Asymptotically nonexpansive mappings in modular function spaces

ASYMPTOTICALLY NONEXPANSIVE MAPPINGS IN MODULAR FUNCTION SPACES T. DOMINGUEZ-BENAVIDES, M.A. KHAMSI AND S. SAMADI ABSTRACT In this paper, we prove that if ρis a convex, σ-finite modular function satisfying a ∆2-type condition, Ca convex, ρ-bounded, ρ-a.e. compact subset of Lρand T:C→Caρ-asymptotically nonexpansive mapping, then Thas a fixed point. In particular, any asymptotically nonexpansive self-map defined on a convex subset of L1(Ω, µ) which is compact for the topology of local convergence in measure has a fixed point. 1991 Mathematics subject classification : Primary 46E30; Secondary 47H09, 47H10. Key Words: asymptotically nonexpansive mappings, fixed point, modular functions. 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 INTRODUCTION Let (M, d) be a metric space. A mapping, T:M→Mis said to be asymptotically nonexpansive if there exists a sequence {kn}of real numbers with lim n→∞ kn= 1 such that d(Tnx, Tny)≤knd(x, y) for any x, y ∈Mand n∈N.In 1970 Goebel and Kirk [5] proved that Thas a fixed point whenever Mis a convex bounded closed subset of a Banach space X. Further generalizations of this result were proved by Yu and Dai [14] when Xis 2-uniformly rotund, by Mart´ınez Ya˜nez [10] and Xu [12] when Xis k-uniformly rotund for some k≥1,by Xu [13] when Xis nearly uniformly convex and by Kim and Xu [9] when Xhas uniform normal stucture. Some special studies on the theory of the fixed point for asymptotically nonexpansive mappings were made by many other authors (see, for example, [2,11]). The first fixed point results in modular function spaces were given by Khamsi, Koz lowski and Reich [7]. Even though a metric is not defined, many problems in metric fixed point theory can be reformulated in modular spaces. For instance, fixed point theorems are proved in [6,7] for nonexpansive mappings, in [3] for asymptotically regular mappings and in [4] for uniformly Lipschitzian mappings. In this paper we will prove the existence of fixed points for asymptotically nonexpansive mappings in modular function spaces when the modular ρsatisfies some convexity and ∆2-type properties. Our results can be, in particular, applied to L1(Ω, µ), showing that asymptotically nonexpansive mappings have a fixed point when they are defined on a convex subset of L1(Ω, µ) which is compact with respect to the topology of convergence local in measure. 1. PRELIMINARIES We start by reviewing some basic facts about modular spaces as formulated by Koz lowski [8]. For more details the reader may consult [6,7]. 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 NONEXPANSIVE MAPPINGS IN MODULAR FUNCTION SPACES 3 Ω = SKn. 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 →Rsuch 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 1.1. 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)|for every ω∈Ω}. Definition 1.2. 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. For the sake of simplicity we write ρ(f) instead of ρ(f, Ω). Definition 1.3. A modular function ρis called σ-finite if there exists an increasing sequence of sets Kn∈ P such that 0 < ρ(Kn)<∞and Ω = SKn. It is easy to see that the functional ρ:M → [0,∞] is a modular and satisfies the following properties: (i) ρ(f) = 0 iff f= 0 ρ-a.e. 4 T. DOMINGUEZ-BENAVIDES, M.A. KHAMSI AND S. SAMADI (ii) ρ(αf) = 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 Lρ given by Lρ={f∈ M;ρ(λf)→0 as λ→0}. When ρis convex, the formula ||f||ρ= inf nα > 0; ρf α≤1o defines a norm in the modular space Lρwhich is frequently called the Luxemburg norm. We can also consider the space Eρ={f∈ M;ρ(αf, An)→0 as n→ ∞ for every An∈ Σ that decreases to ∅and α > 0}. Definition 1.4. A function modular is said to