Banach operator pairs and common fixed points in modular function spaces

Abstract

In this article, we introduce the concept of a Banach operator pair in the setting of modular function spaces. We prove some common fixed point results for such type of operators satisfying a more general condition of nonexpansiveness. We also establish a version of the well-known De Marr's theorem for an arbitrary family of symmetric Banach operator pairs in modular function spaces without Δ₂-condition.

MSC(2000)

primary 06F30; 47H09; secondary 46B20; 47E10; 47H10.

1. Introduction

The purpose of this article is to give an outline of fixed point theory for mappings defined on some subsets of modular function spaces which are natural generalization 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. This article operates within the framework of convex function modulars.

The importance of applications of nonexpansive mappings in modular function spaces lies 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 concepts 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 is rich (see, e.g., [1–4]) and has been well developed since the 1960s and generalized to other metric spaces (see, e.g., [5–7]), and modular function spaces (see, e.g., [8–11]). The corresponding fixed point results were then extended to larger classes of mappings like asymptotic mappings [12, 13], pointwise contractions [14] and asymptotic pointwise contractions and nonexpansive mappings [15–17].

As noted in [16, 18], questions are sometimes asked whether the theory of modular function spaces provides general methods for the consideration of fixed point properties; the situation here is the same as it is in the Banach space setting.

In this article, we introduce the concept of a Banach operator pair in modular function spaces. Then, we investigate the existence of common fixed points for such operators. Believing that the well-known De Marr's theorem [19] is not known yet in the setting of modular function spaces, we establish this classical result in this new setting.

2. Preliminaries

Let Ω be a nonempty set and Σ be a nontrivial σ-algebra of subsets of Ω. Let $\mathcal{P}$ be a δ-ring of subsets of Ω, such that $E\cap A\in \mathcal{P}$ for any $E\in \mathcal{P}$ and A ∈ Σ. Let us assume that there exists an increasing sequence of sets $K_n\in \mathcal{P}$ such that Ω = UK _n . By $\mathcal{E}$ we denote the linear space of all simple functions with supports from $\mathcal{P}$. By ${\mathcal{M}}_{\infty }$ we will denote the space of all extended measurable functions, i.e., all functions f : Ω → [- ∞, ∞] such that there exists a sequence $\left\{g_n\right\}\subset \mathcal{E}$, |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 $\rho :{\mathcal{M}}_{\infty }\to \left[0,\infty \right]$ be a nontrivial, 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 $f,g\in {\mathcal{M}}_{\infty }$;

(iii) ρ is orthogonally subadditive, i.e., ρ(f 1_A∪B) ≤ ρ(f 1 _A ) + ρ(f 1 _B ) for any A,B ∈ Σ such that $A\cap B\ne \varnothing ,f\in \mathcal{M}$;

(iv) ρ has the Fatou property, i.e., |f_n(ω)| ↑ |f(ω)| for all ω ∈ Ω implies ρ(f _n ) ↑ ρ(f), where $f\in {\mathcal{M}}_{\infty }$;

(v) ρ is order continuous in $\mathcal{E}$, i.e., $g_n\in \mathcal{E}$ and |g _n (ω)| ↓, 0 implies ρ(g _n ) ↓, 0.

As in the case of measure spaces, we say that a set A ∈ Σ is ρ-null if ρ(g 1 _A ) = 0 for every $g\in \mathcal{E}$. 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

$$\mathcal{M}\left(\Omega ,\Sigma ,\mathcal{P},\rho \right)=\left\{f\in {\mathcal{M}}_{\infty };\left|f\left(\omega \right)\right|<\infty \rho -\mathsf{\text{a}}.\mathsf{\text{e}}.\right\},$$

(2.1)

where each $f\in \mathcal{M}\left(\Omega ,\Sigma ,\mathcal{P},\rho \right)$ is actually an equivalence class of functions equal ρ-a.e. rather than an individual function. When no confusion arises we will write $\mathcal{M}$. instead of $\mathcal{M}\left(\Omega ,\Sigma ,\mathcal{P},\rho \right)$.

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 $\Re$.

Let us denote ρ(f, E) = ρ(f 1 _E ) for $f\in \mathcal{M}$, E ∈ Σ. It is easy to prove that ρ(f, E) is a function pseudomodular in the sense of Definition 2.1.1 in [20] (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 [20–22]; see also Musielak [23] for the basics of the general modular theory.

Remark 2.1. We limit ourselves to convex function modulars in this article. However, omitting convexity in Definition 2.1 or replacing it by s-convexity would lead to the definition of nonconvex or s-convex regular function pseudomodulars, semimodulars and modulars as in [20].

