In this paper we present a generalization of the Daher's fixed point theorem to countably condensing multimaps. We obtain this result as a consequence of a new M\"onch type theorem for multimaps having weakly closed graph

Cardinali, Tiziana

Rubbioni, Paola

Electronic journal of qualitative theory of differential equations

Tiziana Cardinali

Paola Rubbioni

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Countably condensing multimaps and fixed points

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Electronic Journal of Qualitative Theory of Differential Equations2012, No. 83, 1-9; http://www.math.u-szeged.hu/ejqtde/Countably condensing multimaps andfixed pointsTiziana Cardinali - Paola Rubbioni∗Department of Mathematics and Informatics, University of Perugia1Address: via L. Vanvitelli 1, 06123 Perugia, ITALYPhone: +390755855042; Fax: +390755855024E-mail addresses: tiziana@dmi.unipg.it - rubbioni@dmi.unipg.itAbstract: In this paper we present a generalization of the Daher’s fixed point theorem tocountably condensing multimaps. We obtain this result as a consequence of a new Mo¨nchtype theorem for multimaps having weakly closed graph.Keywords and Phrases: countably condensing multimap, fixed point, measure of non-compactness, weakly closed graph, quasi-convex setAMS Subject Classification : 47H10, 47H04, 47H081 IntroductionCondensing operators have been the object of a wide and deep study in nonlinearfunctional analysis. This research was started in 1967 by Sadovskii [22]. He introducedthe concept of condensing operator for single-valued functions by using the Kuratowskimeasure of noncompactness and proved that a condensing operator from a closedbounded convex subset of a Banach space into itself has a fixed point, extending thewell known Darbo’s fixed point theorem [13].Later, the Sadovskii’s result has been improved in different directions: from one hand,Daher [12] showed that it is still true for countably condensing maps (i.e. condensingonly on countable subsets); from another, Himmelberg, Porter and Van Vleck [17] ex-∗Corresponding author1Supported by the national research project PRIN 2009 ”Ordinary Differential Equations andApplications”EJQTDE, 2012 No. 83, p. 1tended the definition of condensing operator to multimaps and provided a multivaluedversion of the Sadovskii’s theorem.More recently, Agarwal and O’Regan [1] established a multivalued version of the Da-her’s theorem for upper semicontinuous multimaps.The main purpose of the present paper is to state a Daher type theorem (see The-orem 3.2) for multimaps having weakly closed graph, which can take some nonconvexvalues and are countably condensing.By countably condensing we mean the following property:Definition 1.1 Let D be a nonempty subset of a Banach space X and let β be anabstract measure of noncompactness. A map F : D → P(X) is said to be countablycondensing if(I) F (D) is bounded;(II) β(F (B)) < β(B) for all countable bounded subsets B of D with β(B) > 0.As far as we know, this definition is new and it is an improving of the analogous defi-nition given by Agarwal and O’Regan. In fact, they say that a multimap is countablycondensing if it satisfies conditions (I), (II) and the further(III) F is 1-set contractive, i.e. β(F (B)) ≤ β(B) for all countable bounded subsetsB of Dtaking β as the Kuratowski measure of noncompactness.The advantage of our definition is that it is the natural weakening of the classicaldefinition of condensing multimap (see, e.g. [19]) and, moreover, that in the single-valued case it comes down to the Daher’s definition of countably condensing function.To prove our main result, in a preliminary way we provide a new Mo¨nch typetheorem for multimaps having weakly closed graph (cf. Theorem 3.1). The Mo¨nchtype hypothesis we use has been introduced by O’Regan and Precup [21] in order toextend the Mo¨nch theorems to multimaps. We base our proof on a fixed point theoremfor multimaps having weakly closed graph established in [6].We wish to note that our theorems strictly contain respectively the quoted Agarwal-O’Regan’s result [1, Theorem 1.2] and the O’Regan-Precup’s Mo¨nch type theorem [21,Theorem 3.1] (see Remark 3.1).Finally, we remark the relevance of the study of the existence of fixed points dueto its applicability in finding the existence of solutions of various kinds of nonlineardifferential equations or inclusions (see, e.g. [2], [4], [5], [7], [8], [9], [11], [15], [20]) andnonlinear