Research

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

Saleh Abdullah Al-Mezel1*, Abdullah Al-Roqi1 and Mohamed Amine Khamsi2

Author Affiliations

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

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Fixed Point Theory and Applications 2012, 2012:109  doi:10.1186/1687-1812-2012-109

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

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### 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 ρ-nonexpansive mappings defined on a nonempty, ρ-closed and ρ-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 richness 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 of ρ-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

Let Ω be a nonempty set and ∑ be a nontrivial σ-algebra of subsets of Ω. Let be a δ-ring of subsets of Ω, such that for any and A ∈ ∑. Let us assume that there exists an increasing sequence of sets such that Ω = ∪Kn. By ℰ we denote the linear space of all simple functions with supports from . By ℳwe will denote the space of all extended measurable functions, i.e. all functions f: Ω → [-∞, ∞] such that there exists a sequence{gn} ⊂ ℰ, |gn| ≤ | f | and gn(ω) → f(ω) for all ω ∈ Ω By 1A we denote the characteristic function of the set A.

Definition 2.1. Let ρ: ℳ → [0, ∞] be a notrivial, convex and even function. We say that ρ is a regular convex function pseudomodular if:

(i) ρ(0) = 0;

(ii) ρ is monotone, i.e. |f(ω)| |g(ω)| for all ω ∈ Ω implies ρ(f) ≤ ρ(g), where f, g ∈ ℳ;

(iii) ρ is orthogonally subadditive, i.e ρ(f1A∪B) ≤ ρ(f1A)+ρ(f1B) for any A, B ∈ ∑ such that A ∩ B ≠ ∅, f ∈ ℳ;

(iv) ρ has the Fatou property, i.e. |fn(ω)|↑|f(ω)| for all ω ∈ Ω implies ρ(fn) ↑ρ(f), where f ∈ ℳ;

(v) ρ is order continuous in ℰ, i.e. gn ∈ ℰ and |gn(ω)| ↓ 0 implies ρ(gn) ↓ 0.

Similarly as in the case of measure spaces, we we say that a set A ∈ ∑ is ρ-null if ρ(g1A) = 0 for every g ∈ ℰ. We say that a property holds ρ-almost everywhere if the exceptional set is ρ-null. As usual we identify any pair of measurable sets whose symmetric difference is ρ-null as well as any pair of measurable functions differing only on a ρ-null set. With this in mind we define

(2.1)

where each is actually an equivalence class of functions equal ρ-a.e. rather than an individual function. Where no confusion exists we will write ℳ instead of .

Definition 2.2. Let ρ be a regular function pseudomodular.

(1) We say that ρ is a regular convex function semimodular if ρ(αf) = 0 for every α > 0 implies f = 0 ρ - a.e.;

(2) We say that ρ is a regular convex function modular if ρ(f) = 0 implies f = 0 ρ - a.e.;

The class of all nonzero regular convex function modulars defined on Ω will be denoted by ℜ.

Let us denote ρ(f, E) = ρ (f1E) for f ∈ ℳ, E ∈ ∑. It is easy to prove that ρ(f, E) is a function pseudomodular in the sense of Def. 2.1.1 in [13] (more precisely, it is a function 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] Let ρ be a convex function modular.

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

(b) The following formula defines a norm in Lρ (frequently called Luxemburg norm):

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]Let ρ ∈ ℜ.

(1) Lρ, || f ||ρ is complete and the norm || · ||ρ is monotone w.r.t. the natural order in ℳ.

(2) || fn ||ρ → 0 if and only if ρ(afn) → 0 for every α > 0.

(3) If ρ (αfn) → 0 for an α > 0 then there exists a subsequence {gn} of {fn} such that gn → 0 ρ - a.e.

(4) If {fn} converges uniformly to f on a set then ρ (α (fn - f), E) → 0 for every α > 0.

(5) Let fn f ρ - a.e. There exists a nondecreasing sequence of sets such that Hk ↑ Ω and {fn} converges uniformly to f on every Hk (Egoroff Theorem).

(6) ρ(f) ≤ lim inf ρ(fn) whenever fn f ρ - a.e. (Note: this property is equivalent to the Fatou Property).

