Asymptotically regular mappings in modular function spaces

ASYMPTOTICALLY REGULAR MAPPINGS IN MODULAR FUNCTION SPACES T. DOMINGUEZ-BENAVIDES, M.A. KHAMSI AND S. SAMADI 1. ABSTRACT Let ρbe a modular function satisfying a ∆2-type condition and Lρthe corresponding modular space. The main result in this paper states that if Cis a ρ-bounded and ρ-a.e sequentially compact subset of Lρand T:C→Cis an asymptotically regular mapping such that lim inf n→∞ |Tn|<2, where |S|denotes the Lipschitz constant of S, then Thas a fixed point. We show that the estimate lim inf n→∞ |Tn|<2 cannot be, in general, improved. 1991 Mathematics subject classification: Primary 46E30; Secondary 47H09, 47H10. Key Words: asymptotically regular mappings, fixed point, modular functions, Opial property. The first author is partially supported by PB-96-1338-C01-C02 and PAI-FMQ-0127. 1 2 T. DOMINGUEZ-BENAVIDES, M.A. KHAMSI AND S. SAMADI 2. INTRODUCTION Let Xbe a metric space. A mapping T:X→Xis said to be asymptotically regular if lim nd(Tn+1x, Tnx) = 0 for each x∈X. This notion was defined by Browder and Petryshyn [2]. The existence of fixed points for asymptotically regular mappings has been widely studied [3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 17]. When Xis a convex, closed, subset of a Banach space it is known [12] that the problem of the existence of fixed point for a nonexpansive mapping is equivalent to the same problem for a nonexpansive asymptotically regular mapping. On the other hand, the theory of modular spaces was initiated by Nakano [21] in 1950 in connection with the theory of order spaces and redefined and generalized by Musielak and Orlicz [20] in 1959. Besides the idea of defining a norm and considering particular Banach spaces of functions, another direction is based on considering an abstractly given funtional defined on a linear space of functions which controls the growth of members of the space. Even though a metric is not defined, many problems in metric fixed point theory can be reformulated in modular spaces (see, for instance [13] and references therein). In this paper, we study the existence of fixed points for asymptotically regular mapping defined from a ρ-bounded and ρ-a.e sequentially compact set Cof a modular space Lρ into C. We actually prove that Thas a fixed point if lim inf n|Tn|<2, |T|being the exact Lipschitz constant of T. It is worthed to notice the simplicity of this statement in comparison with similar results in Banach spaces, where a different upper bound for lim inf n|Tn|must be considered for each space. Even in Hilbert spaces (see [1, chapter IX]) a best estimate is unknown. We also give an example showing that this result can not be, in general, improved. 