 Advances in Pure Mathematics, 2011, 1, 163-169 doi:10.4236/apm.2011.14030 Published Online July 2011 (http://www.SciRP.org/journal/apm) Copyright © 2011 SciRes. APM On Some Fixed Point Theorems for 1-Set Weakly Contractive Multi-Valued Mappings with Weakly Sequentially Closed Graph Afif Ben Amar1, Aneta Sikorska-Nowak2 1Département de Mathéma tiques Université de Gafsa Faculté des sciences d e Gafsa Cité Universitaire Zarrou k, Gafsa Tunisie 2Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Poznan, Poland E-mail: afif.benamar@ipeis.rnu.tn, anetas@a mu.edu.pl Received February 20, 2011; revised March 11, 20 1 1; acce pted March 21, 2011 Abstract In this paper we prove Leray-Schauder and Furi-Pera types fixed point theorems for a class of multi-valued mappings with weakly sequentially closed graph. Our results improve and extend previous results for weakly sequentially closed maps and are very important in applications, mainly for the investigating of boundary value problems on noncompact intervals. Keywords: Measures of Weak Noncompactness, Weakly Condensing, Weakly Nonexpansive 1. Introduction  bdcvcl,cv:,=: ,=: ,,PXAXA is nonemptyandboundedPXAXA isnonemptyandconvexPXXA isnonemptyclosedandconvex  Fixed point theory for weakly completely continuous multi-valued mappings takes an important role for the existence of solutions for operator inclusions, positive solutions of elliptic equation with discontinuous non- linearities and periodic and boundary value problems for second order differential inclusions (see [1-3]) and others. In  O’Regan has proved a number of fixed point theorems for multi-valued maps defined on bounded domains with weakly compact convex values and which are weakly contractive and have weakly sequentially closed graph. The aim of the present paper is to extend and improve these theorems to the case of weakly condensing and 1-set weakly contractive multi-valued maps with weakly sequentially closed graph. Further- more, we do not assume that they are from a point into weakly compact convex set. We note that the domains of all of the multi-valued maps discussed here are not assumed to be bounded. Our results generalize and extend relevant and recent ones (see [4-10]). The main condition in our results is formulated in terms of axioma t ic measures of w eak noncom- pactness. ANow we shall introduce the notation and give preli- minary results which will be need ed in the paper. Let X be a Hausdorff linear topological space, then we define Let Z a non-empty subset of a Banach space Y and be a multi-valued mapping. We denote :2XFZ==,:yZRFFyand GrFzxZXxFz  the range and the graph of F respectively. Moreover, for every subset A of X, we put 1=:FAzZFzA and =:AzZFzA. F F is called upper semicontinuous on Z if 1FA is closed, for every closed subset A of X (or, equivalently, if FA is open, for every open subset A of X).  F is called weakly upper semicontinuous if F is upper semicontinuous with respect to the weak topologies of Z and X. Now we suppose that X is a Banach space and Z is weakly closed in Y. F is said to be weakly compact if the set RF is relatively weakly compact in X. Moreover, F is said 164 A. B. AMAR ET AL. to have weakly sequentially closed graph if for every sequence nnxZ with wnxx in Z and for every sequence with nny,,nn ynnx wyF y in X implies yFx, where w denotes weak convergence. F is called weakly completely continuous if F has a weakly sequentially closed graph and, if A is bounded subset of Z, then FA is a weakly relative compact subset of X. Definition 1.1 Let be a Banach space and a lattice with a least elemen t, which is denoted by . By a measure of weak non-compactness ()E C0MNWC E on , we mean a function defined on the set of all bounded subsets of with values in , such that for any : EbdPE C12,(1) 1=1conv , where conv denotes the closed convex hull of . (2) 1 2aE 112=a =0 EC , (3) for all . 11(4) 1 if and only if is weakly rela- tively compact in . If the lattice is a cone of vector space, then the MNWC is said positive homogenous provided for all = >0 and PEbdThe notion above is a generalization of the important well known DeBlasi measure of weak non-compactness  .  (see ) defined on each bounded set  of by Eweakly= iheret sas hatnf>0:tcompact setexisuch tEDDB  where EB is the closed unit ball of . EIt is well known that  enjoys these properties: for any , bdPE12,(5). 1 2,1>012=max 1= . (6) for all . (7) . 12 1 2 :2Now let us introduce the following definitions. Definition 1.2 Let be a nonempty subset of Banach space E and a MNWC on E. If EF , we say that (a) F is -condensing if 0 0=0cv has weakly sequentially closed graph and weakly compact map. In addition, assume the following conditions are satisfied: There exists a weakly sequentially continuous retra- ction (3) There exists  and a weakly compact set M with K=:, ,=xEdxMMheredxyx y  (4) =:,,00dNMx y xNyM,0< . Thus, there exists , with , here =KN=,,KxEdxMK. We have  is closed, convex, so weakly closed and KM. Using (4), we obtain K is weakly compact. Because is separable, the weak topology on EK is metrizable (see ), and Copyright © 2011 SciRes. APM 168 A. B. AMAR ET AL. let  denote this metric. For , let 1,2,i ,M Since , then =wi. Applying Corollary 3.2, we get that there exists NU0,1iKyi and iU such that iiyFr yy. In particular, since , then iKiU1,2,irsome[0,1]RforM=:ExFreachiiFr y (6) Now we investigate ,foRx x R is non-empty, because . Also, MRconvF, so by the Krein-Šmulian theorem (see [14, p. 434]), R is relatively weakly com- pact. Since Fr has weakly sequentially closed graph and is compact, we deduce by Lemma 3.1 that [0,1]R is weakly sequentially closed. This together with for 1,2,j,=, [0,1]jjyM j implies that we may assume without loss of generality that 00and \wwjjyyMK=wKM M Also since jjiyFry we have 000yFr y. If 0=1, then 00yFr y[0,1), which contradicts =MN 0. Thus, =,=ji. But (5) with 00=KxryMxyryM implies jiFry M for sufficiently large. This contra- j=dicts (6). Thus . Accordingly, there exists MNxM such that  =xFrxF x()a*E Q.E.D. Remark 3.4 If is separable and M is weakly compact, then there exists a weakly continuous and so weakly sequentially continuous retraction onto M (see ). ()bE If is reflexive then it suffices to take FM is bounded, since a subset of a reflexive Banach space is weakly compact iff it is closed in the weak topology and bounded in the norm topology. Remark 3.5 Theorem 3.3 extends Theorem 2.5 in . ()a()b As a corollary of Theorem 3.3 we find Theorem 2.6 in . Remark 3.6 If in the statements of Theorem 3.3, the convex set  is weakly compact, then a special case of (5) which is useful in applications is If =1,jjjx[0,1]M,wjjxx is a sequence in with  and xFx0<1 with , then jjFxM for j sufficiently large.. (7) Remark 3.7 The set M can be with void weak inte- rior. 4. References  G. Bonanno and S. A. Marano, “Positive Solutions of Elliptic Equations with Discontinuous Nonlinearities”, Topological Methods in Nonlinear Analysis, Vol. 8, 1996, pp. 263-273.  D. Averna and S. A. 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