Banach operator pairs and common fixed points in modular function spaces

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Research

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.

Keywords

Banach operator pair fixed point modular function space nearest point projection asymptotically pointwise ρ-nonexpansive mapping 

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, 2, 3, 4]) and has been well developed since the 1960s and generalized to other metric spaces (see, e.g., [5, 6, 7]), and modular function spaces (see, e.g., [8, 9, 10, 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, 16, 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 P Open image in new window be a δ-ring of subsets of Ω, such that E A P Open image in new window for any E P Open image in new window and A ∈ Σ. Let us assume that there exists an increasing sequence of sets K n P Open image in new window such that Ω = UK n . By E Open image in new window we denote the linear space of all simple functions with supports from P Open image in new window. By M Open image in new window 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 Open image in new window, |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 , ] Open image in new window 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 M Open image in new window;

(iii) ρ is orthogonally subadditive, i.e., ρ(f 1AB) ≤ ρ(f 1 A ) + ρ(f 1 B ) for any A,B ∈ Σ such that A B , f M Open image in new window;

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

(v) ρ is order continuous in E Open image in new window, i.e., g n E Open image in new window 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 E Open image in new window. 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
M ( Ω , Σ , P , ρ ) = { f M ; | f ( ω ) | < ρ - a . e . } , Open image in new window
(2.1)

where each f M ( Ω , Σ , P , ρ ) Open image in new window is actually an equivalence class of functions equal ρ-a.e. rather than an individual function. When no confusion arises we will write M Open image in new window. instead of M ( Ω , Σ , P , ρ ) Open image in new window.

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 Open image in new window.

Let us denote ρ(f, E) = ρ(f 1 E ) for f M Open image in new window, 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, 21, 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, 21, 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 ρ = { f M ; ρ ( λ f ) 0 a s λ 0 } . Open image in new window
(b) The following formula defines a norm in L ρ (frequently called Luxemurg norm):
f ρ = inf { α > 0 ; ρ ( f α ) 1 } . Open image in new window

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 ρ Open image in new window.

(1) L ρ , ||f|| ρ is complete and the norm || · || ρ is monotone w.r.t. the natural order in M Open image in new window.

(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 P Open image in new window, 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 P Open image in new window 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 ρ 0 = { f L ρ ; ρ ( f , ) i s o r d e r c o n t i n u o u s } Open image in new window and E ρ = { f L ρ ; λ f L ρ 0 f o r e v e r y λ > 0 } Open image in new window, we have:

(a) L ρ L ρ 0 E ρ Open image in new window,

(b) E ρ has the Lebesgue property, i.e., ρ(αf,D k ) → 0 for α > 0, fE ρ and D k Open image in new window.

(c) E ρ is the closure of E Open image in new window (in the sense of || · || ρ ).

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

Definition 2.4. Let ρ Open image in new window. We say that ρ has the Δ2-property if sup n ρ(2f n , D k ) → 0 whenever D k Open image in new window and sup n ρ(f n , D k ) → 0.

Theorem 2.2. Let ρ Open image in new window. The following conditions are equivalent:

(a) ρ has Δ2-property,

(b) L ρ 0 Open image in new window is a linear subspace of L ρ ,

(c) L ρ = L ρ 0 = E ρ Open image in new window,

(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 ρ Open image in new window.

(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 BL 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 BL ρ is called ρ-bounded if its ρ-diameter δ ρ (B) = sup{ρ(f - g); fB,gB} < ∞.

(e) Let fL ρ and CL ρ . The ρ-distance between f and C is defined as
d ρ ( f , C ) = int { ρ ( f - g ) ; g C } . Open image in new window

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 Open image in new window.

(i) L ρ is ρ-complete,

(ii) ρ-balls B ρ (x, r) = {yL 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 , α ∈ Γ} of ρ-closed subsets with K A α 1 A α n Open image in new window, for any α1,..., αn∈ Γ, we have
K α Γ A α . Open image in new window

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 ρ Open image in new window and let CL ρ be nonempty and ρ-closed. A mapping T : CC is called an asymptotic pointwise mapping if there exists a sequence of mappings α n : C → [0, ∞) such that
ρ T n ( f ) - T n ( g ) α n ( f ) ρ ( f - g ) f o r a n y f , g L ρ . Open image in new window
(i) If lim supn→∞α n (f) ≤ 1 for any fL ρ , then T is called asymptotic point-wise ρ-nonexpansive.
  1. (ii)
    If supn∈ℕα n (f) ≤ 1 for any fL ρ , then T is called ρ-nonexpansive. In particular, we have
    ρ T ( f ) - T ( g ) ρ ( f - g ) f o r a n y f , g C . Open image in new window
     

The fixed point set of T is defined by Fix(T) = {fC; 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 ρ Open image in new window and let CL 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 ρ Open image in new window. Let CL ρ be a nonempty ρ-closed subset. Let T : CL 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
liminf n ρ ( g - T ( f n ) ) = ρ ( g - T ( f ) ) Open image in new window

for any sequence {f n } ⊂ C which ρ-converges to f and for any gC.