3. PRELIMINARIES We start by recalling some basic concepts and facts of modular spaces as formulated by Kozlowski. For more details the reader is refered to [13], [14], [15] and [19]. Let Ω be a nonempty set and Σ be a nontrivial σ-algebra of subsets of Ω. Let Pbe a δ-ring of subsets of Σ, such that E∩A∈ P for any E∈ P and A∈Σ. Let us assume that there exists an increasing sequence of sets Kn∈ P such that ASYMPTOTICALLY REGULAR MAPPINGS IN MODULAR FUNCTION SPACES 3 Ω = SKn. In other words, the family Pplays the role of the δ-ring of subsets of finite measure. By Ewe denote the linear space of all simple functions with supports from P. By Mwe will denote the space of all 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 1Awe denote the characteristic function of the set A. Definition 3.1. A functional ρ:E × Σ→[0,∞] is called a function modular if (P1)ρ(0, E) = 0 for any E∈Σ, (P2)ρ(f, E)≤ρ(g, E) whenever |f(ω)|≤|g(ω)|for any ω∈Ω, f, g ∈ E and E∈Σ, (P3)ρ(f, .) : Σ →[0,∞] is a σ-subadditive measure for every f∈ E, (P4)ρ(α, A)→0 as αdecreases to 0 for every A∈ P, where ρ(α, A) = ρ(α1A, A), (P5) if there exists α > 0 such that ρ(α, A) = 0, then ρ(β, A) = 0 for every β > 0, (P6) for any α > 0ρ(α, .) is order continuous on P, that is ρ(α, An)→0 if {An} ∈ P and decreases to ∅. The definition of ρis then extended to f∈ M by ρ(f, E) = sup{ρ(g, E); g∈ E,|g(ω)| ≤ |f(ω)|ω∈Ω}. This will enable us to define ρ(α, E) for sets Enot in P; for the sake of simplicity, we write ρ(f) instead of ρ(f, Ω). Definition 3.2. A set Eis said to be ρ-null if and only if ρ(α, E) = 0 for α > 0. A property p(ω) is said to hold ρ-almost everywhere (ρ-a.e.) if the set {ω∈Ω; p(ω) does not hold }is ρ-null. For example we will say frequently fn→f ρ-a.e. Note that a countable union of ρ-null sets is still ρ-null. In the sequel we will identify sets Aand Bwhose symmetric difference A∆Bis ρ-null; similarly we will identify measurable functions which differ only on a ρ-null set. 4 T. DOMINGUEZ-BENAVIDES, M.A. KHAMSI AND S. SAMADI It is easy to see that the functional ρ:M → [0,∞] is a modular because it satisfies the following properties: (i) ρ(f) = 0 iff f= 0 ρ-a.e. (ii) ρ(α f) = ρ(f) for every scalar αwith |α|= 1 and f∈ M. (iii) ρ(αf +βg)≤ρ(f) + ρ(g) if α+β= 1, α≥0, β ≥0 and f, g ∈ M. In addition, if the following property is satisfied (iii)’ ρ(αf +βg)≤αρ(f) + βρ(g) if α+β= 1 ; α≥0, β ≥0 and f, g ∈ M, we say that ρis a convex modular. The modular ρdefines a corresponding modular space, i.e the vector space Lρgiven by Lρ={f∈ M;ρ(λf)→0 as λ→0}. The modular space Lρcan be equipped with an F-norm defined by ||f||ρ= inf nα > 0; ρµf α¶≤αo. When ρis convex the formula ||f||ρ= inf nα > 0; ρµf α¶≤1o Definition 3.3. (a) The sequence {fn} ⊂ Lρis said to be ρ-convergent to f∈Lρif ρ(fn−f)→ 0 as n→ ∞, (a’) The sequence {fn} ⊂ Lρis said to be ρ-a.e convergent to f∈Lρif the set {ω∈Ω; fn(ω)6→ f(ω)}is ρ-null. (b) The sequence {fn} ⊂ Lρis said to be ρ-Cauchy if ρ(fn−fm)→0 as n and mgo to ∞, (b’) The sequence {fn} ⊂ Lρis said to be ρ-a.e Cauchy if {ω∈Ω; {fn(ω)}is not a Cauchy sequence }is ρ-null. (c) A subset Cof Lρis called ρ-closed if the ρ-limit of a