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

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.

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.

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 = ∪K_n. By ℰ we denote the linear space of all simple functions with supports from . By ℳ_∞ we will denote the space of all extended measurable functions, i.e. all functions f: Ω → [-∞, ∞] such that there exists a sequence{g_n} ⊂ ℰ, |g_n| ≤ | f | and g_n(ω) → f(ω) for all ω ∈ Ω By 1_A 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(f1_A∪B) ≤ ρ(f1_A)+ρ(f1_B) for any A, B ∈ ∑ such that A ∩ B ≠ ∅, f ∈ ℳ;

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

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

Similarly as in the case of measure spaces, we we say that a set A ∈ ∑ is ρ-null if ρ(g1_A) = 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

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 (f1_E) 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 L_ρ (Ω, ∑), or briefly L_ρ, defined by

(b) The following formula defines a norm in L_ρ (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) L_ρ, || f ||_ρ is complete and the norm || · ||_ρ is monotone w.r.t. the natural order in ℳ.

(2) || f_n ||_ρ → 0 if and only if ρ(af_n) → 0 for every α > 0.

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

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

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

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

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

(a) ,

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

(c) E_ρ 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 D_k ↓ ∅ and .

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

(a) ρ has Δ₂,

(b) is a linear subspace of L_ρ,

(c) ,

(d) if ρ (f_n) → 0 then ρ(2f_n) → 0,

(e) if ρ(αf_n) → 0 for an α > 0 then || f_n||_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 {f_n} is ρ-convergent to f and write f_n → 0 (ρ) if and only if ρ(f_n - f) → 0.

(b) A sequence {f_n} where f_n ∈ L_ρ is called ρ-Cauchy if ρ (f_n - f_m) → 0 as n, m → ∞.

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

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

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

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

Proposition 2.1. Let ρ ∈ ℜ.

(i) L_ρ is ρ-complete,

(ii) ρ-balls B_ρ(x, r) = {y ∈ L_ρ ; 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 L_ρ has property (R) if and only if every nonincreasing sequence {C_n} of nonempty, ρ-bounded, ρ-closed, convex subsets of L_ρ has nonempty intersection.

Throughout this paper we will need the following.

Definition 2.7. Let ρ ∈ ℜ and C ⊂ L_ρ be nonempty.

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

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

(b) The quantity r_ρ(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 R_ρ(C) = inf {r_ρ (f, C); f ∈ C}.

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

Note that R_ρ(C) ≤ r_ρ (f, C) ≤ δ_ρ(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 ⊂ L_ρ 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).

The following definition is needed.

Definition 2.9. Let ρ ∈ ℜ and C ⊂ L_ρ be nonempty and ρ-bounded. We say that A is an admissible subset of C if

where b_i ∈ C, r_i ≥ 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 ⊂ L_ρ be nonempty.

(1) We will say that is ρ-normal if for any nonempty , which has more than one point, we have R_ρ(A) < δ_ρ(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 L_ρ 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 ∈ L_ρ 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 ⊂ L_ρ , and any f ∈ L_ρ such that d = d_ρ (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 {C_n} of nonempty, convex, and ρ-closed subsets of L_ρ, we have ∩_{n ≥ 1}C_n ≠ ∅, provided there exists f ∈ L_ρ 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 {C_α}_α∈Γ be a decreasing family of nonempty, convex, ρ-closed subsets of L_ρ, where (Γ,≺) is upward directed. Assume that there exists f ∈ L_ρ such that . Then, ∩_α∈ΓC_α ≠ ∅.

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

Since (Γ,≺) is upward directed, we may assume α_n ≺ α_n+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 d_ρ(f, C₀) = ρ (f - g₀). Let us prove that g₀ ∈ C_α for any α ∈ Γ. Fix α ∈ Γ. If for some n ≥ 1 we have α ≺ α_n, then obviously we have .

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

Hence, d_ρ(f, C₁) = d which implies the existence of a unique point g₁ ∈ C₁ such that d_ρ(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 ≺ β_n, we get , for any n ≥ 1, which implies g₀ ∈ C_α. Since α was taking arbitrary in Γ, we get g₀ ∈ ∩_α∈Γ C_α, which implies ∩_α∈Γ C_α ≠ ∅.    □

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

Corollary 2.1. Let ρ ∈ ℜ and C ⊂ L_ρ 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 R_ρ(A) < δ_ρ(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 ⊂ L_ρ 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 r_ρ(f, K), we have K ⊂ B_ρ(f, r_ρ(f, K)). Since T is ρ-nonexpansive, we have T(K) ⊂ B_ρ(T(f), r_ρ(f, K)). The definition of K₀ implies K₀ ⊂ B_ρ(T(f), r_ρ(f, K)). Since K = K₀, we get K ⊂ B_ρ (T (f), r_ρ(f, K)), which implies r_ρ(T(f), K) ≤ r_ρ(f, K). Fix f ∈ K and set r = r_ρ (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 r_ρ(f, K) is constant. In particular, we have r_ρ(f, K) = δ_ρ (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.