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, 30, 31, 32, 33, 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 ρ Open image in new window. Let KL p be ρ-compact convex subset. Then, any T : KK 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 fFix(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., fFix(T).

Definition 3.1. Let KL ρ be nonempty subset. The mapping T : KK is called R-map if Fix (T) is a ρ-continuous retract of K. Recall that a mapping R : KFix(T) is a retract if and only if RR = 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 ρ Open image in new window. Let KL ρ be ρ-compact convex subset. Let T : KK be strongly ρ-continuous R-map. Let S : KK 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 : KFix(T). Since (S,T) is a Banach pair of operators, then S(Fix(T)) ⊂ Fix(T). Note that SR : KK 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 gK, we have
liminf n ρ ( g - S ( R ( f n ) ) ) = ρ ( g - S ( R ( f ) ) ) , Open image in new window

which shows that SR is strongly ρ-continuous. Theorem 3.1 implies that Fix(SR) is nonempty and ρ-compact. Note that if fFix(SR), then we have SR(f) = S(R(f)) = fFix(T) since SR(K) ○ Fix(T). In particular, we have R(f) = f. Hence, S(f) = f, i.e., fFix(T) ⋂ Fix(S). It is easy to then see that Fix(T) ⋂ Fix(S) = Fix(SR) = 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 ρ Open image in new window. We define the following uniform convexity type properties of the function modular ρ:

(i) Let r > 0,ε > 0. Define
D 1 ( r , ε ) = { ( f , g ) ; f , g L ρ , ρ ( f ) r , ρ ( g ) r , ρ ( f - g ) ε r } . Open image in new window
Let
δ 1 ( r , ε ) = inf 1 - 1 r ρ f + g 2 ; ( f , g ) D 1 ( r , ε ) i f D 1 ( r , ε ) , Open image in new window

and δ1(r,ε) = 1 if D 1 ( r , ε ) = Open image in new window. 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 ( r , ε ) Open image in new window, for ε > 0 small enough.

(ii) We say that ρ satisfies (UUC 1) if for every s ≥ 0, ε > 0 there exists
η 1 ( s , ε ) > 0 Open image in new window
depending on s and ε such that
δ 1 ( r , ε ) > η 1 ( s , ε ) > 0 f o r r > s . Open image in new window
(iii) Let r > 0, ε > 0. Define
D 2 ( r , ε ) = ( f , g ) ; f , g L ρ , ρ ( f ) r , ρ ( g ) r , ρ f - g 2 ε r . Open image in new window
Let
δ 2 ( r , ε ) = inf 1 - 1 r ρ f + g 2 ; ( f , g ) D 2 ( r , ε ) i f D 2 ( r , ε ) , Open image in new window

and δ2 (r,ε) = 1 if D 2 ( r , ε ) = Open image in new window. 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 ( r , ε ) Open image in new window, for ε > 0 small enough.

(iv) We say that ρ satisfies (UUC 2) if for every s ≥ 0, ε > 0 there exists
η 2 ( s , ε ) > 0 Open image in new window
depending on s and ε such that
δ 2 ( r , ε ) > η 2 ( s , ε ) > 0 f o r r > s . Open image in new window

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 ρ Open image in new window is (UUC 1). Let C be a ρ-closed ρ-bounded convex nonempty subset of L ρ . Then, any T : CC 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,gFix(T), with fg. For every n ∈ ℕ, we have
ρ f - T n f + g 2 α n ( f ) ρ f - g 2 Open image in new window
and
ρ g - T n f + g 2 α n ( g ) ρ f - g 2 . Open image in new window
Set R = ρ f - g 2 Open image in new window. Then,
limsup n ρ f - T n f + g 2 R and limsup n ρ g - T n f + g 2 R . Open image in new window
Since
ρ 1 2 f - T n f + g 2 + 1 2 T n f + g 2 - g = ρ f - g 2 = R , Open image in new window
and ρ is (UUC 2) (since (UUC 1) implies (UUC 2)), then we must have
lim n ρ 1 2 f - T n f + g 2 - 1 2 T n f + g 2 - g = 0 , Open image in new window
and so
lim n ρ f + g 2 - T n f + g 2 = 0 . Open image in new window
Since ρ is convex we get
ρ 1 2 f + g 2 - T f + g 2 1 2 ρ f + g 2 - T n f + g 2 + 1 2 ρ T n f + g 2 - T f + g 2 Open image in new window
which implies
ρ 1 2 f + g 2 - T f + g 2 1 2 ρ f + g 2 - T n f + g 2 + α 1 f + g 2 2 ρ f + g 2 - T n - 1 f + g 2 . Open image in new window

If we let n → ∞, we get ρ 1 2 f + g 2 - T f + g 2 = 0 , i .e . , T f + g 2 = f + g 2 Open image in new window and so f + g 2 F i x ( T ) Open image in new window. This completes the proof of our claim.

