Fixed point theorems for multivalued mappings in modular function spaces

FIXED POINT THEOREMS FOR MULTIVALUED MAPPINGS IN MODULAR FUNCTION SPACES* S. DHOMPONGSA, T. DOMINGUEZ BENAVIDES, A. KAEWCHAROEN, AND B. PANYANAK Abstract. The purpose of this paper is to study the existence of fixed points for multivalued nonexpansive mappings in modular function spaces. We apply our main result to obtain fixed point theorems for multivalued mappings in the Banach spaces L1and l1. 1. Introduction The theory of modular spaces was initiated by Nakano [?] in 1950 in connection with the theory of order spaces and redefined and generalized by Musielak and Orlicz [?] in 1959. Even though a metric is not defined, many problems in metric fixed point theory can be reformulated and solved in modular spaces (see, for instance, [?,?,?,?]). In particular, some fixed point theorems for (singlevalued) nonexpansive mappings in modular function spaces are given in [?]. In 1969, Nadler [?] established the multivalued version of Banach’s contraction principle in metric spaces. Since then the metric fixed point theory for multivalued mappings has been rapidly developed and many of papers have appeared proving the existence of fixed points for multivalued nonexpansive mappings in special classes of Banach spaces (see, for instance, [?,?,?,?]). In this paper, we study similar problems in the setting of modular function spaces. Namely, we prove that every ρ−contraction T:C→Fρ(C) has a fixed point where ρis a convex function modular satisfying the ∆2−type condition and Cis a nonempty ρ−bounded ρ−closed subset of Lρ.By using this result, we can assert the existence of fixed points for multivalued ρ−nonexpansive mappings. Finally, we apply our main result to obtain fixed point theorems in the Banach space L1(resp. l1) for multivalued mappings whose domains are compact in the topology of the convergence locally in measure (resp. w∗−topology). Key words and phrases : Multivalued mappings, fixed points, Modular function spaces. * Supported by Thailand Research Fund under grant BRG4780013. The second author was partially supported by DGES, Grant D.G.E.S. REF. PBMF2003-03893-C02-C01 and Junta de Andalucia, Grant FQM-127. The third and fourth authors were supported by the Royal Golden Jubilee program under grant PHD/0250/2545 and PHD/0251/2545, respectively. 1 2 S. DHOMPONGSA, T. DOMINGUEZ BENAVIDES, A. KAEWCHAROEN, AND B. PANYANAK 2. Preliminaries We start by recalling some basic concept about modular function spaces. For more details the reader is referred to [?,?]. 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 Ω, 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 Ω = SKn(for instance, Pcan be the sets of finite measure in a σ−finite measure space). By Ewe denote the linear space of all simple functions with support 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. Let us recall that a set function µ: Σ →[0,∞] is called a σ−subadditive measure if µ(∅)=0, µ(A)≤µ(B) for any A⊂Band µ(SAn)≤Pµ(An) for any sequence of sets {An} ⊂ Σ.By χAwe denote the characteristic function of the set A. Definition 2.1. 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(αχA, 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 > 0, ρ(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∈Ωª. Definition 2.2. A set Eis said to be ρ−null if ρ(α, E) = 0 for every 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. In the above condition, we define the function ρ:M → [0,∞] by ρ(f) = ρ(f, Ω).We know from [?] that ρsatisfies the following properties : (i) ρ(f) = 0 if and only if f= 0 ρ−a.e. (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, β ≥0 and f, g ∈ M. In addition, if the following property is satisfied FIXED POINT THEOREMS FOR MULTIVALUED MAPPINGS IN MODULAR FUNCTION