Uniformly Lipschitzian mappings in modular function spaces

UNIFORMLY LIPSCHITZIAN MAPPINGS IN MODULAR FUNCTION SPACES T. DOMINGUEZ BENAVIDES, M.A. KHAMSI AND S. SAMADI ABSTRACT Let ρbe a convex modular function satisfying a ∆2-type condition and Lρthe corresponding modular space. Assume that Cis a ρ-bounded and ρ-a.e compact subset of Lρand T:C→Cis a k-uniformly Lipschitzian mapping. We prove that Thas a fixed point if k < (˜ N(Lρ))−1/2where ˜ N(Lρ) is a geometrical coefficient of normal structure. We also show that ˜ N(Lρ)<1 in modular Orlicz spaces for uniformly convex Orlicz functions. 1991 Mathematics subject classification: Primary 46E30; Secondary 47H09, 47H10. Key Words: uniformly Lipschitzian mappings, fixed point, modular functions, uniform normal stucture, uniform convex Orlicz function, modulus of convexity. 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 The theory of modular spaces was initiated by Nakano [14] in 1950 in connection with the theory of order spaces and redefined and generalized by Musielak and Orlicz [13] in 1959. Defining a norm, particular Banach spaces of functions can be considered. Metric fixed theory for these Banach spaces of functions has been widely studied (see, for instance, [15]). Another direction is based on considering an abstractly given functional which controls the growth of the functions. Even though a metric is not defined, many problems in fixed point theory for nonexpansive mappings can be reformulated in modular spaces (see, for instance, [8] and references therein). In this paper, we study the existence of fixed points for a more general class of mappings: uniformly Lipschitzian mappings. Fixed point theorems for this class of mappings in Banach spaces have been studied in [3,4] and in metric spaces in [11,12] (for further information about this subject, see [2, chapter VIII] and references therein). The main tool in our approach is the coefficient of normal structure ˜ N(Lρ). We prove that under suitable conditions a k-uniformly Lipschitzian mapping has a fixed point if k < (˜ N(Lρ))−1/2.In the last section we show a class of modular spaces where ˜ N(Lρ)<1 and so, the above theorem can be successfully applied. 1. PRELIMINARIES We start by recording a brief collection of basic concepts and facts of modular spaces as formulated by Kozlowski. For more details the reader is refered to [7], [8], [10] and [13]. 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 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. UNIFORMLY LIPSCHITZIAN MAPPINGS IN MODULAR FUNCTION SPACES 3 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)|ω∈Ω}. A set Eis said to be ρ-null if ρ(α, E) = 0 for every α > 0.For the sake of simplicity we write ρ(f) instead of ρ(f, Ω). 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(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}. 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}. 