One-local retract and common fixed point in modular function spaces

R E S E A R C H

Open Access

One-local retract and common fixed point in

modular function spaces

Saleh Abdullah Al-Mezel

, Abdullah Al-Roqi

and Mohamed Amine Khamsi

* Correspondence: salmezel@kau. edu.sa

1_{Department of Mathematics, King}

Abdulaziz University, PO Box 80203, Jeddah 21589, Saudi Arabia Full list of author information is available at the end of the article

Abstract

In this paper, we introduce and study the concept of one-local retract in modular function spaces. In particular, we prove that any commutative family of

r-nonexpansive mappings defined on a nonempty,r-closed andr-bounded subset of a modular function space has a common fixed point provided its convexity structure of admissible subsets is compact and normal.

MSC: Primary 47H09; Secondary 46B20, 47H10.

Keywords:convexity structure, fixed point, modular function space, nonexpansive mappings, normal structure, retract

Introduction

The purpose of this paper is to give an outline of a common fixed point theory for nonexpansive mappings defined on some subsets of modular function spaces. These spaces are natural generalizations of both function and sequence variants of many important, from applications perspective, spaces like Lebesgue, Orlicz, Musielak-Orlicz, Lorentz, Orlicz-Lorentz, Calderon-Lozanovskii spaces and many others. The current paper operates within the framework of convex function modulars. The importance for applications of nonexpansive mappings in modular function spaces consists in the rich-ness of structure of modular function spaces, that-besides being Banach spaces (or

F-spaces in a more general settings)-are equipped with modular equivalents of norm or metric notions, and also are equipped with almost everywhere convergence and convergence in submeasure. 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. There are also important results that can be proved only using the apparatus of modular function spaces. From this perspective, the fixed point theory in modular function spaces should be considered as complementary to the fixed point theory in normed spaces and in metric spaces.

The theory of contractions and nonexpansive mappings defined on convex subsets of Banach spaces has been well developed since the 1960s (see e.g. [1-6]), and generalized to other metric spaces (see e.g. [7-9]), and modular function spaces (see e.g. [10-12]).

In this paper, we invesigate the structure of the fixed point set ofr-nonexpansive mappings. In particular, we introduce and investigate the concept of one-local retracts

in the framework of modular function spaces. Then we show a common fixed point in this setting.

Preliminaries

LetΩ be a nonempty set and∑be a nontrivial s-algebra of subsets ofΩ. Let P be a δ-ring of subsets ofΩ, such that E∩A∈P for any E∈P andAÎ∑. Let us assume

that there exists an increasing sequence of sets Kn∈P such thatΩ=∪Kn. Byℰwe

denote the linear space of all simple functions with supports from _P. Byℳ_∞we will denote the space of all extended measurable functions, i.e. all functionsf:Ω® [-∞,∞] such that there exists a sequence{gn}⊂ℰ, |gn|≤ |f| andgn(ω)®f(ω) for allω ÎΩ By 1Awe denote the characteristic function of the setA.

Definition 2.1. Letr:ℳ∞® [0,∞] be a notrivial, convex and even function. We say

thatris a regular convex function pseudomodular if: (i) r(0) = 0;

(ii)ris monotone, i.e. |f(ω)|≤|g(ω)| for all ωÎ Ωimpliesr(f)≤r(g), wheref, gÎ

ℳ∞;

(iii) ris orthogonally subadditive, i.er(f1A∪B)≤r(f1A)+r(f1B) for anyA, BÎ ∑such that A ∩B≠∅,fÎℳ;

(iv) r has the Fatou property, i.e. |fn(ω)|↑|f(ω)| for all ω Î Ωimplies r(fn) ↑r(f), wherefÎℳ_∞;

(v)ris order continuous inℰ, i.e.gnÎℰand|gn(ω)|↓ 0 impliesr(gn)↓0.

Similarly as in the case of measure spaces, we we say that a set AÎ ∑ isr-null if

r(g1A) = 0 for everyg Î ℰ. We say that a property holdsr-almost everywhere if the exceptional set is r-null. As usual we identify any pair of measurable sets whose sym-metric difference isr-null as well as any pair of measurable functions differing only on ar-null set. With this in mind we define

M(, , P, ρ) ={f ∈M_∞;|f(ω)| <∞ρ−a.e}, (2:1)

where each f ∈M(, , P, ρ) is actually an equivalence class of functions equal

r-a.e. rather than an individual function. Where no confusion exists we will writeℳ instead of M(, , P, ρ).

Definition 2.2. Letrbe a regular function pseudomodular.

(1) We say thatris a regular convex function semimodular ifr(af) = 0 for everya> 0 impliesf= 0r-a.e.;

(2) We say thatris a regular convex function modular ifr(f) = 0 impliesf= 0r- a.e.; The class of all nonzero regular convex function modulars defined on Ω will be denoted byℜ.

