈Γ‍Dβ∈ℱ.
Let A=∏β∈Γ‍Aβ⊂D be a minimal element of ℱ. The above proof shows that
(71)⋂β∈Γ‍Aβ⊂⋂β∈Γ‍Dβ≠∅.
The proof of our theorem is complete.The next theorem will be useful to prove the main result of the next section.Theorem 14.
Let(X,ω) be a modular metric space and C be a nonempty ω-closed ω-bounded subset of Xω. Assume that 𝒜ω(C) is normal and compact. Let (Cβ)β∈Γ be a family of one-local retracts of C such that for any finite subset I of Γ. Then ∩β∈ΓCβ is not empty and is one-local retract of C.Proof.
Consider the familyℱ of subsets I⊂Γ such that, for any finite subset J⊂Γ (empty or not), we have ∩α∈I∪JCα that is nonempty one-local retract of C. Note that ℱ is not empty since any finite subset of Γ is in ℱ. Using Theorem 13, we can show that ℱ satisfies the hypothesis of Zorn’s lemma. Hence ℱ has a maximal element I⊂Γ. Assume I≠Γ. Let α∈Γ∖I. Obviously we have I∪{α}∈ℱ. This is a clear contradiction with the maximality of I. Therefore we have I=Γ∈ℱ; that is, ∩β∈ΓCβ is not empty and is a one-local retract of C.


## 4. Common Fixed Point Result

In this section we discuss the existence of a common fixed point of a family of commutativeω-nonexpansive mappings in modular metric space which either generalize or improve the corresponding recent common fixed point results of [26, 27].First, we will need to discuss the case of finite families.Theorem 15.
Let(X,ω) be a modular metric space and C be a nonempty ω-closed ω-bounded subset of Xω. Assume that 𝒜ω(C) is normal and compact. Let ℱ={T1,T2,…,Tn} be a family of commutative ω-nonexpansive mappings defined on C. Then the family ℱ has a common fixed point. Moreover, the common fixed point set Fix(ℱ) is a one-local retract of C.Proof.
First, let us prove thatFix(ℱ) is not empty. Using Theorem 9, Fix(T1) is nonempty one-local retract of C, and then Theorem 12 implies that 𝒜ω(Fix(T1)) is compact and normal. On the other hand since T1 and T2 are commutative, we have
(72)T2(Fix(T1))⊂Fix(T1).
Hence T2 has a fixed point in Fix(T1). If we restrict ourselves to Fix(T1,T2), the common fixed point set of T1 and T2, then one can prove in an identical argument that T3 has a fixed point in Fix(T1,T2). Step by step, we can prove that the common fixed point set Fix(ℱ) of T1,T2,…,Tn is not empty. The same argument used to prove that the fixed point set of ω-nonexpansive map is a one-local retract can be reduced here to prove that Fix(ℱ) is one-local retract.The following result extends [26, Theorem 8] to the setting of modular metric space.Theorem 16.
Let(X,ω) be a modular metric space and let C be a nonempty ω-closed ω-bounded subset of Xω. Assume that 𝒜ω(C) is normal and compact. Let ℱ=(Ti)i∈I be a family of commutative ω-nonexpansive mappings defined on C. Then the family ℱ has a common fixed point. Moreover, the common fixed point set Fix(ℱ) is a one-local retract of C.Proof.
LetΓ={β:βbeanonemptyfinitesubsetofI}. Theorem 15 implies that, for every β∈Γ, the set Fβ=∩i∈βFix(Ti) of common fixed point set of the mappings Ti, i∈β, is nonempty one-local retract of C. Clearly the family (Fβ)β∈Γ is decreasing and satisfies the assumptions of Theorem 14. Therefore, we deduced that ∩β∈ΓFβ is nonempty and is a one-local retract of C.



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*Source: 672069-2013-09-23.xml*