As a corollary, we obtain the following result.

Corollary 3.1. Assume ρ Open image in new window is (UUC 1). Let C be a ρ-closed ρ-bounded convex nonempty subset of L p . Then, any T : CC 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 ρ Open image in new window is (UUC 1). Let C be a ρ-closed ρ-bounded convex nonempty subset of L p . Let T : CC be asymptotically pointwise ρ-nonexpansive mapping. Then, any S : CC 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 ρ Open image in new window is (UUC 1). Let C be a ρ-closed ρ-bounded convex nonempty subset of L p . Let T : CC be ρ-nonexpansive mapping. Then, any S : CC 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 ρ Open image in new window and C be a ρ-closed nonempty subset of L p . Let T Open image in new window be a family of self-maps defined on C. Then, the family T Open image in new window has a common fixed point if it is the fixed point of each member of T Open image in new window. The set of common fixed points is denoted by F i x ( T ) Open image in new window. We have by definition F i x ( T ) = T T F i x ( T ) Open image in new window.

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

Theorem 4.1. Let ρ Open image in new window. Let KL p be nonempty ρ-compact convex subset. Let T Open image in new window be a family of self-maps defined on K such that any map in T Open image in new window is strongly ρ-continuous R-map. Assume that any two mappings in T Open image in new window form a symmetric Banach operator pair. Then, the family T Open image in new window has a common fixed point. Moreover, F i x ( T ) Open image in new window is a ρ-compact subset of K.

Proof. Using Theorem 3.2, we deduce that for any T1,T2,... ,T n in T Open image in new window, we have Fix(T1)⋂Fix(T2)⋂⋯⋂Fix(T n ) is a nonempty ρ-compact subset of K. Therefore, any finite family of the subsets { F i x ( T ) ; T T } Open image in new window has a nonempty intersection. Since these sets are all ρ-closed and K is ρ-compact, we conclude that F i x ( T ) = T T F i x ( T ) Open image in new window is not empty and is ρ-closed. Therefore, F i x ( T ) Open image in new window 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 ρ Open image in new window. Let KL p be nonempty ρ-compact convex subset. Let T Open image in new window be a family of commuting self-maps defined on K such that any map in T Open image in new window is strongly ρ-continuous R-map. Then, the family T Open image in new window has a common fixed point. Moreover, F i x ( T ) Open image in new window 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 ρ Open image in new window 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 ρ Open image in new window 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, α Γ C α Open image in new window.

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 αβ, then C β C α . Note that for any α0 ∈ Γ, we have
α Γ C α = α 0 α C α . Open image in new window
Therefore, without of any generality, we may assume that there exists CL 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 fC. 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
ρ ( f - f α ) = d ρ ( f , C α ) = inf { ρ ( f - g ) ; g C α } . Open image in new window
Set A α = {f β ; αβ}, for any α ∈ Γ. Then, A α C α , for any α ∈ Γ. Notice that
δ ρ ( A α ) = δ ρ c o n v ¯ ρ A α for any α Γ . Open image in new window
Indeed, let gA 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 c o n v ¯ ρ ( A α ) B ( g , δ ρ ( A α ) ) Open image in new window. Hence, for any h c o n v ¯ ρ ( A α ) Open image in new window, 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 c o n v ¯ ρ ( A α ) B ( h , δ ρ ( A α ) ) Open image in new window. Hence, for any g, h c o n v ¯ ρ ( A α ) Open image in new window we have ρ(g - h) ≤ δ ρ (A α ), which implies δ ρ c o n v ¯ ρ A α δ ρ ( A α ) Open image in new window. This is enough to have δ ρ c o n v ¯ ρ A α = δ ρ ( A α ) Open image in new window. Set R = supα∈Γρ(f - f α ). Without loss of any generality, we may assume R > 0. Let us prove that infα∈Γδ ρ (A α ) = 0. Assume not. Then, infα∈Γδ ρ (A α ) > 0. Set δ = 1 2 inf α Γ δ ρ ( A α ) Open image in new window. Then, for any α ∈ Γ, there exist β,γ ∈ Γ such that αβ and αγ and
ρ ( f β - f γ ) > δ . Open image in new window
Since ρ(f - f γ ) ≤ R and ρ(f - f β ) ≤ R, then we have
ρ f - f β + f γ 2 R 1 - δ 1 R , δ R . Open image in new window
Since f β , f γ C α and C α is convex, we get
ρ ( f - f α ) R 1 - δ 1 R , δ R , Open image in new window
using the definition of f α . Since α was chosen arbitrarily in Γ we get
R = sup α Γ ρ ( f - f α ) R 1 - δ 1 R , δ R . Open image in new window