(7) Defining is order continuous} and we have:

(a) ,

(b) Eρ has the Lebesgue property, i.e. ρ (αf, Dk) → 0 for α > 0, f Eρ and Dk ↓ ∅.

(c) Eρ is the closure of ℰ (in the sense of || · ||ρ).

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

Definition 2.4. Let ρ ∈ ℜ. We say that ρ has the Δ2-property if whenever Dk ↓ ∅ and .

Theorem 2.2. Let ρ ∈ ℜ. The following conditions are equivalent:

(a) ρ has Δ2,

(b) is a linear subspace of Lρ,

(c) ,

(d) if ρ (fn) → 0 then ρ(2fn) → 0,

(e) if ρ(αfn) → 0 for an α > 0 then || fn||ρ → 0, i.e. the modular convergence is equivalent to the norm convergence.

The following definition is crucial throughout this paper.

Definition 2.5. Let ρ ∈ ℜ.

(a) We say that {fn} is ρ-convergent to f and write fn → 0 (ρ) if and only if ρ(fn - f) → 0.

(b) A sequence {fn} where fn Lρ is called ρ-Cauchy if ρ (fn - fm) → 0 as n, m → ∞.

(c) A set B Lρ is called ρ-closed if for any sequence of fn B, the convergence fn → f (ρ) implies that f belongs to B.

(d) A set B Lρ is called ρ-bounded if sup{ρ (f - g); f B, g B} <

(e) Let f Lρ and C Lρ. The ρ-distance between f and C is defined as

Let us note that ρ-convergence does not necessarily imply ρ-Cauchy condition. Also, fn f does not imply in general λfn λf, λ > 1. Using Theorem 2.1 it is not difficult to prove the following

Proposition 2.1. Let ρ ∈ ℜ.

(i) Lρ is ρ-complete,

(ii) ρ-balls Bρ(x, r) = {y Lρ ; ρ(x - y) ≤ r} are ρ-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 that Lρ has property (R) if and only if every nonincreasing sequence {Cn} of nonempty, ρ-bounded, ρ-closed, convex subsets of Lρ has nonempty intersection.

Throughout this paper we will need the following.

Definition 2.7. Let ρ ∈ ℜ and C Lρ be nonempty.

(a) By the ρ-diameter of C, we will understand the number

The subset C is said to be ρ-bounded whenever δρ(C) < ∞.

(b) The quantity rρ(f, C) = sup{ρ(f - g);g C} will be called the ρ-Chebyshev radius of C with respect to f.

(c) The ρ-Chebyshev radius of C is defined by Rρ(C) = inf {rρ (f, C); f C}.

(d) The ρ-Chebyshev center of C is defined as the set

Note that Rρ(C) ≤ rρ (f, C) ≤ δρ(C) for all f C and observe that there is no reason, in general, for to be nonempty.

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

Definition 2.8. Let ρ ∈ ℜ and C Lρ be nonempty and ρ-closed. A mapping T: C C is called a ρ-nonexpansive mapping if

A point f C is called a fixed point of T whenever T(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. Let ρ ∈ ℜ and C Lρ be nonempty and ρ-bounded. We say that A is an admissible subset of C if

where bi C, ri ≥ 0 and I is an arbitrary index set. By we denote the family of all admissible subsets of C.

Note that if C is ρ-bounded, then . In order to prove an analogue of Kirk's fixed point theorem [3], Penot [17] introduced the following definition.

Definition 2.10. Let ρ ∈ ℜ and C Lρ be nonempty.

(1) We will say that is ρ-normal if for any nonempty , which has more than one point, we have Rρ(A) < δρ(A).

(2) We will say that is compact if for any family we have

provided that for any finite subset F of Γ.

Clearly if is compact, then Lρ has 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. Let ρ ∈ ℜ.

(i) Let r > 0, ε > 0. Define

Let

and δ(r, ε) = 1 if D(r, ε) = ∅. We say that ρ satisfies (UC) if for every r > 0, ε > 0, δ(r, ε) > 0. Note, that for every r > 0, D(r, ε) ≠ ∅, for ε > 0 small enough.

