Countably condensing multimaps and fixed points

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 Monch


Introduction
Condensing operators have been the object of a wide and deep study in nonlinear functional analysis.This research was started in 1967 by Sadovskii [22].He introduced the concept of condensing operator for single-valued functions by using the Kuratowski measure of noncompactness and proved that a condensing operator from a closed bounded convex subset of a Banach space into itself has a fixed point, extending the well 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.condensing only on countable subsets); from another, Himmelberg, Porter and Van Vleck [17] ex-tended the definition of condensing operator to multimaps and provided a multivalued version of the Sadovskii's theorem.More recently, Agarwal and O'Regan [1] established a multivalued version of the Daher's theorem for upper semicontinuous multimaps.
The main purpose of the present paper is to state a Daher type theorem (see Theorem 3.2) for multimaps having weakly closed graph, which can take some nonconvex values 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 an abstract measure of noncompactness.A map F : D → P(X) is said to be countably condensing if As far as we know, this definition is new and it is an improving of the analogous definition given by Agarwal and O'Regan.In fact, they say that a multimap is countably condensing 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 subsets B of D taking β as the Kuratowski measure of noncompactness.The advantage of our definition is that it is the natural weakening of the classical definition of condensing multimap (see, e.g.[19]) and, moreover, that in the singlevalued 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 Mönch type theorem for multimaps having weakly closed graph (cf.Theorem 3.1).The Mönch type hypothesis we use has been introduced by O'Regan and Precup [21] in order to extend the Mönch theorems to multimaps.We base our proof on a fixed point theorem for multimaps having weakly closed graph established in [6].

Preliminaries
Let X be a locally convex Hausdorff topological linear space and P(X ) be the family of all nonempty subsets of X .EJQTDE, 2012 No. 83, p. 2 We recall (see [16]) that a set A ⊂ X is said to be quasi-convex if for every balanced convex neighborhood V of zero and for every {a 1 , . . ., a n } ⊂ A there exists {z 1 , . . ., z n } ⊂ A such that z i − a i ∈ V , i = 1, . . ., n, co{z 1 , . . ., z n } ⊂ 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 (x δ ) δ in D, x δ → x, x ∈ D, and for every net (y δ ) δ , y δ ∈ G(x δ ), y δ → y, then S(x, y) ∩ G(x) = ∅, 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 linear space, K be a nonempty compact subset of X and G : K → P(K) be a map taking nonempty closed values and with the properties (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 use the following notations: We recall (see, e.g.[3], [19]) that a function β : P b (X) → IR + 0 is said to be a measure of noncompactness (MNC, for short) if β(co(Ω)) = β(Ω) , for every Ω ∈ P b (X) . (1) In the sequel, we consider a measure of noncompactness β verifying the following properties: (β 1 ) regularity: β(Ω) = 0 if and only if Ω is compact; (β 2 ) monotonicity: where As examples of measures of noncompactness which satisfy all the previous properties, we recall the Hausdorff and the Kuratowski measures of noncompactness.In the single-valued case, this is the definition of countably condensing map sometimes adopted in the literature.

The fixed point theorems
In this section, we will provide a Daher type theorem for multimaps.In a preliminary way, we give a fixed point theorem for multimaps by using a Mönch type assumption which has been introduced by O'Regan and Precup [21] in order to extend the classical Mönch theorem to set-valued maps.
Theorem 3.1 Let D be a closed convex subset of a Banach space X and F : D → P k (D) be a map such that (i) there exists A ⊂ D, A quasi-convex, A = D such that F (x) is convex for every x ∈ A; (ii) F has weakly closed graph; (iii) F maps compact sets into relatively compact sets; (M) there exists x 0 ∈ D such that Then there exists x ∈ D such that x ∈ F (x).
Proof.Let x 0 ∈ D be as by hypothesis (M).We consider the iterative sequence (M n ) n∈IN of sets: Clearly, Let us prove by induction that M n , n ∈ IN + , is relatively compact.First, the Mazur Theorem (see, e.g.[18,Theorem A.3.68]) implies that co({x . By (iii) and the Mazur's theorem, we can say that also M n is relatively compact.By induction again, we can say that Let us consider the subset of D defined as and its countable subset First of all, it is easy to show that Now, being C n a subset of M n , we have and so, from (7), we deduce Therefore, by ( 5), ( 4) and ( 8), we can say that Further, some easy considerations, which make use of ( 3) and ( 5), lead us to write that Since M is a set verifying the property of hypothesis (M), we can claim that the set Now, we consider the map G : M → P(M ) defined by As a matter of fact, the multimap G has nonempty values.In fact, fixed x ∈ M, there exists a sequence (x n ) n∈IN in M such that x n → x.Let us consider a sequence (y n ) n∈IN , y n ∈ F (x n ), n ∈ IN.By (10), {y n } n∈IN is included in the compact set M (see (11)).Therefore, w.l.o.g., we can say that y n → y ∈ M .So, by applying hypothesis (ii), S(x, y) ∩ F (x) = ∅; by the convexity of M , we have S(x, y) ⊂ M .Therefore G(x) = F (x) ∩ M = ∅.Now, let us prove that G has weakly closed graph in M × X.We fix a sequence (x n ) n∈IN in M converging to a point x and a sequence (y n ) n∈IN , y n ∈ G(x n ), n ∈ IN, converging to a point ȳ.By hypothesis (ii) and by the fact that S(x, ȳ) ⊂ M , we can conclude that ∅ = S(x, ȳ) ∩ F (x) = S(x, ȳ) ∩ F (x) ∩ M = S(x, ȳ) ∩ G(x).
Finally, F takes closed values and satisfies hypothesis (i), so we can conclude that the map G verifies all the assumptions of Theorem 2.1.Therefore, there exists x ∈ D such that x ∈ G(x) ⊂ F (x) . 2 Now, we can provide our fixed point theorem for Daher type multimaps.

Remark 2 . 1
We note that if β is a regular MNC, property (II) of Definition 1.1 can be equivalently formulated as (II)' for every countable bounded subset B of D the relation β(B) ≤ β(F (B)) implies that B is compact.
[14] for every n ∈ IN, let us consider the space (M n , d), where d is the metric induced on M n by the metric generated by • .The compactness of M n implies that (M n , d) is a separable space (cf., e.g.Corollary 1.4.29 and Corollary 1.4.12in[14]).Hence, there exists a countable set C n ⊂ M n such that