Let us denote r(f, E) =r(f1E) forfÎ ℳ, EÎ ∑. It is easy to prove thatr(f, E) is a function pseudomodular in the sense of Def. 2.1.1 in [13] (more precisely, it is a func-tion pseudomodular with the Fatou property). Therefore, we can use all results of the standard theory of modular function spaces as per the framework defined by Kozlowski in [13-15], see also Musielak [16] for the basics of the general modular theory.

Definition 2.3. [13-15] Letrbe a convex function modular.

(a) A modular function space is the vector space Lr(Ω,∑), or brieflyLr, defined by

(b) The following formula defines a norm in Lr(frequently called Luxemburg norm): f_ρ= inf{α >0;ρ(f/α)≤1}.

In the following theorem, we recall some of the properties of modular spaces that will be used later on in this paper.

Theorem 2.1. [13-15]LetrÎℜ.

(1)Lr, ||f||ris complete and the norm|| · ||ris monotone w.r.t. the natural order in

ℳ.

(2) || fn||r®0if and only if r(afn)®0for everya> 0.

(3) Ifr(afn)®0 for ana> 0then there exists a subsequence{gn}of{fn}such that gn® 0r- a.e.

(4)If{fn}converges uniformly to f on a set E∈P thenr(a(fn- f), E)® 0for every

a > 0.

(5)Let fn®fr- a.e. There exists a nondecreasing sequence of sets Hk∈P such that Hk↑Ωand{fn}converges uniformly to f on every Hk(Egoroff Theorem).

(6)r(f)≤lim infr(fn)whenever fn®fr- a.e. (Note: this property is equivalent to the

Fatou Property).

(7) Defining L0

ρ={f ∈Lρ; ρ(f,·) is order continuous} and

Eρ={f ∈Lρ; λf ∈L0_ρfor everyλ >0} we have:

(a) L_ρ⊃L0_ρ⊃E_ρ,

(b)Erhas the Lebesgue property, i.e.r(af, Dk)®0 fora> 0,fÎErand Dk↓∅. (c)Eris the closure ofℰ(in the sense of|| · ||r).

The following definition plays an important role in the theory of modular function spaces.

Definition 2.4. LetrÎ ℜ. We say thatrhas theΔ2-property if sup

n ρ

(2fn, Dk)→0

wheneverDk↓∅and sup

n ρ

(fn, Dk)→0_.

Theorem 2.2.LetrÎℜ.The following conditions are equivalent:

(a)rhasΔ2,

(b) L0

ρ is a linear subspace of Lr,

(c) L_ρ=L0

ρ=Eρ,

(d) ifr(fn)® 0thenr(2fn)®0,

(e) if r(afn) ® 0 for an a > 0 then || fn||r ® 0, i.e. the modular convergence is equivalent to the norm convergence.

The following definition is crucial throughout this paper. Definition 2.5. LetrÎ ℜ.

(a) We say that {fn} isr-convergent tofand writefn®0 (r) if and only ifr(fn- f)®0. (b) A sequence {fn} wherefnÎLris calledr-Cauchy ifr(fn- fm)®0 as n, m®∞. (c) A set B⊂ Lris called r-closed if for any sequence of fnÎ B, the convergence

fn®f(r) implies thatfbelongs toB.

(d) A set B⊂Lris calledr-bounded if sup{r(f - g);fÎB, gÎB}<∞

(e) Let fÎ Lrand C⊂Lr. Ther-distance betweenfandCis defined as

Let us note that r-convergence does not necessarily implyr-Cauchy condition. Also,

fn® fdoes not imply in generallfn®lf,l> 1. Using Theorem 2.1 it is not difficult to prove the following

Proposition 2.1.LetrÎℜ.

(i) Lris r-complete,

(ii)r-balls Br(x, r) = {yÎLr;r(x - y)≤r}arer-closed.

The following property plays in the theory of modular function spaces a role similar to the reflexivity in Banach spaces (see e.g. [11]).

Definition 2.6. We say thatLrhas property (R) if and only if every nonincreasing

sequence {Cn} of nonempty,r-bounded,r-closed, convex subsets ofLrhas nonempty

intersection.

Throughout this paper we will need the following. Definition 2.7. LetrÎ ℜandC⊂Lrbe nonempty.

(a) By the r-diameter ofC, we will understand the number

δρ(C) = sup{ρ(f−g);f,g∈C}.

The subset Cis said to be r-bounded wheneverδr(C) <∞.

(b) The quantityrr(f, C) = sup{r(f-g);gÎC} will be called the r-Chebyshev radius

ofCwith respect tof.

(c) Ther-Chebyshev radius ofCis defined by Rr(C) = inf {rr(f, C);fÎC}.