(ii) We say that ρ satisfies (UUC) if for every s ≥ 0, ε > 0 there exists

depending on s and ε such that

(iii) We say that ρ is Strictly Convex, (SC), if for every f, g Lρ such that ρ(f) = ρ(g) and

there holds f = 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 the Δ2 condition but are uniformly convex are: φ1(t) = e|t|-|t|-1 and . In these cases, the associated modular is (UUC).

It is shown in [18], that if ρ ∈ ℜ is (UUC), then for any nonempty, convex, and ρ-closed C Lρ , and any f Lρ such that d = dρ (f, C) < ∞, there exists a unique best ρ-approximant of f in C, i.e. a unique g0 C such that

Moreover it is also shown in [18] that if ρ ∈ ℜ is (UUC), then for any nonincreasing sequence {Cn} of nonempty, convex, and ρ-closed subsets of Lρ, we have ∩n ≥ 1Cn ≠ ∅, provided there exists f Lρ such that . 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 ρ ∈ ℜ. Assume ρ ∈ ℜ is (UUC). Let {Cα}α∈Γ be a decreasing family of nonempty, convex, ρ-closed subsets of Lρ, where (Γ,≺) is upward directed. Assume that there exists f ∈ Lρ such that . Then, ∩α∈ΓCα ≠ ∅.

Proof. Set . Without loss of generality, we may assume d > 0. For Any n ≥ 1, there exists an ∈ Γ such that

Since (Γ,≺) is upward directed, we may assume αn αn+1. In particular we have for any n ≥ 1. Since ρ is (UUC), we get . Clearly C0 is ρ-closed and

Again using the property (UUC) satisfied by ρ, there exists g0 C0 unique such that dρ(f, C0) = ρ (f - g0). Let us prove that g0 Cα for any α ∈ Γ. Fix α ∈ Γ. If for some n ≥ 1 we have α αn, then obviously we have .

Therefore let us assume that ααn, for any n ≥ 1. Since Γ is upward directed, there exists βn ∈ Γ such that αnβn and α βn for any n ≥ 1. We can also assume that βn βn+1 for any n ≥ 1. Again we have . Since , for any n ≥ 1, we get C1 C0. Moreover we have

Hence, dρ(f, C1) = d which implies the existence of a unique point g1 C1 such that dρ(f, C1) = ρ(f - g1) = d. Since ρ is uniformly convex, it must be (SC). Hence, g0 = g1. In particular, we have , for any n ≥ 1. Since α βn, we get , for any n ≥ 1, which implies g0 Cα. Since α was taking arbitrary in Γ, we get g0 ∈ ∩α∈Γ Cα, which implies ∩α∈Γ Cα ≠ ∅.    □

Since ρ is convex, ρ-closed balls are convex. Theorem 2.3 implies the following.

Corollary 2.1. Let ρ ∈ ℜ and C Lρ be nonempty, convex, ρ-closed, and ρ-bounded. Assume ρ is (UUC). Then is compact.

Remark 2.1. Note that under the above assumptions, is ρ-normal. Indeed let nonempty and not reduced to one point. Let f, g A such that f ≠ g. Then . Since ρ is (UUC), there exists η > 0 such that for any h A, we have

Hence, , which implies Rρ(A) < δρ(A).

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

Theorem 2.4. Let ρ ∈ ℜ and C Lρ be nonempty, ρ-closed, and ρ-bounded. Assume that is compact and ρ-normal. Then any ρ-nonexpansive T: C C has a fixed point.

Proof. Since C is ρ-bounded, then we have . Since is compact, the family has a minimal element K. Set

Note that T(K) ⊂ K0. This implies that K0 is nonempty and belongs to . Moreover since K0 K, we get T(K0) ⊂ T(K) ⊂ K0. Hence K0 ∈ ℱ. The minimality of K implies that K = K0. Next let f K. By definition of the ρ-Chebyshev radius rρ(f, K), we have K Bρ(f, rρ(f, K)). Since T is ρ-nonexpansive, we have T(K) ⊂ Bρ(T(f), rρ(f, K)). The definition of K0 implies K0 Bρ(T(f), rρ(f, K)). Since K = K0, we get K Bρ (T (f), rρ(f, K)), which implies rρ(T(f), K) ≤ rρ(f, K). Fix f K and set r = rρ (f, K). We have

Clearly, we have T(K1) ⊂ K1 and . Since K is minimal, we get K = K1 which implies that the ρ-Chebyshev radius rρ(f, K) is constant. In particular, we have rρ(f, K) = δρ (K), for any f K. Since is ρ-normal, we conclude that K does not have more than one point. Therefore, K = {f} which forces T (f) = f.    □

In the next section, we investigate the structure of the fixed point set of ρ-nonexpansive mappings.