Definition 2.3. [20–22] Let ρ be a convex function modular.

(a) A modular function space is the vector space L _ρ (Ω, Σ), or briefly L _ρ , defined by

$$L_{\rho }=\left\{f\in \mathcal{M};\rho \left(\lambda f\right)\to 0as\lambda \to 0\right\}.$$

(b) The following formula defines a norm in L _ρ (frequently called Luxemurg norm):

$${∥f∥}_{\rho }=inf\left\{\alpha >0;\rho \left(f∕\alpha \right)\le 1\right\}.$$

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

Theorem 2.1. [20, 21, 22] Let $\rho \in \Re$.

(1) L _ρ , ||f|| _ρ is complete and the norm || · || _ρ is monotone w.r.t. the natural order in $\mathcal{M}$.

(2) ||f _n || _ρ → 0 if and only if ρ(αf _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 $E\in \mathcal{P}$, 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 $H_k\in \mathcal{P}$ such that H _k ↑ Ω and {f _n } 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 $L_{\rho }^0=\left\{f\in L_{\rho };\rho \left(f,\cdot \right)isordercontinuous\right\}$ and $E_{\rho }=\left\{f\in L_{\rho };\lambda f\in L_{\rho }^{0}forevery\lambda >0\right\}$, we have:

(a) $L_{\rho }\supset L_{\rho }^0\supset E_{\rho }$,

(b) E _ρ has the Lebesgue property, i.e., ρ(αf,D _k ) → 0 for α > 0, f ∈ E _ρ and $D_k\downarrow \varnothing$.

(c) E _ρ is the closure of $\mathcal{E}$ (in the sense of || · || _ρ ).

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

Definition 2.4. Let $\rho \in \Re$. We say that ρ has the Δ₂-property if sup _n ρ(2f _n , D _k ) → 0 whenever $D_k\downarrow \varnothing$ and sup _n ρ(f _n , D _k ) → 0.

Theorem 2.2. Let $\rho \in \Re$. The following conditions are equivalent:

(a) ρ has Δ₂-property,

(b) $L_{\rho }^0$ is a linear subspace of L _ρ ,

(c) $L_{\rho }=L_{\rho }^0=E_{\rho }$,

(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 article.

Definition 2.5. Let $\rho \in \Re$.

(a) We say that {f _n } is ρ-convergent to f and write f _n → f (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 its ρ-diameter δ _ρ (B) = 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

$$d_{\rho }\left(f,C\right)=\mathsf{\text{int}}\left\{\rho \left(f-g\right);g\in C\right\}.$$

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 $\rho \in \Re$.

(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., [10]).

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. A nonempty subset K of L _ρ is said to be ρ-compact if for any family {A _α ; A _α ∈ 2^Lρ, α ∈ Γ} of ρ-closed subsets with $K\cap A_{\alpha 1}\cap \cdots \cap A_{\alpha n}\ne \varnothing$, for any α₁,..., α_{n∈ Γ}, we have

$$K\cap \left(\bigcap _{\alpha \in \Gamma }{A}_{\alpha }\right)\ne \varnothing .$$

Next, we give the modular definitions of asymptotic pointwise nonexpansive mappings. The definitions are straightforward generalizations of their norm and metric equivalents [13, 15, 17].

Definition 2.7. Let $\rho \in \Re$ and let C ⊂ L _ρ be nonempty and ρ-closed. A mapping T : C → C is called an asymptotic pointwise mapping if there exists a sequence of mappings α _n : C → [0, ∞) such that

$$\rho \left(T^n\left(f\right)-T^n\left(g\right)\right)\le {\alpha }_n\left(f\right)\rho \left(f-g\right)foranyf,g\in L_{\rho }.$$

(i) If lim sup_n→∞α _n (f) ≤ 1 for any f ∈ L _ρ , then T is called asymptotic point-wise ρ-nonexpansive.

• (ii) If sup_n∈ℕα _n (f) ≤ 1 for any f ∈ L _ρ , then T is called ρ-nonexpansive. In particular, we have

  $$\rho \left(T\left(f\right)-T\left(g\right)\right)\le \rho \left(f-g\right)foranyf,g\in C.$$

The fixed point set of T is defined by Fix(T) = {f ∈ C; T(f) = f}.

In the following definition, we introduce the concept of Banach Operator Pairs [24, 25] in modular function spaces.