(d) The r-Chebyshev center ofCis defined as the set

Cρ(C) ={f ∈C;rρ(f, C) =Rρ(C)}.

Note thatRr(C)≤rr(f, C)≤δr(C) for allfÎCand observe that there is no reason,

in general, for Cρ(C) to be nonempty.

Let us finish this section with the modular definitions of r-nonexpansive mappings. The definitions are straightforward generalizations of their norm and metric equivalents.

Definition 2.8. LetrÎℜandC⊂Lrbe nonempty andr-closed. A mappingT: C®C

is called ar-nonexpansive mapping if

ρ(T(f)−T(g))≤ρ(f −g)for any f, g∈C.

A point fÎCis called a fixed point ofT wheneverT(f) =f. The set of fixed point of

T is denoted by Fix(T).

Penot compactness of admissible sets The following definition is needed.

Definition 2.9. LetrÎℜandC⊂Lrbe nonempty and r-bounded. We say thatA

is an admissible subset ofCif

A= i∈I

B_ρ(bi, ri)∩C,

wherebiÎC, ri≥0 andIis an arbitrary index set. By A(C) we denote the family of all admissible subsets ofC.

Definition 2.10. LetrÎℜandC⊂Lrbe nonempty.

(1) We will say that A(C) is r-normal if for any nonempty A∈A(C), which has more than one point, we haveRr(A)<δr(A).

(2) We will say that A(C) is compact if for any family {Aα}α∈ ⊂A(C) we have

α∈

Aα=∅,

provided that

α∈F

A_α=∅ _{for any finite subsetFofΓ.}

Clearly if A(L_ρ)is compact, thenLrhas property (R). In [18], the authors discussed

the concept of uniform convexity in modular function spaces. In particular they proved that uniform convexity implies the property (R). Next, we show that uniform convexity implies compactness in the sense of Penot [17] of the family of convex sets. First, let us recall the definition of uniform convexity in modular function spaces. For more on this, the reader may consult [18].

Definition 2.11. LetrÎℜ. (i) Letr> 0, ε> 0. Define

D(r, ε) =

(f, g);f, g∈Lρ, ρ(f)≤r, ρ(g)≤r, ρ

f−g

2

≥εr

.

Let

δ(r, ε) = inf

1− 1

rρ

f +g

2

; (f, g)∈D(r, ε)

, if D(r,ε)=∅,

andδ(r,ε) = 1if D(r,ε) =∅. We say thatrsatisfies (UC) if for everyr> 0,ε> 0,δ(r,ε) > 0. Note, that for everyr> 0,D(r,ε)≠∅, forε> 0 small enough.

(ii) We say thatrsatisfies (UUC) if for everys≥0,ε> 0 there exists

η(s, ε)>0

depending onsand εsuch that

δ(r, ε)> η(s, ε)>0 for r>s.

(iii) We say that ris Strictly Convex, (SC), if for everyf, gÎLrsuch that r(f) = r(g)

and

ρ

f+g

2

= ρ(f) +ρ(g) 2

there holdsf=g.

Note that in [11], the authors proved that in Orlicz spaces over a finite, atomless measure space, both conditions (UC) and (UUC) are equivalent. Typical examples of Orlicz functions that do not satisfy theΔ2 condition but are uniformly convex are:1

(t) =e|t|-|t|-1 and _ϕ2(t) =et

2

ρ(f −g0) =dρ(f, C) = inf{ρ(f−g);g∈C}.

Moreover it is also shown in [18] that if rÎℜis (UUC), then for any nonincreasing sequence {Cn} of nonempty, convex, andr-closed subsets ofLr, we have∩n≥1Cn≠∅, provided there exists fÎLrsuch that sup

n≥1

d_ρ(f, Cn)<∞_{. The authors in [18] did not}

show that such conclusion is still valid for any decreasing family. A property useful to get the compactness of the admissible subsets.

Theorem 2.3. Let r Î ℜ. Assume r Î ℜ is (UUC). Let{Ca}aÎΓ be a decreasing family of nonempty, convex, r-closed subsets of Lr, where(Γ,≺)is upward directed. Assume that there exists fÎ Lrsuch that sup

α∈ dρ(f, Cα)<∞.Then,∩aÎΓCa≠∅. Proof. Set d= sup

α∈ dρ(f, Cα). Without loss of generality, we may assume d> 0. For

Anyn≥1, there existsanÎΓsuch that

d

1−1

n

<d_ρ(f, C_αn)≤d.

Since (Γ,≺) is upward directed, we may assume an ≺an+1. In particular we have

Cαn+1 ⊂Cαn for anyn≥ 1. Since ris (UUC), we get C0=∩n≥1Cαn =∅. Clearly C0 is

r-closed and

d_ρ(f, C0) = sup

n≥1

d_ρ(f, C_αn) =d.