integral equations or inclusions (see, e.g. [10], [21]).2 PreliminariesLet X be a locally convex Hausdorff topological linear space and P(X ) be the familyof all nonempty subsets of X .EJQTDE, 2012 No. 83, p. 2We recall (see [16]) that a set A ⊂ X is said to be quasi-convex if for everybalanced convex neighborhood V of zero and for every {a1, . . . , an} ⊂ A there exists{z1, . . . , zn} ⊂ A such that zi − ai ∈ V , i = 1, . . . , n, co{z1, . . . , zn} ⊂ A.Moreover, let D be a nonempty subset of X . As in [6], we say that a map G : D →P(X ) has weakly closed graph in D × X if for every net (xδ)δ in D, xδ → x, x ∈ D,and for every net (yδ)δ, yδ ∈ G(xδ), yδ → y, then S(x, y)∩G(x) 6= ∅, where S(x, y) ={x+ λ(y − x) : λ ∈ [0, 1]}.In the sequel we will use the following theorem.Theorem 2.1 [6, Teorema I] Let X be a locally convex Hausdorff topological linearspace, K be a nonempty compact subset of X and G : K → P(K) be a map takingnonempty closed values and with the properties(i) there exists A ⊂ K, A quasi-convex, A = K such that G(x) is convex for everyx ∈ A;(ii) G has weakly closed graph.Under these conditions, there exists x ∈ K such that x ∈ G(x).From now on, X will be a Banach space endowed with the norm ‖ · ‖. We will usethe following notations: Pb(X) = {H ⊂ X : H 6= ∅ , H bounded }; Pk(X) = {H ⊂X : H 6= ∅ , H compact }.We recall (see, e.g. [3], [19]) that a function β : Pb(X) → IR+0 is said to be ameasure of noncompactness (MNC, for short) ifβ(co(Ω)) = β(Ω) , for every Ω ∈ Pb(X) . (1)In the sequel, we consider a measure of noncompactness β verifying the followingproperties:(β1) regularity: β(Ω) = 0 if and only if Ω is compact;(β2) monotonicity: Ω1 ⊂ Ω2 implies β(Ω1) ≤ β(Ω2);(β3) semiadditivity: β(Ω1 ∪ Ω2) = max{β(Ω1), β(Ω2)};where Ω,Ω1,Ω2 ∈ Pb(X).As examples of measures of noncompactness which satisfy all the previous properties,we recall the Hausdorff and the Kuratowski measures of noncompactness.Remark 2.1 We note that if β is a regular MNC, property (II) of Definition 1.1 canbe equivalently formulated as(II)’ for every countable bounded subset B of D the relation β(B) ≤ β(F (B)) impliesthat B is compact.In the single-valued case, this is the definition of countably condensing map sometimesadopted in the literature.EJQTDE, 2012 No. 83, p. 33 The fixed point theoremsIn this section, we will provide a Daher type theorem for multimaps. In a preliminaryway, we give a fixed point theorem for multimaps by using a Mo¨nch type assumptionwhich has been introduced by O’Regan and Precup [21] in order to extend the classicalMo¨nch theorem to set-valued maps.Theorem 3.1 Let D be a closed convex subset of a Banach space X and F : D →Pk(D) be a map such that(i) there exists A ⊂ D, A quasi-convex, A = D such that F (x) is convex for everyx ∈ A;(ii) F has weakly closed graph;(iii) F maps compact sets into relatively compact sets;(M) there exists x0 ∈ D such thatM ⊂ D, M = co({x0} ∪ F (M))and M = C with C ⊂M countable}⇒ M is compact.Then there exists x ∈ D such that x ∈ F (x).Proof. Let x0 ∈ D be as by hypothesis (M). We consider the iterative sequence(Mn)n∈IN of sets:M0 = {x0} ; Mn = co({x0} ∪ F (Mn−1)) , n ∈ IN+ .Clearly,Mn ⊂ D , n ∈ IN . (2)Let us prove by induction that Mn, n ∈ IN+, is relatively compact.First, the Mazur Theorem (see, e.g. [18, Theorem A.3.68]) implies that co({x0} ∪F (M0)) is compact. So, M1 = co({x0} ∪ F (M0)) is relatively compact.Now, suppose that Mn−1 is relatively compact, n ≥ 2. Of course Mn ⊂ co({x0} ∪F (Mn−1)) (see (2)). By (iii) and the Mazur’s theorem, we can say that also Mn isrelatively compact.By induction again, we can say thatMn−1 ⊂ Mn , n ∈ IN+ . (3)Now, for every n ∈ IN , let us consider the space (Mn, d), where d is the metric inducedon Mn by the metric generated by ‖ · ‖. The compactness of Mn implies that (Mn, d)is a separable space (cf., e.g. Corollary 1.4.29 and Corollary 1.4.12 in [14]). Hence,there exists a countable set Cn ⊂Mn such thatCn(Mn,d)= Mn . (4)EJQTDE, 2012 No. 83, p. 4Let us consider the subset of D defined asM = ∪n∈INMn (5)and its countable subsetC = ∪n∈INCn . (6)First of all, it is easy to show thatC = ∪n∈INCn . (7)Now, being Cn a subset of Mn, we haveCn = Cn(Mn,d)and so, from (7), we deduceC = ∪n∈INCn(Mn,d). (8)Therefore, by (5), (4) and (8), we can say thatM = C . (9)Further, some easy considerations, which make use of (3) and (5), lead us to writethatM = co({x0} ∪ F (M)) . (10)Since M is a set verifying the property