4 T. DOMINGUEZ BENAVIDES, M.A. KHAMSI AND S. SAMADI 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 (see [10]) that Eρ=Lρwhen ρsatisfies the ∆2-condition. When ρis convex, the formula ||f||ρ= inf nα > 0; ρf α≤1o definies a norm in the modular space Lρwhich is frequently called the Luxemburg norm. Definition 1.2. (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 sequentially 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 sequentially 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}<∞. Let Bbe a bounded subset of Lρ.We define the ρ-ball of center f∈Lρand radius r > 0 by B(f, r) = {g∈Lρ, ρ(g−f)≤r}.We will denote r(f, B) = sup{ρ(f− g), g ∈B}, δ(B) = sup{r(f, B), f ∈B}, R(B) = inf{r(f, B), f ∈B}.We define the admissible hull of Bas the intersection of all ρ-ball containing B, i.e: ad(B) = \{A:B⊂A⊂Lρ,where Ais a ρ-ball}. UNIFORMLY LIPSCHITZIAN MAPPINGS IN MODULAR FUNCTION SPACES 5 Bis said admissible if ad(B) = B. We define the normal structure coefficient ˜ N(Lρ) of Lρby ˜ N(Lρ) = sup R(B) δ(B), B is admissible, ρ-bounded and ρ-a.e sequentially compact. The useful following proposition is easily seen: Proposition 1.1. Let Bbe a ρ-bounded subset of Lρand f∈Lρ.Then (1) r(f, ad(B)) = r(f, B). (2) δ(ad(B)) = δ(B). We say that ρsatisfies the ∆2-type condition if there exists K > 0 such that ρ(2f)≤Kρ(f) for all f∈Lρ.In general, ∆2-type condition and ∆2-condition are not equivalent, even though it is obvious that ∆2-type condition implies ∆2condition. Assume that ρis convex and satisfies the ∆2-type condition. We define a growth function ωby ω(t) = sup ρ(tf) ρ(f),0< ρ(f)<∞for all 0 ≤t < ∞. The following properties of the growth function can be easily seen. Lemma 1.1. Let ρbe a convex function modular satisfying the ∆2-type condition. Then the growth funtion 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. 6 T. DOMINGUEZ BENAVIDES, M.A. KHAMSI AND S. SAMADI Lemma 1.2. [5] Let ρbe a convex function modular satisfying the ∆2-type condition. Then ||f||ρ≤1 ω−11 ρ(f)whenever f∈Lρ. The following lemma can be found in [7]. Lemma 1.3. Let ρbe a function modular satisfying the ∆2-condition and {fn}n be a sequence in Lρsuch that fn ρ−a.e →f∈Lρand there exists k > 1such that supnρ(k(fn−f)) <∞. Then, lim inf n→∞ ρ(fn−g) = lim inf n→∞ ρ(fn−f) + ρ(f−g)for all g∈Lρ. Lemma 1.4. Let ρbe a modular function satisfying the ∆2-type condition. Let B be a ρ-a.e sequentially closed and ρ-bounded subset of Lρ. Let {gn}nbe a sequence in Bsuch that gn ρ−a.e →g. Then, (1) ρ(g)≤lim infn→∞ ρ(gn). (2) B(0, r)∩Bis ρ-a.e sequentially closed. (3) ad(A)∩Bis ρ-a.e sequentially closed, for all A⊂Lρ. Proof. Condition (1) is a straighforward consequence of Lemma 1.3 applied to the sequence gn ρ−a.e →gand the null function. Condition (2) and (3) can be easily deduced from (1). 2. FIXED POINT FOR UNIFORMLY LIPSCHITZIAN MAPPINGS The following lemma is the key of our fixed point result. Lemma 2.1. Let ρbe a modular function satisfying the ∆2-type condition and Baρ-bounded and ρ-a.e sequentially compact subset of Lρ. Let {fn}nand {gn}n be sequences in B. Then, there exists g∈ ∩∞ n=1ad(gj, j ≥n)∩Bsuch that lim sup n→∞ ρ(g−fn)≤lim sup j→∞ lim sup n→∞ ρ(gj−fn) Proof. Let {fn}nand {gn}nbe sequences in B. We define θ(h) = lim supn→∞ ρ(h− fn) for all h∈B. Since Bis ρ-sequentially compact and ρ-bounded, there exist a subsequence {gφ(n)}n⊂ {gn}nsuch that gφ(n) ρ−a.e →gand a subsequence {fψ(n)}n⊂ {fn}nsuch that limn→∞ ρ(fψ(n)−g) = lim supn→∞ ρ(fn−g) and UNIFORMLY LIPSCHITZIAN MAPPINGS IN MODULAR FUNCTION SPACES 7 fψ(n) ρ−a.e →f∈B. Since gφ(n)∈ad(gj, j ≥n)∩Bwhich