SPACES3 (iii)’ ρ(αf +βg)≤αρ(f) + βρ(g) if α+β= 1, α, β ≥0 and f, g ∈ M, we say that ρis convex modular. The modular ρdefines a corresponding modular space Lρ,which is given by Lρ={f∈ M :ρ(λf)→0 as λ→0}. In general the modular ρis not subadditive and therefore does not behave as a norm or a distance. But one can associate to a modular an F−norm. Recall that a functional k·k:X→[0,∞] defines an F−norm if and only if (1) kxk= 0 if and only if x= 0, (2) kαxk=kxkwhenever |α|= 1, (3) kx+yk ≤ kxk+kyk, (4) kαnxn−αxk → 0 if αn→αand kxn−xk → 0. The modular space Lρcan be equipped with an F−norm defined by kfkρ= inf nα > 0 : ρµf α¶≤αo. We know from [?] that the linear space (Lρ,k·kρ) is a complete metric space. If ρis convex the formula kfkρ= inf nα > 0 : ρµf α¶≤1o defines a norm which is frequently called the Luxemburg norm. The formula kfka= inf ½1 k(1 + ρ(kf)) : k > 0¾ defines a different norm which is called Amemiya norm. Moreover, k · kρand k · ka are equivalent norms. We can also consider the space Eρ={f∈ M :ρ(αf, ·) is order continuous for all α > 0}. Definition 2.3. A function modular ρis said to satisfy the ∆2−condition if sup n≥1 ρ(2fn, Dk)→0 as k→ ∞ whenever {fn} ⊂ M, Dk∈Σ decreases to ∅and sup n≥1 ρ(fn, Dk)→0 as k→ ∞. It is known that the ∆2−condition is equivalent to Eρ=Lρ. Definition 2.4. A function modular ρis 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, the ∆2−type condition and ∆2−condition are not equivalent, even though it is obvious that the ∆2−type condition implies the ∆2−condition. Definition 2.5. Let Lρbe a modular space. 4 S. DHOMPONGSA, T. DOMINGUEZ BENAVIDES, A. KAEWCHAROEN, AND B. PANYANAK (1) The sequence {fn} ⊂ Lρis said to be ρ−convergent to f∈Lρif ρ(fn−f)→ 0 as n→ ∞. (2) The sequence {fn} ⊂ Lρis said to be ρ−a.e. convergent to f∈Lρif the set {ω∈Ω : fn(ω)9f(ω)}is ρ−null. (3) A subset Cof Lρis called ρ−closed if the ρ−limit of a ρ−convergent sequence of Calways belongs to C. (4) 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. (5) A subset Cof Lρis called ρ−compact if every sequence in Chas a ρ−convergent subsequence in 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 diamρ(C) = sup{ρ(f−g) : f, g ∈C}<∞. We know by [?] that under the ∆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 2.6. Let ρbe as above. We define a growth function ωby ω(t) = sup nρ(tf) ρ(f):f∈Lρ,0< ρ(f)<∞ofor all 0 ≤t < ∞. The following properties of the growth function can be found in [?]. Lemma 2.7. Let ρbe as above. Then the growth function ωhas the following properties : (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 2.8 (T. Dominguez Benavides et al. [?]).Let ρbe as above. Then kfkρ≤1 ω−1(1/ρ(f)) whenever f∈Lρ\{0}. The following lemma is a technical lemma which will be need because of lack of the triangular inequality. FIXED POINT THEOREMS FOR MULTIVALUED MAPPINGS IN MODULAR FUNCTION SPACES5 Lemma 2.9 (T. Dominguez Benavides et al. [?]).Let ρbe as above, {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) and lim n→∞ ρ(gn) = 0 =⇒lim inf n→∞ ρ(fn+gn) = lim inf n→∞ ρ(fn) In the same way as the Hausdorff distance defined on the family of bounded closed subsets of a metric space, we can define the analogue to the Hausdorff distance for modular function spaces. We will call ρ−Hausdorff distance even though it is not a metric. Definition 2.10. Let Cbe a nonempty subset of Lρ.We shall denote by Fρ(C) the family of nonempty ρ−closed subsets of Cand by Kρ(C) the family of nonempty ρ−compact subsets of C. Let Hρ(·,·) be the ρ−Hausdorff distance on Fρ(Lρ),i.e., Hρ(A, B) = max nsup f∈A distρ(f, B),sup g∈B distρ(g, A)o, A, B ∈Fρ(Lρ). where distρ(f, B) = inf{ρ(f−g) : g∈B}is the ρ−distance between fand B. A multivalued mapping T:C→Fρ(Lρ) is said to be a ρ−contraction if there exists a constant k∈[0,1) such that (2.1) Hρ(Tf, T g)≤kρ(f−g), f, g ∈C. If (??) is valid when k= 1,then Tis called ρ−nonexpansive. A function f∈Cis called a fixed point for a multivalued mapping Tif f∈Tf. 3. Main results We begin stating the Banach Contraction Principle for multivalued mappings in modular function spaces. Theorem 3.1. Let ρbe a convex function modular satisfying the ∆2−type condition, Ca nonempty ρ−bounded ρ−closed subset of Lρ,and T:C→Fρ(C)a ρ−contraction mapping, i.e., there exists a constant k∈[0,1) such that (3.1) Hρ(Tf, T g)≤kρ(f−g), f, g ∈C. Then Thas a fixed point. Proof. Let f0∈Cand α∈(k, 1).Since Tf0is nonempty, there exists f1∈T f0 such that ρ(f0−f1)>0 (otherwise f0is a fixed point of T). In view of (??), we have distρ(f1, Tf1)≤Hρ(Tf0, Tf1)≤kρ(f0−f1)< αρ(f0−f1). Since distρ(f1, Tf1) = inf{ρ(f1−g) : g∈Tf1},it follows that there exists f2∈Tf1 such that ρ(f1−f2)< αρ(f0−f1). 6 S. DHOMPONGSA, T. DOMINGUEZ BENAVIDES, A. KAEWCHAROEN, AND B. PANYANAK Similarly, there exists f3∈T f2such that ρ(f2−f3)< αρ(f1−f2). Continuing in this way, there exists a sequence {fn}in Csatisfying fn+1 ∈T fn and ρ(fn−fn+1)< αρ(fn−1−fn) < α2(ρ(fn−2−fn−1)) < ... < αn−1(ρ(f1−f2)) < αn(ρ(f0−f1)) ≤αndiamρ(C), Let M= diamρ(C),then 1 αnM<1 ρ(fn−fn+1). By Lemma ??, we have ³ω−1¡1 α¢´n ω−1³1 M´< ω−1³1 ρ(fn−fn+1)´, It follows that 1 ω−1³1 ρ(fn−fn+1)´<1 ³ω−1¡1 α¢´n ω−1³1 M´. By Lemma ??, we obtain kfn−fn+1kρ<³1 ω−1¡1 α¢´n·1 ω−1³1 M´. Since ω−1is strictly increasing, we have 1 ω−1¡1 α¢<1.This implies that {fn}is a Cauchy sequence in (Lρ,k · kρ).Since (Lρ,k·kρ) is a complete metric space, there exists f∈Lρsuch that {fn}is k · kρ−convergent to f. Since under the ∆2−type condition, norm convergence and modular convergence are identical, {fn} is ρ−convergent to fand f∈Cbecause Cis ρ−closed. Since fn∈T fn−1, we have (3.2) distρ(fn, Tf)≤Hρ(T fn−1, T f)≤kρ(fn−1−f)−→ 0. We observe that, for each n, there exists gn∈Tf such that (3.3) ρ(fn−gn)≤distρ(fn, Tf) + 1 n. Thus, (??) and (??) imply that lim n→∞ ρ(fn−gn) = 0.By Lemma ??, lim sup n→∞ ρ(gn−f) = lim sup n→∞ ρ(gn−fn+fn−f) = lim sup n→∞ ρ(fn−f) = 0. Since Tf is ρ−closed, we can conclude that f∈Tf. ¤ FIXED POINT THEOREMS FOR MULTIVALUED MAPPINGS IN MODULAR FUNCTION SPACES7 The following results will be very useful in the proof of our main theorem. Theorem 3.2 (M. A. Khamsi [?]).Let {fn} ⊂ Lρbe ρ−a.e. convergent to 0. Assume there exists k > 1such that sup n≥1 ρ(kfn) = M < ∞. Let g∈Eρ,then we have lim inf n→∞ ρ(fn+g) = lim inf n→∞ ρ(fn) + ρ(g). The following lemma guarantee that every nonempty ρ−compact subset of Lρ attains a nearest point. Lemma 3.3. Let ρbe a convex function modular satisfying the ∆2−type condition, f∈Lρ,and Ka nonempty ρ−compact subset of Lρ.Then there exists g0∈Ksuch that ρ(f−g0) = distρ(f, K). Proof. Let m= distρ(f, K).For each n∈N,there exists gn∈Ksuch that m−1 n≤ρ(f−gn)≤m+1 n. By the ρ−compactness of K, we can assume, by passing through a subsequence, that gn ρ −→ g0∈K. By Lemma ??, we obtain m= lim sup n→∞ ρ(gn−f) = lim sup n→∞ ρ(gn−g0+g0−f) = lim sup n→∞ ρ(g0−f) =ρ(g0−f). ¤ We can now state our main theorem. Theorem 3.4. Let ρbe a convex function modular satisfying the ∆2−type condition, Ca nonempty ρ−a.e. compact ρ−bounded convex subset of Lρ,and T:C→ Kρ(C)aρ−nonexpansive mapping. Then Thas a fixed point. Proof. Fix f0∈C. For each n∈N,the ρ−contraction Tn:C→Fρ(C) is defined by Tn(f) = 1 nf0+ (1 −1 n)Tf, f ∈C. By Theorem ??, we can conclude that Tnhas a fixed point, say fn.It is easy to see that distρ(fn, Tfn)≤1 ndiamρ(C)−→ 0. 