Definition 2.8. Let $\rho \in \Re$ and let C ⊂ L _ρ be nonempty. The ordered pair (S, T) of two self-maps of the subset C is called a Banach operator pair, if the set Fix(T) is S-invariant, namely S(Fix(T)) ⊆ Fix(T).

In [26], a result similar to Ky Fan's fixed point theorem in modular function spaces was proved. The following definition is needed:

Definition 2.9. Let $\rho \in \Re$. Let C ⊂ L _ρ be a nonempty ρ-closed subset. Let T : C → L _ρ be a map. T is called ρ-continuous if {T(f _n )} ρ-converges to T(f) whenever {f _n } ρ-converges to f. Also, T will be called strongly ρ-continuous if T is ρ-continuous and

$$\underset{n\to \infty }{liminf}\rho \left(g-T\left(f_n\right)\right)=\rho \left(g-T\left(f\right)\right)$$

for any sequence {f _n } ⊂ C which ρ-converges to f and for any g ∈ C.

3. Common fixed points for Banach operator pairs

The study of a common fixed point of a pair of commuting mappings was initiated as soon as the first fixed point result was proved. This problem becomes more challenging and seems to be of vital interest in view of historically significant and negatively settled problem that a pair of commuting continuous self-mappings on the unit interval [0,1] need not have a common fixed point [27]. Since then, many fixed point theorists have attempted to find weaker forms of commutativity that may ensure the existence of a common fixed point for a pair of self-mappings on a metric space. In this context, the notions of weakly compatible mappings [28] and Banach operator pairs [24, 25, 29–34] have been of significant interest for generalizing results in metric fixed point theory for single valued mappings. In this section, we investigate some of these results in modular function spaces.

We first prove the following technical result.

Theorem 3.1. Let $\rho \in \Re$. Let K ⊂ L _p be ρ-compact convex subset. Then, any T : K → K strongly ρ-continuous has a nonempty fixed point set Fix(T). Moreover, Fix(T) is ρ-compact.

Proof. The existence of a fixed point is proved in [26]. Hence, Fix(T) is nonempty. Let us prove that Fix(T) is ρ-compact. It is enough to show that Fix(T) is ρ-closed since K is ρ-compact. Let {f _n } be a sequence in Fix(T) such that {f _n } ρ-converges to f. Let us prove that f ∈ Fix(T). Since T is ρ-continuous, so {T(f _n )} ρ-converges to T(f). Since T(f _n ) = f _n , we get {f _n } ρ-converges to f and T(f). The uniqueness of the ρ-limit implies T(f) = f, i.e., f ∈ Fix(T).

Definition 3.1. Let K ⊂ L _ρ be nonempty subset. The mapping T : K → K is called R-map if Fix (T) is a ρ-continuous retract of K. Recall that a mapping R : K → Fix(T) is a retract if and only if R ○ R = R.

Note that in general the fixed point set of ρ-continuous mappings defined on any ρ-compact convex subset of L _ρ may not be a ρ-continuous retract.

Theorem 3.2. Let $\rho \in \Re$. Let K ⊂ L _ρ be ρ-compact convex subset. Let T : K → K be strongly ρ-continuous R-map. Let S : K → K be strongly ρ-continuous such that (S,T) is a Banach operator pair. Then, F(S,T) = Fix(T) ⋂ Fix(S) is a nonempty ρ-compact subset of K.

Proof. From Theorem 3.1, we know that Fix(T) is not empty and ρ-compact subset of K. Since T is an R-map, then there exists a ρ-continuous retract R : K → Fix(T). Since (S,T) is a Banach pair of operators, then S(Fix(T)) ⊂ Fix(T). Note that S○R : K → K is strongly ρ-continuous. Indeed, if {f _n } ⊂ K ρ-converges to f, then {R(f _n )} ⊂ K ρ-converges to R(f) since R is ρ-continuous. And since S is strongly ρ-continuous, then for any g ∈ K, we have

$$\underset{n\to \infty }{liminf}\rho \left(g-S\left(R\left(f_n\right)\right)\right)=\rho \left(g-S\left(R\left(f\right)\right)\right),$$

which shows that S ○ R is strongly ρ-continuous. Theorem 3.1 implies that Fix(S ○ R) is nonempty and ρ-compact. Note that if f ∈ Fix(S ○ R), then we have S ○ R(f) = S(R(f)) = f ∈ Fix(T) since S ○ R(K) ○ Fix(T). In particular, we have R(f) = f. Hence, S(f) = f, i.e., f ∈ Fix(T) ⋂ Fix(S). It is easy to then see that Fix(T) ⋂ Fix(S) = Fix(S ○ R) = F(S,T) which implies F(S,T) is nonempty and ρ-compact subset of K.