is ρ-a.e sequentially closed (by property (3) of Lemma 1.4) and gφ(n) ρ−a.e →g, we obtain g∈ad(gj, j ≥ n)∩Bfor all n≥1.We will see that θ(g)≤lim sup j→∞ θ(gj).Indeed, from Lemma 2.3 we have θ(gj) = lim supn→∞ ρ(fn−gj)≥lim infn→∞ ρ(fψ(n)−gj) = lim infn→∞ ρ(fψ(n)−f) + ρ(f−gj).Thus, again using Lemma 2.3, we obtain lim sup j→∞ θ(gj)≥lim inf n→∞ ρ(fψ(n)−f) + lim sup j→∞ ρ(f−gj) ≥lim inf n→∞ ρ(fψ(n)−f) + lim inf j→∞ ρ(f−gφ(j)) = lim inf n→∞ ρ(fψ(n)−f) + lim inf j→∞ ρ(gφ(j)−g) + ρ(f−g). On the other hand θ(g) = lim supn→∞ ρ(fn−g) = lim infn→∞ ρ(fψ(n)−g) = lim infn→∞ ρ(fψ(n)−f) + ρ(f−g). Therefore, θ(g)≤lim supj→∞ θ(gj). The following lemma is inspired on [3] where a similar lemma is proved in reflexive Banach spaces (see also [12, Lemma 6] for a version in metric spaces with additional properties). Lemma 2.2. Let ρbe a function modular satisfying the ∆2-type condition and Ban be admissible, ρ-a.e sequentially compact and ρ-bounded subset of Lρ.Let {fn}be a sequence in Band ca constant such that c > ˜ N(Lρ). Then there exists f∈Bsuch that (1) lim sup n→∞ ρ(f−fn)≤c δ({fn}n). (2) ρ(f−g)≤lim sup n→∞ ρ(fn−g)for all g∈B. Proof. Let {fn}nbe a sequence of B. Denote Am=ad(fj:j≥m)⊂Band A=T∞ m=1 Am.Since Bis ρ-a.e sequentially compact, there exists a subsequence of {fn}nρ−a.e convergent, say to h. It is clear that h∈Aand so A6=∅. Furthermore, from Proposition 1.1 (2), we have δ(An)≤δ({fn}n).On the other hand, for any f∈Aand g∈Bwe have ρ(g−f)≤r(g, A)≤r(g, An) = r(g, {fj: j≥n}) = supj≥nρ(g−fj).Therefore, ρ(g−f)≤lim supn→∞ ρ(g−fn) and (2) holds for any f∈A. We will prove that there exists f∈Asatisfying (1). Without loss of generality we may assume that δ({fn}n)>0.Choose ε > 0 such that 8 T. DOMINGUEZ BENAVIDES, M.A. KHAMSI AND S. SAMADI ˜ N(Lρ)δ({fn}n)+ε≤c δ({fn}n).By definition of R(An), there exists gn∈Ansuch that r(gn, An)< R(An)+ε≤˜ N(Lρ)δ(An)+ε≤˜ N(Lρ)δ({fn}n)+ε≤c δ({fn}n). Since r(gn, An) = r(gn,{fj}j≥n) = supj≥nρ(gn−fj),we have, lim sup j→∞ ρ(gn−fj)≤c δ({fn}n) (A). Using Lemma 2.5, there exists f∈ ∩∞ n=1ad(gi, i ≥n) such that lim sup j→∞ ρ(f−fj)≤lim sup n→∞ lim sup j→∞ ρ(gn−fj) (B). We will check that f∈A. Indeed, for all i, n integers such that i≥nwe have gi∈Ai⊂An.Thus, {gi}i≥n⊂Anwhich implies ad(gi, i ≥n)⊂Anand f∈A. Using (B) and (A) it is clear that lim supj→∞ ρ(f−fj)≤c δ({fn}n). Theorem 2.1. Let ρbe a convex function modular satisfying the ∆2-condition and B an admissible, ρ-a.e sequentially compact and ρ-bounded subset of Lρ.Suppose that ˜ N(Lρ)<1and let T:B→Bbe a k-uniformly lipschitzian mapping satisfying k < (˜ N(Lρ))−1/2.Then, Thas a fixed point. Proof. We can assume that k > 1; otherwise Twill be nonexpansive and the existence of a fixed point is a consequence of [8, Theorem 3.5]. Choose a constant c,˜ N(Lρ)< c < 1 such that 1 < k < c−1/2.Fix f0∈B. By Lemma 2.6, we can inductively construct a sequence {fj}j≥0⊂Bsuch that for each j≥0 (1) lim supn→∞ ρ(Tn(fj)−fj+1)≤c δ({Tn(fj)}n). (2) ρ(fj+1 −g)≤lim supn→∞ ρ(Tn(fj)−g) for all g∈B. Denote Dj= lim supn→∞ ρ(Tn(fj)−fj+1) and h=ck2<1.For n≥m≥0,we have ρ(Tmfj−Tnfj)≤kρ(fj−Tn−mfj) ≤klim sup i→∞ ρ(Tifj−1−Tn−mfj) ≤k2lim sup i→∞ ρ(Ti−(n−m)fj−1−fj) ≤k2Dj−1. UNIFORMLY LIPSCHITZIAN MAPPINGS IN MODULAR FUNCTION SPACES 9 Since Dj= lim supn→∞ ρ(Tn(fj)−fj+1)≤c