Again using the property (UUC) satisfied byr, there existsg0Î C0 unique such that

dr(f, C0) = r(f - g0). Let us prove thatg0ÎCafor any aÎΓ. FixaÎΓ. If for somen

≥1 we havea≺an, then obviously we have g0∈C_αn ⊂Cα.

Therefore let us assume that a⊀ an, for anyn≥1. SinceΓ is upward directed, there exists bnÎΓ such thatan≺bnanda ≺bn for anyn≥1. We can also assume thatbn

≺bn+1for any n≥ 1. Again we have C1=∩n≥1Cβn=∅. Since Cβn⊂Cαn, for anyn≥ 1, we get C1 ⊂C0. Moreover we have

d=dρ(f, C0)≤dρ(f, C1) = sup n≥1

dρ(f, Cβn)≤d.

Hence, dr(f, C1) =dwhich implies the existence of a unique pointg1 ÎC1such that

dr(f, C1) =r(f - g1) =d. Since ris uniformly convex, it must be (SC). Hence, g0 =g1.

In particular, we have g0∈C_βn, for any n≥1. Since a≺bn, we get C_βn ⊂Cα, for any

n ≥1, which impliesg0 Î Ca. Since awas taking arbitrary in Γ, we getg0Î ∩aÎΓCa,

which implies∩_aÎΓC_a≠∅. □

Since ris convex,r-closed balls are convex. Theorem 2.3 implies the following. Corollary 2.1. LetrÎℜand C⊂Lrbe nonempty, convex,r-closed, andr-bounded. Assume ris (UUC).Then A(C) is compact.

Remark2.1. Note that under the above assumptions, A(C) isr-normal. Indeed let A∈A(C) nonempty and not reduced to one point. Letf, g ÎAsuch thatf≠g. Then

ρ(f−g

ρ

h−f +g

2

≤(1−η)δ_ρ(A).

Hence, r_ρ

f+g

2 ,A

≤(1−η)δ_ρ(A), which impliesRr(A)<δr(A).

Finally, we state Penot’s formulation of Kirk’s fixed point theorem in modular func-tion spaces. For the sake of completeness we will give its proof.

Theorem 2.4. LetrÎℜand C⊂Lrbe nonempty,r-closed, andr-bounded. Assume that A(C) is compact and r-normal. Then anyr-nonexpansive T: C® C has a fixed point.

Proof. Since Cisr-bounded, then we have C∈A(C). Since A(C) is compact, the family F={A∈A(C);T(A)⊂A} has a minimal elementK. Set

K0= ∩{A;A∈A(C) andT(K)⊂A}∩K.

Note that T(K)⊂K0. This implies thatK0is nonempty and belongs to A(C).

More-over since K0 ⊂ K, we getT(K0)⊂T(K)⊂ K0. Hence K0 Î ℱ. The minimality of K

implies that K=K0. Next letf ÎK. By definition of ther-Chebyshev radius rr(f, K),

we have K⊂ Br(f, rr(f, K)). SinceT isr-nonexpansive, we haveT(K)⊂Br(T(f), rr(f, K)). The definition of K0 impliesK0 ⊂Br(T(f),rr(f, K)). SinceK=K0, we getK⊂Br

(T (f), rr(f, K)), which implies rr(T(f), K)≤ rr(f, K). Fix fÎKand setr =rr (f, K). We

have

K1={g∈K;r_ρ(g, K)≤r}= h∈K

B_ρ(h, r)∩K.

Clearly, we have T(K1)⊂ K1 and K1∈A(C). Since K is minimal, we get K= K1

which implies that ther-Chebyshev radiusrr(f, K) is constant. In particular, we have rr(f, K) =δr(K), for anyfÎ K. Since A(C) isr-normal, we conclude thatKdoes not

have more than one point. Therefore, K= {f} which forcesT(f) =f. □

In the next section, we investigate the structure of the fixed point set ofr -nonexpan-sive mappings.

One-local retract subsets in modular function spaces

Let rÎ ℜandC⊂Lrbe nonempty. A nonempty subset DofCis said to be a one-local retract ofCif for every family {Bi; i ÎI} of r-balls centered inDsuch thatC∩

(∩iÎIBi)≠∅, it is the case thatD∩(∩iÎIBi)≠∅. It is immediate that eachr -nonex-pansive retract ofLr is a one-local retract (but not conversely). Recall thatD⊂Cis a r-nonexpansive retract of Cif there exists ar-nonexpansive map R: C®Dsuch that

R(f) =f, for every fÎ D.

The following result will shed some light on the interest generated around this concept.

Theorem 2.5. LetrÎℜand C⊂Lrbe nonempty,r-closed, andr-bounded. Assume that A(C) is compact andr-normal. Then for anyr-nonexpansive mapping T: C®C, the fixed point set Fix(T)is a nonempty one-local retract of C.