Before we state next result which deals with ρ-nonexpansive mappings, let us recall the definition of uniform convexity in modular function spaces [18].

Definition 3.2. Let $\rho \in \Re$. We define the following uniform convexity type properties of the function modular ρ:

(i) Let r > 0,ε > 0. Define

$$D_1\left(r,\epsilon \right)=\left\{\left(f,g\right);f,g\in L_{\rho },\rho \left(f\right)\le r,\rho \left(g\right)\le r,\rho \left(f-g\right)\ge \epsilon r\right\}.$$

Let

$${\delta }_1\left(r,\epsilon \right)=inf\left\{1-\frac{1}{r}\rho \left(\frac{f+g}{2}\right);\left(f,g\right)\in D_1\left(r,\epsilon \right)\right\}if{D}_1\left(r,\epsilon \right)\ne \varnothing ,$$

and δ₁(r,ε) = 1 if $D_1\left(r,\epsilon \right)=\varnothing$. We say that ρ satisfies (UC 1) if for every r > 0,ε > 0, δ₁(r,ε) > 0. Note that for every r > 0, $D_1\left(r,\epsilon \right)\ne \varnothing$, for ε > 0 small enough.

(ii) We say that ρ satisfies (UUC 1) if for every s ≥ 0, ε > 0 there exists

$${\eta }_1\left(s,\epsilon \right)>0$$

depending on s and ε such that

$${\delta }_1\left(r,\epsilon \right)>{\eta }_1\left(s,\epsilon \right)>0forr>s.$$

(iii) Let r > 0, ε > 0. Define

$$D_2\left(r,\epsilon \right)=\left\{\left(f,g\right);f,g\in L_{\rho },\rho \left(f\right)\le r,\rho \left(g\right)\le r,\rho \left(\frac{f-g}{2}\right)\ge \epsilon r\right\}.$$

Let

$${\delta }_2\left(r,\epsilon \right)=inf\left\{1-\frac{1}{r}\rho \left(\frac{f+g}{2}\right);\left(f,g\right)\in D_2\left(r,\epsilon \right)\right\}if{D}_2\left(r,\epsilon \right)\ne \varnothing ,$$

and δ₂ (r,ε) = 1 if $D_2\left(r,\epsilon \right)=\varnothing$. We say that ρ satisfies (UC 2) if for every r > 0,ε > 0, δ₂ (r,ε) > 0. Note that for every r > 0, $D_2\left(r,\epsilon \right)\ne \varnothing$, for ε > 0 small enough.

(iv) We say that ρ satisfies (UUC 2) if for every s ≥ 0, ε > 0 there exists

$${\eta }_2\left(s,\epsilon \right)>0$$

depending on s and ε such that

$${\delta }_2\left(r,\epsilon \right)>{\eta }_2\left(s,\epsilon \right)>0forr>s.$$

In [18], it is proved that any asymptotically pointwise ρ-nonexpansive mapping defined on a ρ-closed ρ-bounded convex subset has a fixed point. The next result improves their result by showing that the fixed point set is convex.

Theorem 3.3. Assume $\rho \in \Re$ is (UUC 1). Let C be a ρ-closed ρ-bounded convex nonempty subset of L _ρ . Then, any T : C → C asymptotically pointwise ρ-nonexpansive has a fixed point. Moreover, the set of all fixed points Fix(T) is ρ-closed and convex.

Proof. In [18], it is proved that Fix(T) is a ρ-closed nonempty subset of C. Let us prove that Fix(T) is convex. Let f,g ∈ Fix(T), with f ≠ g. For every n ∈ ℕ, we have

$$\rho \left(f-T^n\left(\frac{f+g}{2}\right)\right)\le {\alpha }_n\left(f\right)\rho \left(\frac{f-g}{2}\right)$$

and

$$\rho \left(g-T^n\left(\frac{f+g}{2}\right)\right)\le {\alpha }_n\left(g\right)\rho \left(\frac{f-g}{2}\right).$$

Set $R=\rho \left(\frac{f-g}{2}\right)$. Then,

$$\underset{n\to \infty }{limsup}\rho \left(f-T^n\left(\frac{f+g}{2}\right)\right)\le R\mathsf{\text{and}}\underset{n\to \infty }{limsup}\rho \left(g-T^n\left(\frac{f+g}{2}\right)\right)\le R.$$