This is a contradiction. Therefore, we have infα∈Γ δ ρ (A α ) = 0. Since Γ is directed, there exists {α n } ⊂ Γ such that α n αn+1and infn ≥ 1 δ ρ (A αn ) = 0. In particular, we have Aαn+ 1A αn which implies c o n v ¯ ρ A α n + 1 c o n v ¯ ρ A α n Open image in new window. Using the property (R) satisfied by L ρ , we conclude A = n 1 c o n v ¯ ρ A α n Open image in new window. Since inf α Γ δ ρ c o n v ¯ ρ A α n = inf α Γ δ ρ ( A α ) = 0 Open image in new window, we conclude that A = {h} for some hC. Let us prove that for any α ∈ Γ we have h c o n v ¯ ρ ( A α ) Open image in new window. Indeed, let α ∈ Γ. If there exists n ≥ 1 such that αα n , then we have A αn A α . Hence, c o n v ¯ ρ A α n c o n v ¯ ρ A α Open image in new window. This clearly implies h c o n v ¯ ρ ( A α ) Open image in new window. Otherwise, assume that for any n ≥ 1 such that α n α, so AαA αn . Hence, c o n v ¯ ρ A α c o n v ¯ ρ A α n Open image in new window. In particular, we have c o n v ¯ ρ ( A α ) n 1 c o n v ¯ ρ A α n = { h } Open image in new window. Which forces h c o n v ¯ ρ ( A α ) Open image in new window. Therefore, h α Γ c o n v ¯ ρ A α Open image in new window. Since α Γ c o n v ¯ ρ ( A α ) α Γ C α Open image in new window, we conclude that h α Γ C α Open image in new window. Hence, α Γ C α Open image in new window.

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

Theorem 4.3. Assume ρ Open image in new window. is (UUC 1). Let C be a ρ-closed ρ-bounded convex nonempty subset of L p . Let T Open image in new window be a family of self-maps defined on C such that any map in T Open image in new window is asymptotically pointwise ρ-nonexpansive. Assume that any two mappings in T Open image in new window form a symmetric Banach operator pair. Then, the family T Open image in new window has a common fixed point. Moreover, F i x ( T ) Open image in new window is a ρ-closed convex subset of C.

Proof. Using Theorem 3.4, we deduce that for any T1,T2,...,T n in T Open image in new window, 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 { F i x ( T ) ; T T } Open image in new window has a nonempty intersection. Since these sets are all ρ-closed and convex subsets of C, then Theorem 4.2 implies that F i x ( T ) = T T F i x ( T ) Open image in new window is not empty and is ρ-closed and convex. Therefore, F i x ( T ) Open image in new window 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 ρ Open image in new window is (UUC 1). Let C be a ρ-closed ρ-bounded convex nonempty subset of L ρ . Let T Open image in new window be a family of self-maps defined on C such that any map in T Open image in new window is ρ-nonexpansive. Assume that any two mappings in T Open image in new window form a symmetric Banach operator pair. Then, the family T Open image in new window has a common fixed point. Moreover, F i x ( T ) Open image in new window is a ρ-closed convex subset of C.

Corollary 4.3. Assume ρ Open image in new window is (UUC 1). Let C be a ρ-closed ρ-bounded convex nonempty subset of L p . Let T Open image in new window be a family of commuting self-maps defined on C such that any map in T Open image in new window is asymptotically pointwise ρ-nonexpansive. Then, the family T Open image in new window has a common fixed point. Moreover, F i x ( T ) Open image in new window is a ρ-closed convex subset of C.

Notes

Acknowledgements

The authors gratefully acknowledge the financial support from the Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU) represented by the Unit of Research Groups through the Grant number (11/31/Gr) for the group entitled "Nonlinear Analysis and Applied Mathematics".

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© Hussain et al; licensee Springer. 2011

This article is published under license to BioMed Central Ltd. 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.

Authors and Affiliations

  1. 1.Department of MathematicsKing Abdulaziz UniversityJeddahSaudi Arabia
  2. 2.Department of Mathematical SciencesThe University of Texas at El PasoEl PasoUSA

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