of hypothesis (M), we can claim that the setM is compact . (11)Now, we consider the map G :M → P(M ) defined byG(x) = F (x) ∩M , x ∈M .As a matter of fact, the multimap G has nonempty values. In fact, fixed x ∈ M ,there exists a sequence (xn)n∈IN in M such that xn → x. Let us consider a sequence(yn)n∈IN , yn ∈ F (xn), n ∈ IN . By (10), {yn}n∈IN is included in the compact set M (see(11)). Therefore, w.l.o.g., we can say that yn → y ∈ M . So, by applying hypothesis(ii), S(x, y) ∩ F (x) 6= ∅; by the convexity of M , we have S(x, y) ⊂ M . ThereforeG(x) = F (x) ∩M 6= ∅.Now, let us prove that G has weakly closed graph in M × X. We fix a sequence(xn)n∈IN in M converging to a point x¯ and a sequence (yn)n∈IN , yn ∈ G(xn), n ∈ IN ,converging to a point y¯. By hypothesis (ii) and by the fact that S(x¯, y¯) ⊂M , we canconclude that ∅ 6= S(x¯, y¯) ∩ F (x¯) = S(x¯, y¯) ∩ F (x¯) ∩M = S(x¯, y¯) ∩G(x¯).Finally, F takes closed values and satisfies hypothesis (i), so we can conclude thatthe map G verifies all the assumptions of Theorem 2.1. Therefore, there exists x ∈ Dsuch that x ∈ G(x) ⊂ F (x) . 2Now, we can provide our fixed point theorem for Daher type multimaps.EJQTDE, 2012 No. 83, p. 5Theorem 3.2 Let D be a closed, convex subset of a Banach space X and F : D →Pk(D) be a map verifying hypotheses (i), (ii), (iii) of Theorem 3.1 and the following(c) F is countably condensing.Then there exists x ∈ D with x ∈ F (x).Proof. We prove that the multimap F satisfies hypothesis (M) of Theorem 3.1.Let us fix x0 ∈ D and let M be a subset of D such that M = co({x0} ∪ F (M))andM = C , (12)with C countable subset of M .Being C ⊂ co({x0} ∪ F (M)), every point of C can be written as a finite combinationof points belonging to the set {x0} ∪ F (M). Therefore, there exists a countable setM⊂M such thatC ⊂ co({x0} ∪ F (M)) . (13)By hypothesis (c), F (D) is bounded, then also sets M , C and M are bounded.Let us prove that β(C) = 0. First of all, by using (13), (1), (β2) and (β3), we haveβ(C) ≤ β(co({x0} ∪ F (M))) = β({x0} ∪ F (M)) = β(F (M)) . (14)Now, suppose that β(M) 6= 0. Then, hypothesis (c) yieldsβ(F (M)) < β(M) . (15)Combining (14) with (15), by (β2), (1) and (12), we obtainβ(C) < β(M) ≤ β(M) = β(M) = β(C) = β(C) , (16)that is a contradiction.Hence, it must be β(M) = 0. So, by the regularity of β, M is compact. Thus,assumption (iii) provides thatβ(F (M)) = 0 .Then, since F (M) ⊂ F (M) and by using (β2), we haveβ(F (M)) = 0 . (17)Therefore, by (14) and (17), we can concludeβ(C) = 0 .Now, thanks to the definition of β and by (12), we can writeβ(M) = β(M) = β(C) = β(C) = 0EJQTDE, 2012 No. 83, p. 6and so β(M) = 0 too. Therefore, the set M is compact. Hence, hypothesis (M) ofTheorem 3.1 is verified.Consequently, by applying Theorem 3.1, we can conclude that there exists a fixedpoint for F in D. 2Remark 3.1 We note that our Theorem 3.1 and Theorem 3.2 strictly contain re-spectively Theorem 3.1 in [21] and Theorem 1.2 in [1]. In fact, let us consider themapF : [0, 2]→ P([0, 2]) defined by F (x) ={{1} , x ∈ [0, 2[{2} , x = 2 .It is easy to check that F satisfies either all the assumptions of Theorems 3.1 and 3.2,but it has not closed graph. Therefore, it is not possible to apply to F neither Theorem3.1 in [21] nor Theorem 1.2 in [1].Remark 3.2 We wish to observe that if we strengthen of Definition 1.1 by the fol-lowingDefinition 3.1 Let D be a nonempty subset of X. For k ∈ [0, 1[, we say that a mapF : D → P(X) is countably k-condensing if F (D) is bounded and β(F (B)) ≤ kβ(B)for all countable bounded subsets B of D .then an immediate consequence of Theorem 3.2 is the next Darbo type result:Corollary 3.1 Let D be a closed, convex subset of a Banach space X and F : D →Pk(D) be a map verifying hypotheses (i), (ii), (iii) of Theorem 3.1 and the following(c)’ F is countably k-condensing, for k ∈ [0, 1[.Then there exists x ∈ D with x ∈ F (x).References[1] R.P. Agarwal, D. O’Regan, A generalization of the Krasnoselskii-Petryshyn compres-sion and expansion theorem: an essential map approach, J. Korean Math. Soc. 38(2001), no. 3, 669-681.[2] M. Arias, J. Campos, C. Marcelli, Fastness and continuous dependence in front prop-agation in Fisher-KPP equations, Discr. Cont. Dyn. Syst., Ser. B, 11 (2009), no. 1,11-30.[3] J.M. Ayerbe Toledano, T. Dominguez Benavides, G. 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Countably condensing multimaps and fixed points

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