Since

$$\rho \left(\frac{1}{2}\left[f-T^n\left(\frac{f+g}{2}\right)\right]+\frac{1}{2}\left[T^n\left(\frac{f+g}{2}\right)-g\right]\right)=\rho \left(\frac{f-g}{2}\right)=R,$$

and ρ is (UUC 2) (since (UUC 1) implies (UUC 2)), then we must have

$$\underset{n\to \infty }{lim}\rho \left(\frac{1}{2}\left[f-T^n\left(\frac{f+g}{2}\right)\right]-\frac{1}{2}\left[T^n\left(\frac{f+g}{2}\right)-g\right]\right)=0,$$

and so

$$\underset{n\to \infty }{lim}\rho \left(\frac{f+g}{2}-T^n\left(\frac{f+g}{2}\right)\right)=0.$$

Since ρ is convex we get

$$\begin{array}{c}\rho \left(\frac{1}{2}\left[\frac{f+g}{2}-T\left(\frac{f+g}{2}\right)\right]\right)\le \frac{1}{2}\rho \left(\frac{f+g}{2}-T^n\left(\frac{f+g}{2}\right)\right)\\ +\frac{1}{2}\rho \left(T^n\left(\frac{f+g}{2}\right)-T\left(\frac{f+g}{2}\right)\right)\end{array}$$

which implies

$$\begin{array}{c}\rho \left(\frac{1}{2}\left[\frac{f+g}{2}-T\left(\frac{f+g}{2}\right)\right]\right)\le \frac{1}{2}\rho \left(\frac{f+g}{2}-T^n\left(\frac{f+g}{2}\right)\right)\\ +\frac{{\alpha }_1\left(\frac{f+g}{2}\right)}{2}\rho \left(\frac{f+g}{2}-T^{n-1}\left(\frac{f+g}{2}\right)\right).\end{array}$$

If we let n → ∞, we get $\rho \left(\frac{1}{2}\left[\frac{f+g}{2}-T\left(\frac{f+g}{2}\right)\right]\right)=0,\mathsf{\text{i}}\mathsf{\text{.e}}.,T\left(\frac{f+g}{2}\right)=\frac{f+g}{2}$ and so $\frac{f+g}{2}\in Fix\left(T\right)$. This completes the proof of our claim.

As a corollary, we obtain the following result.

Corollary 3.1. Assume $\rho \in \Re$ is (UUC 1). Let C be a ρ-closed ρ-bounded convex nonempty subset of L _p . Then, any T : C → C ρ-nonexpansive has a fixed point. Moreover, the set of all fixed points Fix(T) is ρ-closed and convex.

Next, we discuss the existence of common fixed points for Banach operator pairs of pointwise asymptotically ρ-nonexpansive mappings.

Theorem 3.4. Assume $\rho \in \Re$ is (UUC 1). Let C be a ρ-closed ρ-bounded convex nonempty subset of L _p . Let T : C → C be asymptotically pointwise ρ-nonexpansive mapping. Then, any S : C → C pointwise asymptotically ρ-nonexpansive mapping such that (S, T) is a Banach operator pair has a common fixed point with T. Moreover F(S, T) = Fix(T) ⋂ Fix(S) is a nonempty ρ-closed convex subset of C.

Proof. Since T is asymptotically pointwise ρ-nonexpansive, then Fix(T) is nonempty ρ-closed convex subset of C. Since (S, T) is a Banach operator pair, then we must have S(Fix(T)) ⊂ Fix(T). Theorem 3.3 implies that the restriction of S to Fix(T) has a nonempty fixed point set which is ρ-closed and convex, i.e., F(S,T) = Fix(T) ⋂ Fix(S) is a nonempty ρ-closed convex subset of C. This completes the proof of our claim.

As a corollary, we get the following result.

Corollary 3.2. Assume $\rho \in \Re$ is (UUC 1). Let C be a ρ-closed ρ-bounded convex nonempty subset of L _p . Let T : C → C be ρ-nonexpansive mapping. Then, any S : C → C ρ-nonexpansive mapping such that (S,T) is a Banach operator pair has a common fixed point with T. Moreover, F(S,T) = Fix(T) ⋂ Fix(S) is a nonempty ρ-closed convex subset of C.

4. Common fixed point of Banach operator family

The aim of this section is to extend the common fixed point results found in the previous section to a family of Banach operator mappings. In particular, we prove an analogue of De Marr's result in modular function spaces. In order to obtain such extension we need to introduce the concept of symmetric Banach operator pairs.