satisfy the ∆2-condition if sup n≥1 ρ(2fn, Dk)→0 as k→ ∞ whenever {fn}n≥1⊂ M, Dk∈ Σ decreases to ∅and sup n≥1 ρ(fn, Dk)→0 as k→ ∞. We know from [8] that Eρ=Lρwhen ρsatisfies the ∆2-condition. Definition 1.5. A function modular is said to satisfy the ∆2-type condition if there exists K > 0 such that for any f∈Lρwe have ρ(2f)≤Kρ(f). In general, ∆2-type condition and ∆2-condition are not equivalent, even though it is obvious that ∆2-type condition implies ∆2-condition on the modular space Lρ. Definition 1.6. Let Lρbe a modular space. ASYMPTOTICALLY NONEXPANSIVE MAPPINGS IN MODULAR FUNCTION SPACES 5 (1) The sequence {fn}n⊂Lρis said to be ρ-convergent to f∈Lρif ρ(fn− f)→0 as n→ ∞. (2) The sequence {fn}n⊂Lρis said to be ρ-a.e. convergent to f∈Lρif the set {ω∈Ω; fn(ω)6→ f(ω)}is ρ-null. (3) The sequence {fn}n⊂Lρis said to be ρ-Cauchy if ρ(fn−fm)→0 as n and mgo to ∞. (4) A subset Cof Lρis called ρ-closed if the ρ-limit of a ρ-convergent sequence of Calways belongs to C. (5) A subset Cof Lρis called ρ-a.e. closed if the ρ-a.e. limit of a ρ-a.e. convergent sequence of Calways belongs to C. (6) A subset Cof Lρis called ρ-a.e. compact if every sequence in Chas a ρ-a.e. convergent subsequence in C. (7) A subset Cof Lρis called ρ-bounded if δρ(C) = sup{ρ(f−g); f, g ∈C}<∞. We recall two basic results (see [7]) in the theory of modular spaces. (i) If there exists a number α > 0 such that ρ(α(fn−f)) →0,then there exists a subsequence {gn}nof {fn}nsuch that gn→f ρ-a.e. (ii) (Lebesgue’s Theorem) If fn, f ∈ M,fn→f ρ-a.e. and there exists a function g∈Eρsuch that |fn| ≤ |g|ρ-a.e. for all n, then ||fn−f||ρ→0. We know, by [6,7] that under ∆2-condition the norm convergence and modular convergence are equivalent, which implies that the norm and modular convergence are also the same when we deal with the ∆2-type condition. In the sequel we will assume that the modular function ρis convex and satisfies the ∆2-type condition. Definition 1.7. Let ρbe as above. We define a growth function ωby: ω(t) = sup ρ(tf) ρ(f), f ∈Lρ\{0}for all 0 ≤t < ∞. We have the following: Lemma 1.1. [3] Let ρbe as above. Then the growth function ωhas the following properties: 6 T. DOMINGUEZ-BENAVIDES, M.A. KHAMSI AND S. SAMADI (1) ω(t)<∞,∀t∈[0,∞) (2) ω: [0,∞)→[0,∞)is a convex, strictly increasing function. So, it is continuous. (3) ω(αβ)≤ω(α)ω(β); ∀α, β ∈[0,∞) (4) ω−1(α)ω−1(β)≤ω−1(αβ);∀α, β ∈[0,∞),where ω−1is the function inverse of ω. The following lemma shows that the growth function can be used to give an upper bound for the norm of a function. Lemma 1.2. [3] Let ρbe a convex function modular satisfying the ∆2-type condition. Then ||f||ρ≤1 ω−11 ρ(f)whenever f∈Lρ. The next lemma will be of major interest throughout this work. Lemma 1.3. [6] Let ρbe a function modular satisfying the ∆2-condition and {fn}nbe a sequence in Lρsuch that fn ρ−a.e →f∈Lρand there exists k > 1such that sup n ρ(k(fn−f)) <∞. Then, lim inf n→∞ ρ(fn−g) = lim inf n→∞ ρ(fn−f) + ρ(f−g)for all g∈Lρ. Moreover, we have ρ(f)≤lim inf n→∞ ρ(fn). 2. AN EQUIVALENT TOPOLOGY The concept of ρ-a.e. closed, compact sets have been studied extensively in the sequential case. One of the problem that many authors have found hard to circumvent is whether these notions are related to a topology. In this section we will discuss this problem. In particular, we will construct a topology τfor which the ρ-a.e. compactness is equivalent to the usual compactness for τ. This is crucial when we try to use Zorn’s lemma. ASYMPTOTICALLY NONEXPANSIVE MAPPINGS IN MODULAR FUNCTION SPACES 7 From now on, we assume that the modular function ρis, in addition, σ-finite. Set d(f, g) = ∞ X k=1 1 2k 1 ρ(1Kk)ρ|f−g| 1 + |f−g|1Kkfor any f, g ∈Lρ. Some basic properties satisfied by dare discussed in the following proposition. Proposition 2.1. The functional dsatisfies the following: (1) d(f, g) = 0 if and only if f=g ρ-a.e.; (2) d(f, g) = d(g, f); (3) d(f, g)≤ω(2) 2d(f, h) + d(h, g); for any f, g and hin Lρ. Proof. (1) and (2) are obvious. To prove (3) we only need to recall the inequality |a+b| 1 + |a+b|≤|a| 1 + |a|+|b| 1 + |b| for all positive numbers a, b and use the definition of the growth function ω.  