δ({Tn(fj)}n),we obtain Dj≤ c k2Dj−1=hDj−1.Thus, Dj≤hjD0and we have ρ(fj+1 −fj)≤ω(2)ρ(fj+1 −Tnfj) + ρ(fj−Tnfj) ≤ω(2)ρ(fj+1 −Tnfj) + lim sup m→∞ ρ(Tmfj−1−Tnfj) ≤ω(2)ρ(fj+1 −Tnfj) + klim sup m→∞ ρ(Tm−nfj−1−fj) ≤ω(2)ρ(fj+1 −Tnfj) + kDj−1. Taking limsup as n→ ∞,we obtain ρ(fj+1 −fj)≤ω(2)(Dj+kDj−1) ≤ω(2)(hj+khj−1)D0 ≤ω(2)(h+k)hj−1D0 ≤Ahj,where A=ω(2)h+k hD0. Hence, there exists an integer Nand some β < 1 such that for j > N we have ρ(fj+1 −fj)≤βj,which implies 1 βj≤1 ρ(fj+1 −fj).Using properties (2) and (3) of Lemma 1.1 we obtain ω−11 βj≤ω−11 ρ(fj+1 −fj) and ω−11 βj ≤ω−11 ρ(fj+1 −fj). Therefore, by Lemma 2.2 we have ||fj+1 −fj||ρ≤1 ω−11 ρ(fj+1−fj)≤1 ω−1(1 β)j. Hence {fj}is a Cauchy sequence in (Lρ,||.||ρ),there exists f∈Lρsuch that ||fj−f||ρ→0, because (Lρ,||.||ρ) is complete. Since under ∆2-condition normconvergence and modular-convergence are identical, {fj}is modular convergent to f. Thus, there exists a subsequence of {fj}jρ-a.e convergent to f[1, Theorem 16 T. DOMINGUEZ BENAVIDES, M.A. KHAMSI AND S. SAMADI ACKNOWLEDGEMENTS The first and third authors are very grateful to the Department of Mathematical Sciences at the University of Texas at El Paso for their hospitality while completing this work. References [1] A.Kaminska. On uniform convexity of Orlicz spaces. Mathematics, Proceedings. Konink. Nederl, Ak. Wet. Amsterdam, A 85 (1) (1982), 2736. [2] J.M.Ayerbe, T.Dominguez Benavides, G.Lopez Acedo. Measures of Noncompactness in Metric Fixed Point Theory. Bikhauser: Basel (1997). [3] E.Casini, E.Maluta. Fixed points of uniformly Lipchitzian mappins in spaces with uniformly normal structure. Nonlinear Anal. 9(1985), 103-108. [4] T.Dominguez Benavides. Fixed point theorems for uniformly Lipschitzian mappings and asymptotically regular mappings. Nonlinear Anal. 32 (1) (1998), 15-27. [5] T.Dominguez Benavides, M.A.Khamsi, S.Samadi. Asymptotically regular mappings in modular function spaces, (preprint). [6] H.Hudzik, A.Kaminska, J.Musielak. On the convexity coefficient of Orlicz spaces. Math. Z. 197 (1988), 291-295. [7] M.A.Khamsi. Fixed point theory in modular function spaces. Recent Advances on Metric Fixed Point Theory. 31-58. Universidad de Sevilla, Sevilla (1996). [8] M.A.Khamsi, W.M.Kozlowski, S.Reich. Fixed point theory in modular function spaces. Nonlinear Anal. 14 (1990), 935-953. [9] M.A.Khamsi, W.M.Kozlowski, C.Shutao. Some geometrical properties and fixed point theorems in Orlicz Spaces. J. Math. Anal. Appl. 155 (2) (1991), 393-412. [10] W.M.Koslowski. Modular function spaces. Dekker: New York, Basel (1988). [11] E.A.Lifschitz. Fixed point theorems for operators in strongly convex spaces (Russian) Voronez. Gos. Univ. Trudy Mat. Fak. 16 (1975), 23-28. [12] T.C.Lim, H.K.Xu. Uniformly Lipschitzian mappings in metric spaces with uniform normal structure. Nonlinear Anal. 25 (11) (1995), 1231-1235. [13] J.Musielak, W.Orlicz. On modular spaces. Studia Math. 18 (1959), 591-597. [14] H. Nakano. Modulared semi-ordered spaces, Tokyo (1950). [15] D.Van Dultst, V.de Valk. (KK)-properties, normal structure and fixed points of nonexpansive mappings in Orlicz spaces. Canad. J. Math. 38 (1986), 728-750. UNIFORMLY LIPSCHITZIAN MAPPINGS IN MODULAR FUNCTION SPACES 17 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]