Definition 4.1. Let T and S be two self-maps of a set C. The pair (S,T) is called symmetric Banach operator pair if both (S, T) and (T, S) are Banach operator pairs, i.e., T(Fix(S)) ⊆ Fix(S) and S(Fix(T)) ⊆ Fix(T).

Let $\rho \in \Re$ and C be a ρ-closed nonempty subset of L _p . Let $\mathcal{T}$ be a family of self-maps defined on C. Then, the family $\mathcal{T}$ has a common fixed point if it is the fixed point of each member of $\mathcal{T}$. The set of common fixed points is denoted by $Fix\left(\mathcal{T}\right)$. We have by definition $Fix\left(\mathcal{T}\right)={\bigcap }_{T\in \mathcal{T}}Fix\left(T\right)$.

Next, we state an analogue of De Marr's result in modular function spaces.

Theorem 4.1. Let $\rho \in \Re$. Let K ⊂ L _p be nonempty ρ-compact convex subset. Let $\mathcal{T}$ be a family of self-maps defined on K such that any map in $\mathcal{T}$ is strongly ρ-continuous R-map. Assume that any two mappings in $\mathcal{T}$ form a symmetric Banach operator pair. Then, the family $\mathcal{T}$ has a common fixed point. Moreover, $Fix\left(\mathcal{T}\right)$ is a ρ-compact subset of K.

Proof. Using Theorem 3.2, we deduce that for any T₁,T₂,... ,T _n in $\mathcal{T}$, we have Fix(T₁)⋂Fix(T₂)⋂⋯⋂Fix(T _n ) is a nonempty ρ-compact subset of K. Therefore, any finite family of the subsets $\left\{Fix\left(T\right);T\in \mathcal{T}\right\}$ has a nonempty intersection. Since these sets are all ρ-closed and K is ρ-compact, we conclude that $Fix\left(\mathcal{T}\right)={\bigcap }_{T\in \mathcal{T}}Fix\left(T\right)$ is not empty and is ρ-closed. Therefore, $Fix\left(\mathcal{T}\right)$ is a ρ-compact subset of K which finishes the proof of our theorem.

As commuting operators are symmetric Banach operators, so we obtain:

Corollary 4.1. Let $\rho \in \Re$. Let K ⊂ L _p be nonempty ρ-compact convex subset. Let $\mathcal{T}$ be a family of commuting self-maps defined on K such that any map in $\mathcal{T}$ is strongly ρ-continuous R-map. Then, the family $\mathcal{T}$ has a common fixed point. Moreover, $Fix\left(\mathcal{T}\right)$ is a ρ-compact subset of K.

Next, we discuss a similar conclusion in modular function spaces L _p when ρ is (UUC 1). Prior to obtain such result we will need an intersection property which seems to be new. Indeed, it is well known [18] that if $\rho \in \Re$ is (UUC 2), then any countable family {C _n } of ρ-bounded ρ-closed convex subsets of L _p has a nonempty intersection provided that the intersection of any finite subfamily has a nonempty intersection. Such intersection property is known as property (R). This intersection property is parallel to the well-known fact that uniformly convex Banach spaces are reflexive. The property (R) is essential for the proof of many fixed point theorems in metric and modular function spaces. But since it is not clear that this intersection property is related to any topology, we did not know if such intersection property is in fact valid for any family. Therefore, the next result seems to be new.

Theorem 4.2. Assume $\rho \in \Re$ is (UUC 1). Let {C _a }_a∈Γbe a nonincreasing family of nonempty, convex, ρ-closed ρ-bounded subsets of L _p , where Γ is a directed index set. then, ${\bigcap }_{\alpha \in \Gamma }{C}_{\alpha }\ne \varnothing$.

Proof. Recall that Γ is directed if there exists an order ≼ defined on Γ such that for any α,β ∈ Γ, there exists γ ∈ Γ such that α ≼ γ and β ≼ γ. And {C _a }_a∈Γis nonincreasing if and only if for any α, β ∈ Γ such that α ≼ β, then C _β ⊂ C _α . Note that for any α₀ ∈ Γ, we have

$$\bigcap _{\alpha \in \Gamma }{C}_{\alpha }=\bigcap _{{\alpha }_0\preccurlyeq \alpha }{C}_{\alpha }.$$

