On Some Fixed Point Theorems for 1-Set Weakly Contractive Multi-Valued Mappings with Weakly Sequentially Closed Graph

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.

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 nonlinearities and periodic and boundary value problems for second order differential inclusions (see [1][2][3]) and others.
In [4] 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.Furthermore, 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][5][6][7][8][9][10]). The main condition in our results is formulated in terms of axiomatic measures of weak noncom-pactness.

A
Now we shall introduce the notation and give preliminary results which will be needed 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 :  the range and the graph of F respectively.Moreover, for every subset A of X , we put , where conv denotes the closed convex hull of .  ) .
  Now let us introduce the following definitions.Definition 1.2 Let be a nonempty subset of Banach space E and a MNWC on E.
for all bounded sets .D   In the sequel, we shall need also the following multivalued fixed point theorem.
Theorem 1.1 (see [12], p. 206).Let be a nonvoid, convex and closed subset of a locally convex space .Let    be an upper semicontinuous multi-valued mapping such that is relatively compact.Then F has a fixed point.

Sadovskii Type Fixed Point Theorems
We begin with the following interesting property of multi-valued maps with weakly sequentially closed graph which is a basic tool for achieving our aim. Theorem Set .It follows, using the Krein-Šmulian theorem (see [14, p. 434]) that is a weakly compact convex set.We have . By Theorem 2.1 F : F has weakly closed graph, and so   F x is weakly closed for every x K  .Thus, by [15, Proposition 14.5, p. 69] F is weakly upper semicontinuous.Because endowed with its weak topology is a Hausdorff locally convex space, we apply Theorem 1.1 to get that Theorem 2.2 improves Theorem 2.2 in [4] and Theorem 2.1 in [1] for the case of Banach spaces.
Because every single-valued and weakly sequentially continuous map has weakly sequentially closed   F graph, then Theorem 2.2 extends and improves a fixed point theorem of Arino, Gautier and Penot for the case of Banach spaces [5], Theorem 2.5 in [9] and Corollary 2.3 in [10].

F is  -nonexpansive and
Now, we are ready to prove some fixed point theorems for a broader class of multi-valued mappings with weakly sequentially closed graph, in which the operators have the property that the image of any set is in a certain sense more weakly compact than the original set itself.
Theorem 2.3 Let be a non-empty, closed, convex subset of a Banach space .Assume a MNWC on and cv has weakly sequentially closed graph.In addition, suppose that for all and hence  .We will prove that K is weakly compact.Denoting by and so is relatively weakly compact.Now,  Q.E.D.

Leray-Schauder and Furi-Pera Types Fixed Point Theorems
In applications, the construction of a set such that    is very difficult and sometimes impossible.In that line, we investigate maps Proof.For all , there exists is non-empty and bounded, because .For all , there exists a , we can a subsequence   x y


, where . Applying Lemma 3.1, we deduce that endowed with its weak topology is a Hausdorff locally convex space, we have that is completely regular [17, p. 16].Since , then by [16, p. 146], there is a weakly continuous function . Since  is convex,   , and   F with nonempty convex values, we can define the multi-valued map by:  , which contradicts the hypothesis   .
Then 0 x U  and 0 0 0 , which implies that 0 x D  , and so and the proof is complete.
Remark 3.2 (a) Theorem 3.1 extends Theorem 2.5 in [4] and shows that the condition F has weakly closed graph can be replaced by F has weakly sequentially closed graph.
(b) Theorem 3.1 extends and improves Theorem 2.4 in [4] and shows that the condition w U E is weakly compact in the statement of this theorem is redundant.
(c) Theorem 3.1 extends Theorem 3.3 in [8] in the context of single-valued and weakly sequentially continuous maps to the case of multi-valued maps with weakly sequentially closed graph. .Now, applying Corollary 3.1, the remainder of the proof is similar to that of Theorem 2.4.Q.E.D.
In applications, it is extremely difficult to construct a weakly open set U as in Theorem 3.1, so we are motivated to construct a Furi-Pera type fixed point theorems ( [18]) for a multi-valued mapping : 2 E F M  with weakly sequentially closed graph.Here M is a closed convex subset of with (possible) an empty weak interior.The last mentioned case is very important in applications, mainly for the investigation of boundary value problems on noncompact intervals (see [18]).
  .Hence, x N  and is weakly sequentially closed.Applying again the Eberlein-Šmulian theorem [13, theorem 8.12.4,p. 549], we obtain that is weakly compact.We now show .To do this, we argue by contradiction and we use some ideas in [18].Suppose is compact and M is closed we have by [19, p. 65 . Using (4), we obtain K  is weakly compact.Because is separable, the weak topology on E K  is metrizable (see [20]), and let  denote this metric.For , let . Applying Corollary 3.2, we get that there exists Now we investigate If is separable and M is weakly compact, then there exists a weakly continuous and so weakly sequentially continuous retraction onto M (see [21]).
is a generalization of the important well known DeBlasi measure of weak non-compactness   .(see[11]) defined on each bounded set  of by is the closed unit ball of .E It is well known that  enjoys these properties: for any ,

R
is non-empty, because   .Also, the Krein-Šmulian theorem (see[14, p. 434]), R is relatively weakly compact.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

3 . 6 Remark 3 . 7
( ) b E Ifis reflexive then it suffices to take F M 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.As a corollary of Theorem 3.3 we find Theorem 2.6 in[8].Remark 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 The set M can be with void weak interior.
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 have weakly sequentially closed graph if for every sequence F is said to be weakly compact if the set  R F is relatively weakly compact in X .Moreover, condensing on  .From then without loss all of the assumptions of