Let ρ ∈ ℜ and C ⊂ L_ρ be nonempty. A nonempty subset D of C is said to be a one-local retract of C if for every family {B_i; i ∈ I} of ρ-balls centered in D such that C ∩ (∩_i∈I B_i) ≠ ∅, it is the case that D ∩ (∩_i∈I B_i) ≠ ∅. It is immediate that each ρ-nonexpansive retract of L_ρ 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 ⊂ L_ρ 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 {B_ρ(f_i, r_i)}_i∈I be any family of ρ-closed balls such that f_i ∈ Fix(T), for any i ∈ I, and

Let us prove that Fix (T) ∩ (∩_i∈I B_ρ(f_i, r_i)) ≠ ∅. Since {fi}_i∈I ⊂ 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 ⊂ L_ρ 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 B_ρ(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 {B_ρ(f_i, r_i)}_i∈I be any family of ρ-closed balls such that f_i ∈ D, for any i ∈ I, and

Let us prove that D ∩ (∩_i∈I B_ρ(f_i, r_i)) = ∅. 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 ∩ (∩_i∈I B_ρ(f_i, r_i)) = ∅, 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 ⊂ L_ρ be nonempty, and ρ-bounded. Let D be a nonempty one-local retract of C. Set and D ⊂ A}). Then

(i) r_ρ(f, D) = r_ρ(f, co_C(D)), for any f ∈ C;

(ii) R_ρ(co_C(D)) = R_ρ(D);

(iii) δ_ρ(co_C(D)) = δ_ρ(D).

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

Next, we prove (ii). Let f ∈ D. We have f ∈ co_C(D). Using (i), we get r_ρ(f, D) = r_ρ(f, co_C(D)) ≥ R_ρ(co_C(D)). Hence, R_ρ(D) ≥ R_ρ(co_C(D)). Next, let f ∈ co_C(D) and set r = r_ρ(f, co_C(D)). We have D ⊂ co_C(D) ⊂ B_ρ(f, r). Hence, f ∈ ∩_g∈DB_ρ(g, r). Hence, C ∩ (∩_{g∈ D}B_ρ(g, r)) = ∅. Since D is a one-local retract of C, we get D₀ = D ∩ (∩_{g∈ D}B_ρ(g, r)) = ∅. Let g ∈ D₀. Then it is easy to see that r_ρ(g, D) ≤ r. Hence, R_ρ(D) ≤ r. Since f was arbitrary taken in co_C(D), we get R_ρ(D) ≤ R_ρ(co_C(D)), which implies R_ρ(D) = R_ρ(co_C(D)).

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

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

Theorem 2.7. Let ρ ∈ ℜ and C ⊂ L_ρ 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 R_ρ(co_C(A₀)) < δ_ρ(co_C(A₀)) because is ρ-normal. Therefore, we have R_ρ(A₀) < δ_ρ(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 ⊂ L_ρ be nonempty, ρ-closed, and ρ-bounded. Assume that is compact and ρ-normal. Let (C_β)_β∈Γ be a decreasing family of one-local retracts of C, where (Γ, ≺) is totally ordered. Then ∩_β∈ΓC_β is not empty and is a one-local retract of C.

Proof. First, let us prove that ∩_β∈ΓC_β is not empty. Consider the family

ℱ is not empty since . ℱ will be ordered by inclusion, i.e., if and only if A_β ⊂ B_β for any β ∈ Γ. From Theorem 2.7, we know that 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 is minimal, then there exists β₀ ∈ Γ such that δ(A_β) = 0 for every β ≻ β₀. Assume not, i.e., δ(A_β) > 0 for every β ∈ Γ. Fix β ∈ Γ. For every K ⊂ C, set

Consider where

The family is decreasing since A ∈ℱ. Let α ≤ γ ≤ β. Then since A_γ ⊂ A_α and A_β = co_β (A_β) ∩ A_β. Hence the family is decreasing. On the other hand if α ≺ β, then since C_β ⊂ C_α. Hence . Therefore, we have A' ∈ ℱ. Since A is minimal, then A = A'. Hence

Let f ∈ C_β and a ≺ β. Since A_β ⊂ A_α, then r_ρ(f, A_β) ≤ r_ρ(f, A_α). Because , then we have co_β(A_β) ⊂ B_ρ(g, r_ρ(g, A_β)) which implies r_ρ(g, A_β) ≤ r_ρ(g, A_α). Since A_α ⊂ co_β(A_β), then

Therefore, we have r_ρ(g, A_α) ≤ r_ρ(g, A_β) for every g ∈ C_β. Using the definition of the ρ-Chebyshev radius R_ρ, we get