### One-local retract subsets in modular function spaces

Let ρ ∈ ℜ and C Lρ be nonempty. A nonempty subset D of C is said to be a one-local retract of C if for every family {Bi; i ∈ I} of ρ-balls centered in D such that C ∩ (∩iI Bi) ≠ ∅, it is the case that D ∩ (∩iI Bi) ≠ ∅. It is immediate that each ρ-nonexpansive retract of Lρ is a one-local retract (but not conversely). Recall that D C is a ρ-nonexpansive retract of C if there exists a ρ-nonexpansive map R: C D such that R(f) = f, for every f D.

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

Theorem 2.5. Let ρ ∈ ℜ and C Lρ be nonempty, ρ-closed, and ρ-bounded. Assume that is compact and ρ-normal. Then for any ρ-nonexpansive mapping T: C C, the fixed point set Fix(T) is a nonempty one-local retract of C.

Proof. Theorem 2.4 shows that Fix(T) is nonempty. Let us complete the proof by showing it is a one-local retract of C. Let {Bρ(fi, ri)}iI be any family of ρ-closed balls such that fi ∈ Fix(T), for any i I, and

Let us prove that Fix (T) ∩ (∩iI Bρ(fi, ri)) ≠ ∅. Since {fi}iI ⊂ Fix(T), and T is ρ-nonexpansive, then T(C0) ⊂ C0. Clearly, and is nonempty. Then we have . Therefore, is compact and ρ-normal. Theorem 2.4 will imply that T has a fixed point in C0 which will imply

□

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. Let ρ ∈ ℜ and C Lρ be 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 ρ-nonexpansive retract of D ∪ {f}, for every f C.

Proof. let us prove (i) ⇒ (ii). Let f C. We may assume that f D. In order to construct a ρ-nonexpansive retract R: D ∪ {f} → D, we only need to find R(f) ∈ D such that

Since f ∈ ∩g∈D Bρ(g, ρ(f-g)) and f C, then

Since D is a one-local retract of C, we get

Any point in D0 will work as R(f).

Next, we prove that (ii) ⇒ (i). In order to prove that D is a one-local retract of C, let {Bρ(fi, ri)}iI be any family of ρ-closed balls such that fi D, for any i I, and

Let us prove that D ∩ (∩iI Bρ(fi, ri)) = ∅. Let f C0. If f D, we have nothing to prove. Assume otherwise that f D. Property (ii) implies the existence of a ρ-nonexpansive retract R: D ∪ {f} → C. It is easy to check that R(f) ∈ D ∩ (∩iI Bρ(fi, ri)) = ∅, which completes the proof of our theorem. □

The following technical lemma will be useful for the next results.

Lemma 2.1. Let ρ ∈ ℜ and C Lρ be nonempty, and ρ-bounded. Let D be a nonempty one-local retract of C. Set and D A}). Then

(i) rρ(f, D) = rρ(f, coC(D)), for any f C;

(ii) Rρ(coC(D)) = Rρ(D);

(iii) δρ(coC(D)) = δρ(D).

Proof. Let us first prove (i). Fix f C. Since D coC(D), we get rρ(f, D) ≤ rρ(f, coC(D)). Set r = rρ(f, D). We have . The definition of coC(D) implies coC(D) ⊂ Bρ(f, r). Hence rρ(f, coC(D)) ≤ r = rρ(f, D), which implies rρ(f, D) = rρ(f, coC(D)).