C0=C∩

i∈I

B_ρ(fi, ri)

=∅.

Let us prove that Fix (T)∩(∩iÎIBr(fi, ri)) ≠∅. Since {fi}iÎI⊂Fix(T), andT isr -non-expansive, then T(C0) ⊂ C0. Clearly, C0∈A(C) and is nonempty. Then we have

A(C0)⊂A(C). Therefore, A(C0) is compact andr-normal. Theorem 2.4 will imply thatT has a fixed point inC0which will imply

Fix(T)∩

i∈I

B_ρ(fi, ri)

=∅.

□

This result gives some information to the structure of the fixed point set. To the best of our knowledge this is the first attempt done in modular function spaces. Next we discuss some properties of one-local retract subsets.

Theorem 2.6.LetrÎ ℜand C⊂Lrbe nonempty. Let D be a nonempty subset of C. The following are equivalent.

(i) D is a one-local retract of C.

(ii)D is a r-nonexpansive retract of D∪{f},for every fÎ C.

Proof. let us prove (i)⇒(ii). LetfÎ C. We may assume thatf∉D. In order to con-struct ar-nonexpansive retractR: D ∪{f}® D, we only need to findR(f) Î Dsuch that

ρ(R(f)−g)≤ρ(f−g), for everyg∈D.

Since fÎ ∩gÎDBr(g,r(f-g)) andfÎC, then

C∩

⎛ ⎝

g∈D

B_ρ(g, ρ(f−g)) ⎞ ⎠=∅.

Since Dis a one-local retract ofC, we get

D0=D∩

⎛ ⎝

g∈D

B_ρ(g, ρ(f −g)) ⎞ ⎠=∅.

Any point inD0will work asR(f).

Next, we prove that (ii)⇒ (i). In order to prove thatDis a one-local retract ofC, let {Br(fi, ri)}iÎIbe any family ofr-closed balls such thatfiÎD, for anyiÎ I, and

C0=C∩

i∈I

B_ρ(fi, ri)

=∅.

Let us prove that D∩(∩iÎIBr(fi, ri)) = ∅. Let fÎ C0. If fÎ D, we have nothing to

prove. Assume otherwise that f∉ D. Property (ii) implies the existence of ar -nonex-pansive retractR: D∪ {f}® C. It is easy to check thatR(f)ÎD∩(∩iÎIBr(fi, ri)) = ∅, which completes the proof of our theorem. □

Lemma 2.1.Letr Îℜand C⊂Lrbe nonempty, andr-bounded. Let D be a none-mpty one-local retract of C. Set coC(D) =C∩(∩{A;A∈A(C) and D⊂A}).Then

(i) rr(f, D) =rr(f, coC(D)),for any fÎC;

(ii)Rr(coC(D)) =Rr(D);

(iii)δr(coC(D)) =δr(D).

Proof. Let us first prove (i). FixfÎ C. Since D⊂coC(D), we getrr(f, D)≤rr(f, coC (D)). Setr =rr(f, D). We have D⊂Bρ(f, r)∈A(C). The definition of coC(D) implies

coC(D)⊂Br(f, r). Hence rr(f, coC(D))≤r =rr(f, D), which implies rr(f, D) =rr(f, coC (D)).

Next, we prove (ii). LetfÎ D. We havefÎ coC(D). Using (i), we getrr(f, D) = rr(f, coC(D))≥Rr(coC(D)). Hence,Rr(D)≥ Rr(coC(D)). Next, letfÎ coC(D) and setr =rr(f, coC(D)). We haveD⊂coC(D)⊂Br(f, r). Hence,fÎ ∩gÎDBr(g, r). Hence,C∩(∩gÎDBr (g, r)) =∅. Since Dis a one-local retract of C, we get D0=D∩ (∩gÎDBr(g, r)) =∅. LetgÎ D0. Then it is easy to see that rr(g, D)≤r. Hence,Rr(D)≤r. Sincefwas

arbi-trary taken incoC(D), we getRr(D)≤Rr(coC(D)), which implies Rr(D) =Rr(coC(D)).

Finally, let us prove (iii). Since D⊂coC(D), we getδr(D)≤δr(coC(D)). Next setd=

δr(D). Then, for anyfÎD, we have D⊂Br(f, d). HencecoC(D)⊂Br(f, d). This implies

f ∈ ∩g∈coC(D))Bρ(g,d). Sice fwas taken arbitrary in D, we get D⊂ ∩g∈coC(D))Bρ(g, d).

The definition ofcoC(D) implies coC(D)⊂ ∩g∈coC(D))Bρ(g, d). So for anyf, gÎ coC(D), we have r(f - g)≤d. Hence δr(coC(D))≤d=δr(D), which impliesδr(D) =δr(coC(D)).