Remark 2.1. The functional dis not a distance because of (3). But there are many mathematical objects which fail the triangle inequality but are very useful tools. That is the case with d. In the next proposition, we discuss the relationship between ρ-a.e. convergence and the convergence for the functional d. Proposition 2.2. Let ρbe a convex, σ-finite modular satisfying the ∆2-type condition and {fn}nbe a sequence of measurable functions. If {fn}nis ρ-a.e. convergent to f, then lim n→∞ d(fn, f) = 0. Moreover, if lim n→∞ d(fn, f) = 0, 8 T. DOMINGUEZ-BENAVIDES, M.A. KHAMSI AND S. SAMADI then there exists a subsequence {fnk}kwhich converges ρ-a.e. to f. Proof. Assume that {fn}nρ-a.e. converges to f. We will show that lim n→∞ d(fn, f) = 0.Let ε > 0,and choose N∈Nsuch that ∞ X k=N+1 1 2k< ε. We have lim n→∞ d(fn, f)≤lim n→∞ N X k=1 1 2k 1 ρ(1Kk)ρ|fn−f| 1 + |fn−f|1Kk+ε = N X k=1 lim n→∞ 1 2k 1 ρ(1Kk)ρ|fn−f| 1 + |fn−f|1Kk+ε. Since |fn−f| 1 + |fn−f|1Kk ρ−a.e −→ 0 as n→ ∞ for any k∈Nand |fn−f| 1 + |fn−f|1Kk≤1Kk,from Lebesgue’s Theorem we obtain lim n→∞ ρ|fn−f| 1 + |fn−f|1Kk= 0 for every non null integer k. Thus lim n→∞ d(fn, f)≤ε for each ε > 0 which means that lim n→∞ d(fn, f) = 0. Assume now that lim n→∞ d(fn, f) = 0.For every non null integer kwe have lim n→∞ ρ|fn−f| 1 + |fn−f|1Kk= 0. Thus, there exists a subsequence {f1 n}nof {fn}nsuch that |f1 n−f| 1 + |f1 n−f|1K1 ρ−a.e −→ 0 and so f1 n ρ−a.e −→ fin K1i.e. lim n→∞ f1 n(x) = f(x) whenever x∈K1\A1where A1⊂K1and ρ(1A1) = 0. By induction and using a diagonal argument we obtain a subsequence of {fn}n which converges ρ-a.e. to f.  Definition 2.1. Let Cbe a subset of Lρ. ASYMPTOTICALLY NONEXPANSIVE MAPPINGS IN MODULAR FUNCTION SPACES 9 (a) Cis said to be d-closed iff for any sequence {fn}nin Cwhich d-converges to f, then we have f∈C. (b) Cis d-open iff Lρ\Cis d-closed. (c) Cis said to be d-sequentially compact if for each sequence {fn}nthere exists a subsequence {fnk}kwhich dconverges to a point in C. It is easily seen that the family of all d-open subsets of Lρform a topology on Lρ.Furthermore, from proposition (2.2) d-sequentially compact sets and ρ-a.e. compact sets are identical. On the other hand, even though dsatisfies (3) instead of the triangular inequality, the usual arguments which prove that sequential compactness and compactness are identical in metric spaces hold in this setting. We also have d-sequential compactness and d-compactness are identical. 3. TECHNICAL LEMMAS In the sequel we assume that ρis a convex, σ-finite modular function satisfying the ∆2-type condition, Cis a convex, ρ-bounded and ρ-a.e. compact subset of the modular function space Lρand T:C→Cis a ρ-asymptotically nonexpansive mapping, i.e. there exists a sequence of positive integers {kn}nwhich converge to 1 such that for every n∈Nand f, g ∈Cwe have ρ(Tnf−Tng)≤knρ(f−g). Lemma 3.1. Under the above assumptions, let {fn}nbe a sequence of elements of C. Consider the functional Φ : C→Rdefined by Φ(g) = lim sup n→∞ ρ(fn−g). Then, for any sequence {gm}min Cwhich ρ-a.e. converges to g∈Cwe have Φ(g)≤lim inf m→∞ Φ(gm). Proof. Since Cis ρ-a.e. compact, there exists a subsequence {fφ(n)}nof {fn}n such that fφ(n) ρ−a.e −→ f∈Cand lim n→∞ ρ(fφ(n)−g) = lim sup n→∞ ρ(fn−g).Hence Φ(gm) = lim sup n→∞ ρ(fn−gm) ≥lim sup n→∞ ρ(fφ(n)−gm) ≥lim inf n→∞ ρ(fφ(n)−gm). 