Next, we prove (ii). Let f D. We have f coC(D). Using (i), we get rρ(f, D) = rρ(f, coC(D)) ≥ Rρ(coC(D)). Hence, Rρ(D) ≥ Rρ(coC(D)). Next, let f coC(D) and set r = rρ(f, coC(D)). We have D coC(D) ⊂ Bρ(f, r). Hence, f ∈ ∩gDBρ(g, r). Hence, C ∩ (∩gDBρ(g, r)) = ∅. Since D is a one-local retract of C, we get D0 = D ∩ (∩gDBρ(g, r)) = ∅. Let g D0. Then it is easy to see that rρ(g, D) ≤ r. Hence, Rρ(D) ≤ r. Since f was arbitrary taken in coC(D), we get Rρ(D) ≤ Rρ(coC(D)), which implies Rρ(D) = Rρ(coC(D)).

Finally, let us prove (iii). Since D coC(D), we get δρ(D) ≤ δρ(coC(D)). Next set d = δρ(D). Then, for any f D, we have D Bρ(f, d). Hence coC(D) ⊂ Bρ(f, d). This implies . Sice f was taken arbitrary in D, we get . The definition of coC(D) implies . So for any f, g coC(D), we have ρ(f - g) ≤ d. Hence δρ(coC(D)) ≤ d = δρ(D), which implies δρ (D) = δρ (coC(D)). □

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

Theorem 2.7. Let ρ ∈ ℜ and C Lρ be nonempty, ρ-closed, and ρ-bounded. Assume that is compact and ρ-normal. If D is a nonempty one-local retract of C, then is compact and ρ-normal.

Proof. Using the definition of one-local retract, it is easy to see that is compact. Let us show that is ρ-normal. Let nonempty and not reduced to one point. Set . Then from the Lemma 2.1, we get

Since , then we must have Rρ(coC(A0)) < δρ(coC(A0)) because is ρ-normal. Therefore, we have Rρ(A0) < δρ(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. Let ρ ∈ ℜ and C Lρ be nonempty, ρ-closed, and ρ-bounded. Assume that is compact and ρ-normal. 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 a one-local retract of C.

Proof. First, let us prove that β∈ΓCβ is not empty. Consider the family

is not empty since . will be ordered by inclusion, i.e., if and only if Aβ Bβ for any β ∈ Γ. From Theorem 2.7, we know that is compact, for every β ∈ Γ. Therefore, satisfies the hypothesis of Zorn's lemma. Hence for every D , there exists a minimal element A such that A D. We claim that if is minimal, then there exists β0 ∈ Γ such that δ(Aβ) = 0 for every β β0. Assume not, i.e., δ(Aβ) > 0 for every β ∈ Γ. Fix β ∈ Γ. For every K C, set

Consider where

The family is decreasing since A . Let α γ β. Then since Aγ Aα and Aβ = coβ (Aβ) ∩ Aβ. Hence the family is decreasing. On the other hand if α β, then since Cβ Cα. Hence . Therefore, we have A' ∈ ℱ. Since A is minimal, then A = A'. Hence

Let f Cβ and a β. Since Aβ Aα, then rρ(f, Aβ) ≤ rρ(f, Aα). Because , then we have coβ(Aβ) ⊂ Bρ(g, rρ(g, Aβ)) which implies rρ(g, Aβ) ≤ rρ(g, Aα). Since Aα coβ(Aβ), then

Therefore, we have rρ(g, Aα) ≤ rρ(g, Aβ) for every g Cβ. Using the definition of the ρ-Chebyshev radius Rρ, we get

Let f Aα and set s = rρ(f, Aα). Then f coβ(Aβ) since Aα coβ(Aβ). Hence, . Since Cβ is a one-local retract of C, then

Since Aβ = Cβ coβ(Aβ), then we have

Let h Sβ, then . Hence, rρ(h, Aβ) ≤ s which implies Rρ(Aβ) ≤ s = rρ(f, Aα), for every f Aα. Hence Rρ(Aβ) ≤ Rρ(Aα). Therefore we have