Therefore, without of any generality, we may assume that there exists C ⊂ L _p ρ-closed ρ-bounded convex subset such that C _α ⊂ C for any α ∈ Γ. If δ _P (C) = 0, then all subsets C _α are reduced to a single point. In this case, we have nothing to prove. Hence, let us assume δ _P (C) > 0. Let f ∈ C. Then, the proximinality of ρ-closed convex subsets of L _p when ρ is (UUC 2) (see [18]) implies the existence of f _α ∈ C _α such that

$$\rho \left(f-f_{\alpha }\right)=d_{\rho }\left(f,C_{\alpha }\right)=inf\left\{\rho \left(f-g\right);g\in C_{\alpha }\right\}.$$

Set A _α = {f _β ; α ≼ β}, for any α ∈ Γ. Then, A _α ⊂ C _α , for any α ∈ Γ. Notice that

$${\delta }_{\rho }\left(A_{\alpha }\right)={\delta }_{\rho }\left({\overline{conv}}^{\rho }\left(A_{\alpha }\right)\right)\mathsf{\text{for}}\mathsf{\text{any}}\alpha \in \Gamma .$$

Indeed, let g ∈ A _α , then A _α ⊂ B(g,δ _ρ (A _α )). Since B(g,δ _ρ (A _α )) is ρ-closed and convex, then we must have ${\overline{conv}}^{\rho }\left(A_{\alpha }\right)\subset B\left(g,{\delta }_{\rho }\left(A_{\alpha }\right)\right)$. Hence, for any $h\in {\overline{conv}}^{\rho }\left(A_{\alpha }\right)$, we have ρ(g - h) ≤ δ _ρ (A _α ). Since g was arbitrary in A _α we conclude that A _α ⊂ B(h, δ _ρ (A _α )). Again for the same reason we get ${\overline{conv}}^{\rho }\left(A_{\alpha }\right)\subset B\left(h,{\delta }_{\rho }\left(A_{\alpha }\right)\right)$. Hence, for any g, $h\in {\overline{conv}}^{\rho }\left(A_{\alpha }\right)$ we have ρ(g - h) ≤ δ _ρ (A _α ), which implies ${\delta }_{\rho }\left({\overline{conv}}^{\rho }\left(A_{\alpha }\right)\right)\le {\delta }_{\rho }\left(A_{\alpha }\right)$. This is enough to have ${\delta }_{\rho }\left({\overline{conv}}^{\rho }\left(A_{\alpha }\right)\right)={\delta }_{\rho }\left(A_{\alpha }\right)$. Set R = sup_α∈Γρ(f - f _α ). Without loss of any generality, we may assume R > 0. Let us prove that inf_α∈Γδ _ρ (A _α ) = 0. Assume not. Then, inf_α∈Γδ _ρ (A _α ) > 0. Set $\delta =\frac{1}{2}{inf}_{\alpha \in \Gamma }{\delta }_{\rho }\left(A_{\alpha }\right)$. Then, for any α ∈ Γ, there exist β,γ ∈ Γ such that α ≼ β and α ≼ γ and

$$\rho \left(f_{\beta }-f_{\gamma }\right)>\delta .$$

Since ρ(f - f _γ ) ≤ R and ρ(f - f _β ) ≤ R, then we have

$$\rho \left(f-\frac{f_{\beta }+f_{\gamma }}{2}\right)\le R\left(1-{\delta }_1\left(R,\frac{\delta }{R}\right)\right).$$

Since f _β , f _γ ∈ C _α and C _α is convex, we get

$$\rho \left(f-f_{\alpha }\right)\le R\left(1-{\delta }_1\left(R,\frac{\delta }{R}\right)\right),$$

using the definition of f _α . Since α was chosen arbitrarily in Γ we get

$$R=\underset{\alpha \in \Gamma }{sup}\rho \left(f-f_{\alpha }\right)\le R\left(1-{\delta }_1\left(R,\frac{\delta }{R}\right)\right).$$