□

As an application of this lemma we get the following result.

Theorem 2.7. LetrÎℜand C⊂Lrbe nonempty,r-closed, andr-bounded. Assume that A(C) is compact andr-normal. If D is a nonempty one-local retract of C, then

A(D) is compact andr-normal.

Proof. Using the definition of one-local retract, it is easy to see that A(D) is com-pact. Let us show that A(D) isr-normal. Let A0∈A(D) nonempty and not reduced to one point. Set coC(A0) =C∩(∩{A;A∈A(C) andA0⊂A}). Then from the Lemma 2.1, we get

R_ρ(coC(A0)) =Rρ(A0), andδρ(coC(A0)) =δρ(A0).

Since coC(A0)∈A(C), then we must have Rr(coC(A0))<δr(coC(A0)) because A(C)

is r-normal. Therefore, we haveRr(A0)<δr(A0), which completes the proof of our

claim.□

The next result is amazing and has found many applications in metric spaces. Most of the ideas in its proof go back to Baillon’s work [8].

Theorem 2.8. LetrÎℜand C⊂Lrbe nonempty,r-closed, andr-bounded. Assume that A(C) is compact andr-normal. Let(Cb)bÎΓ be a decreasing family of one-local retracts of C, where (Γ,≺)is totally ordered. Then∩bÎΓCbis not empty and is a one-local retract of C.

Proof. First, let us prove that∩bÎΓCbis not empty. Consider the family

F= ⎧ ⎨ ⎩

β∈

ℱ is not empty since

β∈ Cβ ∈F. ℱ will be ordered by inclusion, i.e.,

β∈ Aβ ⊂

β∈ Bβ if and only ifAb ⊂Bb for any bÎ Γ. From Theorem 2.7, we know

that A(C_β) is compact, for everybÎ Γ. Therefore,ℱsatisfies the hypothesis of Zorn’s lemma. Hence for every DÎℱ, there exists a minimal elementAÎℱsuch thatA⊂D.

We claim that if A=

β∈ Aβ is minimal, then there existsb0 ÎΓ such thatδ(Ab) = 0

for everyb≻b0. Assume not, i.e.,δ(Ab) > 0 for everybÎΓ. FixbÎΓ. For everyK⊂ C, set

co_β(K) = f∈Cβ

B_ρ(f, r_ρ(f, K)).

Consider A=

α∈

A_{α where}

A_α= co_β(A_β)∩A_α ifα≤β

A_α= A_α ifα≥β.

The family (A_α≥β) is decreasing sinceAÎℱ. Leta≤g≤ b. Then A_γ ⊂A_α sinceAg ⊂AaandAb=cob(Ab)∩Ab. Hence the family (A_α) is decreasing. On the other hand

ifa ≺b, then co_β(A_β)∩A_α∈A(C_α)sinceCb⊂Ca. Hence A_α∈A(Cα). Therefore, we haveA’Îℱ. SinceAis minimal, thenA=A’. Hence

A_α=co_β(A_β)∩A_α, for everyα≺β.

Let f Î Cb and a ≺ b. Since Ab ⊂ Aa, then rr(f, Ab) ≤ rr(f, Aa). Because

co_β(A_β) =∩g∈CβBρ(g, rρ(g, Aβ)), then we havecob(Ab)⊂Br(g, rr(g, Ab)) which implies

rr(g, Ab)≤rr(g, Aa). SinceAa⊂cob(Ab), then

r_ρ(g, A_β)≤r_ρ(g, A_α)≤r_ρ(g, co_β(A_β))≤r_ρ(g, A_β).

Therefore, we have rr(g, Aa)≤rr(g, Ab) for everygÎCb. Using the definition of the r-Chebyshev radiusRr, we get

R_ρ(A_α)≤R_ρ(A_β).

Let f Î Aa and set s = rr(f, Aa). Then f Î cob(Ab) since Aa ⊂ cob(Ab). Hence,

f ∈(∩g∈AβBρ(g, s))∩coβ(Aβ). SinceCbis a one-local retract ofC, then

S_β=C_β∩(∩g∈AβBρ(g, s))∩coβ(Aβ)=∅.

Since Ab=Cb∩cob(Ab), then we have

S_β=A_β∩

⎛

⎝

g∈Aβ

B_ρ(g, s) ⎞ ⎠.

Let hÎSb, then h∈ g∈Aβ

B_ρ(g, s)_{. Hence,}

rr(h, Ab)≤swhich impliesRr(Ab)≤s =rr

R_ρ(A_β) =R_ρ(A_α), for everya,β∈ .