Since δρ(Aβ) > 0 for every β ∈ Γ. Set to be the ρ-Chebyshev center of Aβ, i.e., , for every β ∈ Γ. Since Rρ(Aβ) = Rρ(Aα), for every α, β ∈ Γ, then the family is decreasing. Indeed, let αβ and . Then we have rρ(f, Aβ) = Rρ(Aβ). Since we proved that rρ(g, Aβ) = rρ(g, Aα), for every g Cβ, then

which implies that . Therefore, we have . Since A'' A and A is minimal, we get A = A''. Therefore, we have for every β ∈ Γ. This contradicts the fact that is normal for every β ∈ Γ. Hence there exists β0 ∈ Γ such that

The proof of our claim is therefore complete. Then we have Aβ = {f }, for every β β0. This clearly implies that f ∈ ∩β∈Γ Cβ ≠ ∅. In order to complete the proof, we need to show that S = ∩β∈Γ Cβ is a one-local retract of C. Let (Bi)iI be a family of ρ-balls centered in S such that ∩iI Bi ≠ ∅. Set Dβ = (∩iI Bi) ∩ Cβ, for any β ∈ Γ. Since Cβ is a one-local retract of C, and the family (Bi) is centered in Cβ, then Dβ is not empty and . Therefore, . Let be a minimal element of ℱ. The above proof shows that

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 ρ ∈ ℜ and C Lρ be nonempty, ρ-closed, and ρ-bounded. Assume that is compact and ρ-normal. Let (Cβ)β∈Γ be a family of one-local retracts of C such that for any finite subset I of Γ, ∩β ∈ Γ Cβ is not empty and is a one-local retract of C. Then ∩β ∈ Γ Cβ is not empty and is a one-local retract of C.

Proof. Consider the family ℱ of subsets I ⊂ Γ such that for any finite subset J ⊂ Γ (empty or not), we have ∩αIJCα is a nonempty one-local retract of C. Note that ℱ is not empty since any finite subset of Γ is in ℱ. Using Theorem 2.8, 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, i.e., ∩β∈Γ Cβ is not empty and is a one-local retract of C.

### Common fixed point result

In the previous section, we showed that under suitable conditions, any ρ-nonexpansive mapping has a fixed point. In this section we will discuss the existence of a fixed point common to a family of a commutative ρ-nonexpansive mappings. First we will need to discuss the case of finite families.

Theorem 2.10. Let ρ ∈ ℜ and C Lρ be nonempty, ρ-closed, and ρ-bounded. Assume that is compact and ρ-normal. Then for any finite family ℱ = {T1, T2,...Tn} of commutative ρ-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 mappings T1 and T2. Using Theorem 2.5, we know that Fix(T1) is a nonempty one-local retract of C. Since T1 and T2 are commutative, then T2(Fix(T1)) ⊂ Fix(T1). Theorems 2.4 and 2.7 show that the restriction of T2 to 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 of C. 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 ρ ∈ ℜ and C Lρ be nonempty, ρ-closed, and ρ-bounded. Assume that is compact and ρ-normal. Then for any family ℱ = {Ti; iI}of commutative ρ-nonexpansive mappings defined on C has a common fixed point, i.e., ∩i∈I Fix(Ti) ≠ ∅. Moreover the set of common fixed point set, denoted Fix(ℱ) = i∈I Fix(Ti), is a one-local retract of C.

Proof. Let Γ = {β; β is a nonempty finite subset of I}. Theorem 2.10 implies that for every β ∈ Γ, the set Fβ of common fixed point set of the mappings Ti, i β, is a nonempty one-local retract of C. Clearly the family (Fβ)β∈Γ is decreasing and satisfies the assumptions of Theorem 2.9. Therefore, we have ∩β∈ΓFβ is nonempty and is a one-local retract of C. The proof of Theorem 2.11 is complete.

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

Corollary 2.2. Let ρ ∈ ℜ and C Lρ be nonempty, convex, ρ-closed, andρ-bounded. Assume ρ is (UUC). Then for any family ℱ = {Ti; iI} of commutative ρ-nonexpansive mappings defined on C has a common fixed point, i.e., i∈I Fix(Ti) ≠ ∅. Moreover the set of common fixed point set, denoted Fix(ℱ) = i∈I Fix(Ti), is a one-local retract of C.

### Competing interests

The authors declare that they have no competing interests.

### Authors' contributions

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

### 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.

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