ρ-convergent sequence of Calways belongs to C. (c’) A subset Cof Lρis called ρ-a.e sequentially closed if the ρ-a.e limit of a ρ-a.e convergent sequence of Calways belongs to C. ASYMPTOTICALLY REGULAR MAPPINGS IN MODULAR FUNCTION SPACES 5 (d) A subset Cof Lρis called ρ-sequentially compact if every sequence in C has a ρ-convergent subsequence in C. (d’) A subset Cof Lρis called ρ-a.e sequentially compact if every sequence in Chas a ρ-a.e convergent subsequence in C. (e) A subset Cof Lρis called ρ-bounded if δρ(C) = sup{ρ(f−g); f, g ∈C}<∞. Definition 3.4. Let ρbe a function modular, we define a growth funtion ωρ: [0,∞]→[0,∞] by ωρ(t) = sup ½ρ(tf) ρ(f);f∈Lρ,0< ρ(f)<∞¾, t ≥0. The following technical result [13] is fundamental for this work. Lemma 3.1. Let {fn}nbe a sequence in Eρsuch that fn ρ−a.e →f∈Eρand there exists k > 1 such that supnρ(k(fn−f)) <∞. Then, we have lim inf n→∞ ρ(fn−g) = lim inf n→∞ ρ(fn−f) + ρ(f−g) for all g∈Eρ and, therefore, lim inf n→∞ ρ(fn−f)≤lim inf n→∞ ρ(fn−g) for all g∈Eρ. ¿From this lemma a uniform Opial property-type for Eρcan be derived. Recall [22] that a Banach space is said to satisfy the uniform Opial property with respect to an arbitrary topology τif for every c > 0 there exists r > 0 such that lim inf n→∞ ||xn+x|| ≥ 1 + r, if {xn}nis a τ-null sequence with lim inf n→∞ ||xn|| ≥ 1 and ||x|| ≥ c. From lemma 3.1 it is clear that lim inf n→∞ ρ(fn+f)≥1+cif ρ(f)≥cand {fn}nis a ρ-a.e null sequence which satisfies lim inf n→∞ ρ(fn)≥1 and there exists K > 1 such that supnρ¡K(fn−f)¢<∞. 6 T. DOMINGUEZ-BENAVIDES, M.A. KHAMSI AND S. SAMADI 4. SOME TECHNICAL RESULTS Definition 4.1. Let ρbe a function modular. We say that ρsatisfies the ∆2- type condition if there exists K > 0 such that ρ(2f)≤Kρ(f) for all f∈Lρ. As examples of convex funtion modular with ∆2-type condition we mention, the usual lpspaces and function modular of Orlicz spaces, where the measure space (Ω,Σ, µ) is σ-finite, the measure µis atomless and infinite, and the Orlicz function ψis convex satisfying ∆2-type condition, i.e. lim sup u→∞ ψ(2u) ψ(u)<∞and lim sup u→0 ψ(2u) ψ(u)<∞. Note that the ∆2-type condition implies the ∆2-condition. For this condition we refer to [15] and [19]. In the sequel, we will assume that ρis convex and satisfies the ∆2-condition. In this case we have Lρ=Eρ. The following lemma can be easily proved. Lemma 4.1. The growth funtion ωρhas the following properties: (1) ωρ(t)<∞,∀t∈[0,∞) (2) ωρ: [0,∞)→[0,∞) is a convex, strictly increasing function. So, it is continuous. (3) ωρ(αβ)≤ωρ(α)ωρ(β); ∀α, β ∈[0,∞) (4) ω−1 ρ(α)ω−1 ρ(β)≤ω−1 ρ(αβ);∀α, β ∈[0,∞), where ω−1 ρis the inverse function of ωρ. The following is a technical lemma which will be needed because of lack of the triangular inequality. Lemma 4.2. Let {fn}and {gn}be two sequences in Lρ. Then lim n→∞ ρ(gn) = 0 =⇒lim sup n→∞ ρ(fn+gn) = lim sup n→∞ ρ(fn). ASYMPTOTICALLY REGULAR MAPPINGS IN MODULAR FUNCTION SPACES 7 Proof. By property (iii) of