8 S. DHOMPONGSA, T. DOMINGUEZ BENAVIDES, A. KAEWCHAROEN, AND B. PANYANAK Because of ρ−a.e. compactness of C, we can assume, by passing through a subsequence, that fn ρ−a.e. −→ ffor some f∈C. By Lemma ??, for each n∈N,there exists gn∈Tfnand hn∈Tf such that ρ(fn−gn) = distρ(fn, Tfn) and ρ(gn−hn) = distρ(gn, Tf)≤Hρ(T fn, Tf)≤ρ(fn−f). Because of ρ−compactness of Tf, we can assume, by passing through a subsequence, that hn ρ −→ h∈Tf. Since ρsatisfies the ∆2−type condition, there exists K > 0 such that ρ(2(fn−f)) ≤Kρ(fn−f) for all n∈N. This implies that sup n≥1 ρ(2(fn−f)) ≤Ksup n≥1 ρ(fn−f)<∞. By Theorem ?? and Lemma ??, we obtain lim inf n→∞ ρ(fn−f) + ρ(f−h) = lim inf n→∞ ρ(fn−f+f−h) = lim inf n→∞ ρ(fn−h) = lim inf n→∞ ρ(fn−gn+gn−hn+hn−h) = lim inf n→∞ ρ(gn−hn) ≤lim inf n→∞ ρ(fn−f). It follows that ρ(f−h) = 0 and then we have f=h∈Tf. ¤ Consider the space Lp(Ω, µ) for a σ−finite measure µwith the usual norm. Let Cbe a bounded closed convex subset of Lpfor 1 < p < ∞and T:C→K(C) a multivalued nonexpansive mapping. Because of uniform convexity of Lp,it is known that Thas a fixed point. For p= 1, T can fail to have a fixed point even in the singlevalued case for a weakly compact convex set C(see [?]). However, since L1is a modular space where ρ(f) = RΩ|f|dµ =kfkfor all f∈L1,Theorem ?? implies the existence of a fixed point when we define mappings on a ρ−a.e. compact ρ−bounded convex subset of L1.Thus we can state : Corollary 3.5. Let (Ω, µ)be as above, C⊂L1(Ω, µ)a nonempty bounded convex set which is compact for the topology of the convergence locally in measure, and T:C→K(C)a nonexpansive mapping. Then Thas a fixed point. Proof. Under the above hypothesis ρ−a.e. compact sets and compact sets in the topology of the convergence locally in measure are identical (see [?]). Consequently, Theorem ?? can be applied to obtain a fixed point for T. ¤ In the case of the space l1we also can obtain a bounded closed convex set C and a nonexpansive mapping T:C→Cwhich is fixed point free. Indeed, consider FIXED POINT THEOREMS FOR MULTIVALUED MAPPINGS IN MODULAR FUNCTION SPACES9 the following easy and well known example : Let C=n{xn} ∈ l1: 0 ≤xn≤1 and ∞ X n=1 xn= 1o. Define a nonexpansive mapping T:C→Cby T(x) = (0, x1, x2, x3, ...) where x={xn}. Then Tis a fixed point free. However, if we consider Lρ=l1where ρ(x) = kxk,∀x∈l1.Then ρ−a.e. convergence and w∗−convergence are identical on bounded subsets of l1(see [?]). This fact leads us to obtain the following corollary : Corollary 3.6. Let Cbe a nonempty w∗−compact convex subset of l1and T:C→ K(C)a nonexpansive mapping. Then Thas a fixed point. Proof. By the above argument, we know that ρ−a.e. compact bounded sets and w∗−compact sets are identical. Then we can apply Theorem ?? to assert the existence of a fixed point of T. ¤ In fact ?? and ?? are consequences of a general result: Assume that Xis a linear normed space and τis a Hausdorff topology on X. We say that Xsatisfies the strict τ-Opial property if lim sup n→∞ kxn−xk<lim sup n→∞ kxn−yk for each sequence {xn}in Xwhich converges to xfor the topology τand each y6=x. Following the same argument as in [?] it is easy to prove the following theorem: Theorem 3.7. Let Xbe a Banach space, Ca convex bounded sequentially τcompact subset of X, and T:C→K(C)a nonexpansive mapping. If Xsatisfies the strict τ-Opial property, then Thas a fixed point. When Xis a modular function space equipped with either Luxemburg or Amemiya norm, we can consider the topology τof convergence ρ-a.e. In this case, ?? yields to the following: Theorem 3.8. Let ρbe a convex function modular satisfying the ∆2−type condition. Assume that Lρis equipped either with Luxemburg or Amemiya norm. Let C be a nonempty ρ−a.e. compact ρ−bounded convex subset of Lρ,and T:C→K(C) a nonexpansive mapping. Then Thas a fixed point. Proof. From [?] (Theorem 4.1 and 4.3), Xsatisfies the uniform Opial property with respect to the topology of ρ-a.e. convergence. Since ρ-a.e. compact sets and ρ-a.e. sequentially compact sets are identical for this topology (see [?]), we can deduce the result from ?? ¤