This is a contradiction. Therefore, we have inf_α∈Γ δ _ρ (A _α ) = 0. Since Γ is directed, there exists {α _n } ⊂ Γ such that α _n ≼ α_n+1and inf_{n ≥ 1} δ _ρ (A _αn ) = 0. In particular, we have A_{αn+ 1}⊂ A _αn which implies ${\overline{conv}}^{\rho }\left(A_{{\alpha }_{n+1}}\right)\subset {\overline{conv}}^{\rho }\left(A_{{\alpha }_n}\right)$. Using the property (R) satisfied by L _ρ , we conclude $A={\bigcap }_{n\ge 1}{\overline{conv}}^{\rho }\left(A_{{\alpha }_n}\right)\ne \varnothing$. Since $\underset{\alpha \in \Gamma }{inf}{\delta }_{\rho }\left({\overline{conv}}^{\rho }\left(A_{{\alpha }_n}\right)\right)=\underset{\alpha \in \Gamma }{inf}{\delta }_{\rho }\left(A_{\alpha }\right)=0$, we conclude that A = {h} for some h ∈ C. Let us prove that for any α ∈ Γ we have $h\in {\overline{conv}}^{\rho }\left(A_{\alpha }\right)$. Indeed, let α ∈ Γ. If there exists n ≥ 1 such that α ≼ α _n , then we have A _αn ⊂ A _α . Hence, ${\overline{conv}}^{\rho }\left(A_{{\alpha }_n}\right)\subset {\overline{conv}}^{\rho }\left(A_{\alpha }\right)$. This clearly implies $h\in {\overline{conv}}^{\rho }\left(A_{\alpha }\right)$. Otherwise, assume that for any n ≥ 1 such that α _n ≼ α, so A_α ≼ A _αn . Hence, ${\overline{conv}}^{\rho }\left(A_{\alpha }\right)\subset {\overline{conv}}^{\rho }\left(A_{{\alpha }_n}\right)$. In particular, we have ${\overline{conv}}^{\rho }\left(A_{\alpha }\right)\subset {\bigcap }_{n\ge 1}{\overline{conv}}^{\rho }\left(A_{{\alpha }_n}\right)=\left\{h\right\}$. Which forces $h\in {\overline{conv}}^{\rho }\left(A_{\alpha }\right)$. Therefore, $h\in {\bigcap }_{\alpha \in \Gamma }{\overline{conv}}^{\rho }\left(A_{\alpha }\right)$. Since ${\bigcap }_{\alpha \in \Gamma }{\overline{conv}}^{\rho }\left(A_{\alpha }\right)\subset {\bigcap }_{\alpha \in \Gamma }{C}_{\alpha }$, we conclude that $h\in {\bigcap }_{\alpha \in \Gamma }{C}_{\alpha }$. Hence, ${\bigcap }_{\alpha \in \Gamma }{C}_{\alpha }\ne \varnothing$.

Using Theorem 4.2, we get the following common fixed point result.

Theorem 4.3. Assume $\rho \in \Re$. is (UUC 1). Let C be a ρ-closed ρ-bounded convex nonempty subset of L _p . Let $\mathcal{T}$ be a family of self-maps defined on C such that any map in $\mathcal{T}$ is asymptotically pointwise ρ-nonexpansive. Assume that any two mappings in $\mathcal{T}$ form a symmetric Banach operator pair. Then, the family $\mathcal{T}$ has a common fixed point. Moreover, $Fix\left(\mathcal{T}\right)$ is a ρ-closed convex subset of C.

Proof. Using Theorem 3.4, we deduce that for any T₁,T₂,...,T _n in $\mathcal{T}$, we have Fix(T₁) ⋂ Fix(T₂) ⋂ ⋯ ⋂ Fix(T _n ) is a nonempty ρ-closed convex subset of C. Therefore, any finite family of the subsets $\left\{Fix\left(T\right);T\in \mathcal{T}\right\}$ has a nonempty intersection. Since these sets are all ρ-closed and convex subsets of C, then Theorem 4.2 implies that $Fix\left(\mathcal{T}\right)={\bigcap }_{T\in \mathcal{T}}Fix\left(T\right)$ is not empty and is ρ-closed and convex. Therefore, $Fix\left(\mathcal{T}\right)$ is a ρ-closed convex subset of C which finishes the proof of our theorem.

As corollaries we get the following common fixed point results which seem to be new.

Corollary 4.2. Assume $\rho \in \Re$ is (UUC 1). Let C be a ρ-closed ρ-bounded convex nonempty subset of L _ρ . Let $\mathcal{T}$ be a family of self-maps defined on C such that any map in $\mathcal{T}$ is ρ-nonexpansive. Assume that any two mappings in $\mathcal{T}$ form a symmetric Banach operator pair. Then, the family $\mathcal{T}$ has a common fixed point. Moreover, $Fix\left(\mathcal{T}\right)$ is a ρ-closed convex subset of C.

Corollary 4.3. Assume $\rho \in \Re$ is (UUC 1). Let C be a ρ-closed ρ-bounded convex nonempty subset of L _p . Let $\mathcal{T}$ be a family of commuting self-maps defined on C such that any map in $\mathcal{T}$ is asymptotically pointwise ρ-nonexpansive. Then, the family $\mathcal{T}$ has a common fixed point. Moreover, $Fix\left(\mathcal{T}\right)$ is a ρ-closed convex subset of C.