Since δr(Ab) > 0 for every bÎ Γ. Set A_β to be ther-Chebyshev center ofAb, i.e.,

A_β=C_ρ(A_β), for every bÎ Γ. Since Rr(Ab) = Rr(Aa), for every a, b Î Γ, then the

family (A_β) is decreasing. Indeed, leta≺b and f ∈A_β. Then we haverr(f, Ab) =Rr

(Ab). Since we proved thatrr(g, Ab) =rr(g, Aa), for everygÎ Cb, then

rρ(f, Aα) =rρ(f, Aβ) =Rρ(Aβ) =Rρ(Aα),

which implies that f ∈A_α. Therefore, we have A=

β∈ A

β∈F. SinceA’’⊂ Aand

A is minimal, we getA=A’’. Therefore, we have Cρ(Aβ) =Aβ for every bÎ Γ. This contradicts the fact that A(C_β) is normal for everyb ÎΓ. Hence there existsb0Î Γ

such that

δ(Aβ) = 0, for everyββ0.

The proof of our claim is therefore complete. Then we have Ab= {f}, for every b≻ b0. This clearly implies thatfÎ∩bÎΓCb≠∅. In order to complete the proof, we need

to show thatS =∩bÎΓ Cbis a one-local retract ofC. Let (Bi)iÎIbe a family of r-balls centered inSsuch that ∩iÎIBi ≠∅. SetDb= (∩iÎIBi)∩Cb, for anybÎ Γ. SinceCbis a one-local retract of C, and the family (Bi) is centered inCb, thenDb is not empty

and D_β∈A(C_β). Therefore, D=

β∈ Dβ ∈F. Let A=

β∈ Aβ⊂D be a minimal

ele-ment of ℱ. The above proof shows that

∅ = β∈

A_β⊂

β∈

D_β.

The proof of Theorem 2.8 is therefore complete.□

The next theorem will be useful to prove the main result of the next section.

Theorem 2.9. Let r Î ℜ and C ⊂ Lr be nonempty, r-closed, and r-bounded.

Assume that A(C) is compact and r-normal. Let (Cb)bÎΓ be a family of one-local

retracts of C such that for any finite subsetI ofΓ, ∩b_{Î Γ} Cbis not empty and is a

one-local retract ofC. Then∩b_ÎΓCbis not empty and is a one-local retract ofC. Proof. Consider the familyℱ of subsetsI⊂ Γ such that for any finite subsetJ ⊂Γ

(empty or not), we have ∩a_ÎI∪JCais a nonempty one-local retract of C. Note thatℱis

not empty since any finite subset of Γis inℱ. Using Theorem 2.8, we can show thatℱ satisfies the hypothesis of Zorn’s lemma. Hence ℱ has a maximal element I⊂ Γ. AssumeI≠Γ. Leta ÎΓ \I. Obviously we haveI∪{a}Îℱ. This is a clear contradic-tion with the maximality of I. Therefore we haveI=ΓÎ ℱ, i.e.,∩bÎΓCbis not empty

and is a one-local retract ofC. □

Common fixed point result

Theorem 2.10. Let r Î ℜ and C ⊂ Lr be nonempty, r-closed, and r-bounded. Assume that A(C) is compact andr-normal. Then for any finite familyℱ = {T1,T2,...

Tn}of commutativer-nonexpansive mappings defined on C has a common fixed point, i.e., Fix (T1)∩ ···∩ Fix(Tn)≠∅.Moreover, the set of common fixed point set, denoted

Fix(ℱ) =Fix(T1))∩···∩Fix(Tn),is a one local retract of C.

Proof. Let us first prove Theorem 2.10 for two mappingsT1andT2. Using Theorem

2.5, we know that Fix(T1) is a nonempty one-local retract ofC. Since T1 and T2 are

commutative, thenT2(Fix(T1))⊂Fix(T1). Theorems 2.4 and 2.7 show that the

restric-tion ofT2to Fix(T1) has a fixed point. Again Theorem 2.5 will imply that the common

fixed point set Fix(T1)∩ Fix(T2) is a nonempty one-local retract ofC. Using the same

argument will show that the conclusion of Theorem 2.10 is valid for any finite number of mappings.□

Next we prove the main result of this section.

Theorem 2.11. Let r Î ℜ and C ⊂ Lr be nonempty, r-closed, and r-bounded. Assume that A(C) is compact andr-normal. Then for any family ℱ= {Ti; iÎI}of

com-mutativer-nonexpansive mappings defined on C has a common fixed point, i.e.,∩iÎIFix (Ti)≠∅. Moreover the set of common fixed point set, denotedFix(ℱ) =∩iÎIFix(Ti),is a

one-local retract of C.