Let f ∈ A_α and set s = r_ρ(f, A_α). Then f ∈ co_β(A_β) since A_α ⊂ co_β(A_β). Hence, . Since C_β is a one-local retract of C, then

Since A_β = C_β ∩ co_β(A_β), then we have

Let h ∈ S_β, then . Hence, r_ρ(h, A_β) ≤ s which implies R_ρ(A_β) ≤ s = r_ρ(f, A_α), for every f ∈ A_α. Hence R_ρ(A_β) ≤ R_ρ(A_α). Therefore we have

Since δ_ρ(A_β) > 0 for every β ∈ Γ. Set to be the ρ-Chebyshev center of A_β, i.e., , for every β ∈ Γ. Since R_ρ(A_β) = R_ρ(A_α), for every α, β ∈ Γ, then the family is decreasing. Indeed, let α≺ β and . Then we have r_ρ(f, A_β) = R_ρ(A_β). Since we proved that r_ρ(g, A_β) = r_ρ(g, A_α), for every g ∈ C_β, 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 A_β = {f }, for every β ≻ β₀. This clearly implies that f ∈ ∩_β∈Γ C_β ≠ ∅. In order to complete the proof, we need to show that S = ∩_β∈Γ C_β is a one-local retract of C. Let (B_i)_i∈I be a family of ρ-balls centered in S such that ∩_i∈I B_i ≠ ∅. Set D_β = (∩_i∈I B_i) ∩ C_β, for any β ∈ Γ. Since C_β is a one-local retract of C, and the family (B_i) is centered in C_β, then D_β 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 ⊂ L_ρ be nonempty, ρ-closed, and ρ-bounded. Assume that is compact and ρ-normal. Let (C_β)_β∈Γ be a family of one-local retracts of C such that for any finite subset I of Γ, ∩_{β ∈ Γ} C_β is not empty and is a one-local retract of C. Then ∩_{β ∈ Γ} C_β 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 ∩_α∈I∪JC_α 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 C_β is not empty and is a one-local retract of C.

□

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 ⊂ L_ρ be nonempty, ρ-closed, and ρ-bounded. Assume that is compact and ρ-normal. Then for any finite family ℱ = {T₁, T₂,...T_n} of commutative ρ-nonexpansive mappings defined on C has a common fixed point, i.e., Fix (T₁) ∩ ··· ∩ Fix(T_n) ≠ ∅. Moreover, the set of common fixed point set, denoted Fix(ℱ) = Fix(T₁)) ∩ ··· ∩ Fix(T_n), 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 ⊂ L_ρ be nonempty, ρ-closed, and ρ-bounded. Assume that is compact and ρ-normal. Then for any family ℱ = {T_i; i∈I}of commutative ρ-nonexpansive mappings defined on C has a common fixed point, i.e., ∩_i∈I Fix(T_i) ≠ ∅. Moreover the set of common fixed point set, denoted Fix(ℱ) = ∩_i∈I Fix(T_i), 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 F_β of common fixed point set of the mappings T_i, i ∈ β, is a nonempty one-local retract of C. Clearly the family (F_β)_β∈Γ is decreasing and satisfies the assumptions of Theorem 2.9. Therefore, we have ∩_β∈ΓF_β 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 ⊂ L_ρ be nonempty, convex, ρ-closed, andρ-bounded. Assume ρ is (UUC). Then for any family ℱ = {T_i; i∈I} of commutative ρ-nonexpansive mappings defined on C has a common fixed point, i.e., ∩_i∈I Fix(T_i) ≠ ∅. Moreover the set of common fixed point set, denoted Fix(ℱ) = ∩_i∈I Fix(T_i), is a one-local retract of C.

The authors declare that they have no competing interests.

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

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.

1. Browder, FE: Nonexpansive nonlinear operators in a Banach space. Proc Natl Acad Sci USA. 54, 1041–1044 (1965). PubMed Abstract | Publisher Full Text | PubMed Central Full Text

2. Gohde, D: Zum prinzip der kontraktiven abbildung. Math Nachr. 30, 251–258 (1965). Publisher Full Text

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. Lecture Notes in Mathematics, pp. 485–505. Springer, Berlin (1981)

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

6. Dhompongsa, S, Kirk, WA, Sims, B: Fixed points of uniformly lipschitzian mappings. Nonlinear Anal. 65, 762–772 (2006). Publisher Full Text

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). Publisher Full Text

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). Publisher Full Text

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. Series of Monographs and Textbooks in Pure and Applied Mathematics, 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. Lecture Notes in Mathematics, Springer, Berlin (1983)

17. Penot, JP: Fixed point theorem without convexity, Analyse Non Convexe (1977, Pau). In: 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). Publisher Full Text