16 T. DOMINGUEZ-BENAVIDES, M.A. KHAMSI AND S. SAMADI Fix n≥n0. There exists k0≥1 such that for all k≥k0, we have n(k)≥n+n0 and ρ(Tnf−Tn(k)f) = ρTnf−Tn+(n(k)−n)f=ρTnf−Tn(Tn(k)−nf) ≤knρf−Tn(k)−nf< kn(r+η). Note that if fn ρ−a.e −→ fand Sep{fn}n≥ε, then by Lemma (1.3), we have ε≤lim inf m→∞ lim inf n→∞ ρ(fn−fm)≤2 lim inf n→∞ ρ(fn−f). Combined with Lemma (1.3), we get lim inf n→∞ ρ(fn) = lim inf n→∞ ρ(fn−f) + ρ(f)≥ 2+ρ(f). In particular, since {Tn(k)f−Tnf}kis ρ-a.e. convergent to f∞−Tnfas k→ ∞ and satisfies Sep({Tn(k)f−Tnf}k)≥, we get ρ(Tnf−f∞)≤lim inf k→∞ ρ(Tn(k)f−Tnf)− 2. Hence ρ(f∞−Tnf)≤r+η− 2 which implies r= lim sup n→∞ ρ(f∞−Tnf)≤r+η− 2< r. This contradiction completes the proof of Theorem 4.2.  Assume that Lρ=Lp(Ω, µ) for a σ-finite measure µ. If Cis a convex, bounded and closed subset of Lpfor 1 < p < ∞and T:C→Cis asymptotically nonexpansive, it is known that Chas a fixed point because Lpis uniformly convex. However the result does not hold for p= 1 (even for nonexpansive mappings, see [1]). Since L1is a modular space, Theorem (4.1) implies the existence of fixed point if p= 1 when Cis ρ-a.e. compact. Thus we can state. Corollary 4.1. Let (Ω, µ) be as above, C⊂L1(Ω, µ) a convex bounded set which is compact for the topology of local convergence in measure and T:C→C asymptotically nonexpansive. Then, Thas a fixed point. ASYMPTOTICALLY NONEXPANSIVE MAPPINGS IN MODULAR FUNCTION SPACES 17 Proof. Under the above hypothesis ρ-a.e. compact sets and compact sets in the topology of local convergence in measure are identical.  References [1] D.E. Alspach. A fixed point free nonexpansive map. Proc. Am. Math. Soc., (1981), 82, 423-424. [2] S.C. Bose. Weak convergence to the fixed point of an asymptotically nonexpansive map. Proc. Am. Math. Soc., (1978), 68, 305-308. [3] T. Dominguez Benavides, M.A. Khamsi, S. Samadi. Asymptotically regular mappings in modular function spaces. Preprint. [4] T.Dominguez Benavides, M.A. Khamsi, S. Samadi. Uniformly Lipschitzian mappings in modular function spaces. Preprint. [5] K. Goebel and W.A. Kirk. A fixed point theorem for asymptotically nonexpansive mappings. Proc. Am. Math. Soc., (1972), 35, 171-174. [6] M.A. Khamsi. Fixed point theory in modular function spaces. Recent Advances on Metric Fixed Point Theory. Universidad de Sevilla, Sevilla, (1996), 31-58. [7] M.A. Khamsi, W.M. Koz lowski, S. Reich. Fixed point theory in modular function spaces. Nonlinear Anal., (1990), 14, 935-953. [8] W.M. Kos lowski. Modular function spaces. Dekker: New York, Basel, (1988). [9] T.-H. Kim and H.-K. Xu. Remarks on asymptotically nonexpansive mappings. Nonlinear Anal., to appear. [10] C. Mart´ınez Ya˜nez. A fixed point theorem on k-uniformly rotund spaces. Nonlinear Anal., (1988), 13 857-861. [11] G. Passty. Construction of fixed points for asymptotically nonexpansive mappings. Proc. Am. Math. Soc., (1982), 84, 213-216. [12] H.-K. Xu. k-Uniform rotundity and fixed points of mappings of asymptotically nonexpansive type. To appear in Chinese. [13] H.-K. Xu. Existence and convergence for fixed points of mappings of asymptotically nonexpansive type. Nonlinear Anal., (1991) 16(12), 1139-1146. [14] X.T. Yu and X. Dai. A fixed point theorem of asymptotically nonexpansive mappings. J. Math. (PRC), (1986), 6, 255-262. 18 T. DOMINGUEZ-BENAVIDES, M.A. KHAMSI AND S. SAMADI Tomas Dominguez-Benavides, Department of Mathematical Analysis, University of Seville, P.O.Box 1160. 41080. Seville (Spain). E-mail address:[email protected] Mohamed Amine Khamsi, Department of Mathematical Science, The University of Texas at El Paso, El Paso, TX 79968, (U.S.A). E-mail address:[email protected] Sedki Samadi, Department of Mathematical Analysis, University of Seville, P.O.Box 1160. 41080. Seville (Spain). E-mail address:[email protected]