Proof. Let Γ= {b;b is a nonempty finite subset ofI}. Theorem 2.10 implies that for everybÎΓ, the setFbof common fixed point set of the mappingsTi, iÎb, is a none-mpty one-local retract of C. Clearly the family (Fb)bÎΓ is decreasing and satisfies the

assumptions of Theorem 2.9. Therefore, we have ∩b_ÎΓFbis nonempty and is a

one-local retract ofC. The proof of Theorem 2.11 is complete. □

Using Corollary 2.1 and Remark 2.1 we get the following result.

Corollary 2.2. LetrÎℜandC⊂ Lrbe nonempty, convex,r-closed, andr-bounded. Assumer is(UUC).Then for any familyℱ= {Ti; iÎI}of commutativer-nonexpansive

mappings defined on C has a common fixed point, i.e.,∩iÎIFix(Ti)≠∅. Moreover the

set of common fixed point set, denoted Fix(ℱ) =∩iÎIFix(Ti),is a one-local retract of C.

Acknowledgements

The authors gratefully acknowledge the financial support provided by the University of Tabuk through the project of international cooperation with the University of Texas at El Paso.

Author details

1_{Department of Mathematics, King Abdulaziz University, PO Box 80203, Jeddah 21589, Saudi Arabia}2_{Department of}

Mathematical Sciences, The University of Texas at El Paso, El Paso, TX 79968, USA

Authors_’contributions

All authors participated in the design of this work and performed equally. All authors read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

Received: 6 March 2012 Accepted: 5 July 2012 Published: 5 July 2012

References

1. Browder, FE: Nonexpansive nonlinear operators in a Banach space. Proc Natl Acad Sci USA.54, 1041_–1044 (1965). doi:10.1073/pnas.54.4.1041

2. Gohde, D: Zum prinzip der kontraktiven abbildung. Math Nachr.30, 251_–258 (1965). doi:10.1002/mana.19650300312 3. Kirk, WA: A fixed point theorem for mappings which do not increase distances. Am Math Soc.82, 1004–1006 (1965) 4. Kirk, WA: Fixed Point Theory for Nonexpansive Mappings, I and II. In Lecture Notes in Mathematics, vol. 886, pp. 485–

5. Goebel, K, Reich, S: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. In Series of Monographs and Textbooks in Pure and Applied Mathematics, vol. 83,Dekker, New York (1984)

6. Dhompongsa, S, Kirk, WA, Sims, B: Fixed points of uniformly lipschitzian mappings. Nonlinear Anal.65, 762–772 (2006). doi:10.1016/j.na.2005.09.044

7. Goebel, K, Sekowski, T, Stachura, A: Uniform convexity of the hyperbolic metric and fixed points of holomorphic mappings in the Hilbert ball. Nonlinear Anal.4, 1011_–1021 (1980). doi:10.1016/0362-546X(80)90012-7

8. Baillon, JB: Nonexpansive mappings and hyperconvex spaces. Contemp Math.72, 11–19 (1988) 9. Khamsi, MA, Kirk, WA: An Introduction to Metric Spaces and Fixed Point Theory. Wiley, New York (2001)

10. Khamsi, MA, Kozlowski, WK, Reich, S: Fixed point theory in modular function spaces. Nonlinear Anal.14, 935–953 (1990). doi:10.1016/0362-546X(90)90111-S

11. Khamsi, MA, Kozlowski, WM, Shutao, C: Some geometrical properties and fixed point theorems in Orlicz spaces. J Math Anal Appl.155.2, 393–412 (1991)

12. Khamsi, MA: A convexity property in Modular function spaces. Math Japonica.44.2, 269_–279 (1996)

13. Kozlowski, WM: Modular Function Spaces. In Series of Monographs and Textbooks in Pure and Applied Mathematics, vol. 122,Dekker, New York (1988)

14. Kozlowski, WM: Notes on modular function spaces I. Comment Math.28, 91–104 (1988) 15. Kozlowski, WM: Notes on modular function spaces II. Comment Math.28, 105_–120 (1988)

16. Musielak, J: Orlicz Spaces and Modular Spaces. In Lecture Notes in Mathematics, vol. 1034,Springer, Berlin (1983) 17. Penot, JP: Fixed point theorem without convexity, Analyse Non Convexe (1977, Pau). Bull Soc Math France, Memoire.60,

129–152

18. Khamsi, MA, Kozlowski, WK: On asymptotic pointwise nonexpansive mappings in modular function spaces. J Math Anal Appl.380, 697_–708 (2011). doi:10.1016/j.jmaa.2011.03.031

doi:10.1186/1687-1812-2012-109

Cite this article as:Al-Mezelet al.:One-local retract and common fixed point in modular function spaces.Fixed Point Theory and Applications20122012:109.

Submit your manuscript to a

journal and benefi t from:

7Convenient online submission 7Rigorous peer review

7Immediate publication on acceptance 7Open access: articles freely available online 7High visibility within the fi eld

7Retaining the copyright to your article