the modular ρ, we have ρ(fn+gn)≤ρµfn 1−ε¶+ρ³gn ε´,∀ε∈(0,1) Thus, ρ(fn+gn)≤ωµ1 1−ε¶ρ(fn) + ωµ1 ε¶ρ(gn) and lim sup n→∞ ρ(fn+gn)≤ωµ1 1−ε¶lim sup n→∞ ρ(fn) Since εis arbitrary and ωµ1 1−ε¶→1 as ε→0+ we obtain lim sup n→∞ ρ(fn+gn)≤lim sup n→∞ ρ(fn). Furthermore, the same argument proves lim sup n→∞ ρ(fn) = lim sup n→∞ ρ(fn+gn−gn) ≤lim sup n→∞ ρ(fn+gn). The following lemma shows that the growth function can be used to give an upper bound for the norm of a function. Lemma 4.3. Let Lρbe a function modular space satisfying the ∆2-type condition. Then ||f||ρ≤1 ω−1³1 ρ(f)´ Proof. Assume, α < ||f||ρ.We have 1 < ρ(f/α) which implies 1 ρ(f)< ω µ1 α¶ 8 T. DOMINGUEZ-BENAVIDES, M.A. KHAMSI AND S. SAMADI and so ω−1µ1 ρ(f)¶<1 α. Letting α→ ||f||ρ −, we obtain ||f||ρ≤1 ω−1³1 ρ(f)´. 5. MAIN RESULT In this section we present the main result of this work. Indeed, we will prove a fixed point theorem for asymptotically regular mappings. Certainly, one can also consider mappings which are asymptotically regular with respect to the F-norm induced by the modular function. We should like to mention that, generally speaking, there is no natural relation between these two kinds of asymptotically regularness. Indeed all results expressed in terms of modulars are more convenient in the sense that their assumptions are much easier to verify. Let Ca subset of Lρand T:C→C, we denote by |T|the exact Lipschitz constant of T, i.e. |T|= sup ½ρ(Tf −Tg) ρ(f−g):f6=g, f, g ∈C¾and s(T) = lim inf n→∞ |Tn|. Theorem 5.1. Let ρbe a convex modular function satisfying the ∆2-type condition, Caρ-bounded, ρ−a.e sequentially compact subset of Lρ. Let T:C→C be an asymptotically regular mapping such that s(T)<2. Then, T has a fixed point. Proof. Choose a sequence {nk}of positive integers such that s(T) = lim k→∞ |Tnk|= lim inf n→∞ |Tn|and define a function ron C by r(f) = inf{r > 0 : ∃g∈Csuch that lim inf k→∞ ρ(f−Tnkg)≤r}. Since s(T)<2, there exists b∈(1,2) such that s(T)< b < 2.Let ε∈(0,1) such that ωρµ1 ε¶<1 b−1and choose γ∈(0,1) such that ωρµ1 ε¶<γ b−1.Since γ+ (1 −b)ωρ¡1 ε¢>0, we can choose δ∈(0,1) such that ASYMPTOTICALLY REGULAR MAPPINGS IN MODULAR FUNCTION SPACES 9 δ < γ + (1 −b)ωρ¡1 ε¢. Now, we choose a number µ∈(0,1) such that µ < min (δ ωρ¡1 1−ε¢,γ−δ+ωρ¡1 ε¢(1 −b) ωρ¡1 ε¢b) Finally,denote α= max (1 + µ−δ ωρ¡1 1−ε¢, b(1 + µ)−γ−δ ωρ¡1 ε¢). Then 0 < α < 1.Since γ < 1, we can find a positive integer k0such that |Tnk0|< b and ρ(f−Tnk0f)> γ r(f). Since µ > 0, we can also find g∈Csuch that lim inf k→∞ ρ(f−Tnkg)≤r(f)(1 + µ). Consider a subsequence {nk0}of {nk}such that {Tnk0g}is ρ-a.e convergent in C, say to h, and lim k0→∞ ρ(f−Tnk0g) = lim inf k→∞ ρ(f−Tnkg). Therefore, using Lemma 4.2 and the asymptotic regularity of Twe have lim sup k0→∞ ρ(Tnk0f−Tnk0g)≤ |Tnk0|lim sup k0→∞ ρ(f−Tnk0−nk0g) =|Tnk0|lim sup k0→∞ ρ(f−Tnk0g+Tnk0g−Tnk0−nk0g) =|Tnk0|lim sup k0→∞ ρ(f−Tnk0g). We split the proof into two cases: Case 1. Assume that ρ(f−h)≥δ ωρ¡1 1−ε¢r(f)