Some Fixed Point Theorems for Countably Condensing

: Our aim in this article is to establish the principles results of a ﬁxed point theorems for multivalued mappings of Krasnoselskii type setting in general classes M ¨ onch’s type. We seek to do that, we introduce and recall some theorems to aid our study. The beginning of this work has been introduced some properties of the measure of weak noncompactness under the weak topology and the deﬁnitions of countably condensing operators. We have shown that the operator H ( S ) is relatively weakly compact by using some properties of weak topology. We investigate that all hypotheses guarantee that the operator ( B + H )( S ) is relatively weakly compact and than simply to apply Himmelberg’s theorem in Banach spaces. We extended two ﬁxed point theorems for weakly sequentially upper semicontinuous mappings subjected the perturbation map satisﬁes the M ¨ onch’s type and we obtain our results in the second theorem with a less restrictive hypothesis. Using abstract measures of weak noncompactness, these results are applied to derive some ﬁxed point theorems for a weakly sequentially upper semicontinuous countably µ -condensing multivalued mappins.

Our objectif in this work is to establish some fixed point results of multivalued version of the Himmelberg's theorem in Banach spaces for weakly sequentially continuous maps.This work is motivated by some results obtained in [2] essentially, Theorem 2.2.

Preliminaries
Let X be a Banach space.we denote by K(X) = {S ⊂ X : S is nonempty}, K bd (X) = {S ⊂ X : S is nonempty and bounded}, K cv (X) = {S ⊂ X : S is nonempty and convex}, K cl,cv (X) = {S ⊂ X : S is nonempty, convex and closed} and W(X) = {S ⊂ X : S is nonempty weakly compact}.We introduce the notions in Banach space for the measure of weak noncompactness.
Definition 2.1.Let µ : K bd (X) → [0, +∞[ be said the measure of weak noncompactness on X if µ satisfies the following assertions: (1) The family ker µ := S ∈ K bd (X) : µ(S) = 0 is nonempty and elso ker µ is contained in all set of relatively weakly compact subsets of X, ß (2) The monotonicIty: where co denotes the closed convex hull of S, (4) The homogeneity: µ(λS) = λµ(S) ∀λ ∈ R + .The family ker µ is given on first assertion and it is called the kernel of the measure µ.It should be noticed that the inclusions S ⊆ S w ⊆ co(S) together with the item (3) of Definition 2.1 imply (5) µ(S w ) = µ(S).Note that if µ(•) is the full measure of weak noncompactness to having the maximum property, then it is non-singular, that is: (6) µ(S ∪ {x}) = µ(S), for all S ∈ K bd (X) and x ∈ X.Before going further we recall the following definitions.Definition 2.2 Let (X, d) and (Y, d) be two metric spaces and let G : X → K cl,cv (Y ) be a multivalued map.ß • G is called weakly upper semicontinuous (w.u.s.c. for short) if G is upper semi-continuous with respect to the weak topologies of X and Y .ß• F is called weakly sequentially upper semicontinuous (w.s.u.s.c. for short) if for each weakly closed set F of Y, G −1 (F ) is weakly sequentially closed. 2 Definition 2.3 Let µ(•) a measure of weak noncompactness on X and B : S → K(X) a multivalued map and S is a closed, convex, nonempty subset of a Hausdorff locally convex linear topological space X.We say that B is condensing with respect to the measure µ( , for all bounded subset J of S with µ(S) > 0.
2 Now, we recall the following results.Theorem 2.1 [16] Let S be a weakly compact subset on Banach space X.
Theorem 2.2 [2] Let S be a convex nonempty, closed subset on Banach space X.Let B : S → K cl,cv (S) be a w.s.u.s.c.multivalued map and B(S) is relatively weakly compact.Then there exists z ∈ S such that z ∈ B(z).

Main Results
The goal of the following results is to improve some fixed point results of Krasnosel'skii theorems for a w.s.u.s.c.multivalued map subjected the perturbation H satisfies the Mönch's type.
Theorem 3.1 Suppose that B : S → K cl,cv (S) and H : S → K cl,cv (S) be two w.s.u.s.c.multivalued maps.Assume that (a) B(S) is relatively weakly compact; (b) There exists z 0 ∈ S satisfies H(S) + B(S) ⊂ S .Thus, there exists x ∈ S such that x ∈ H(z) + B(z).Proof Let P m be defined by Assume that P n be a relatively weakly compact for each n ∈ {1, 2, • • • }.We note that according to the Theorem 2.1, B : P w n → K cl,cv (S) is weakly semicontinuous, and so [6, p.464] ensures that B P w n is weakly compact.Theorem of Krein-Smulian [9, p.82] ensures that P n+1 is relatively weakly compact.by (b) there exists a sequence of countable sets Thus, P is also convex, since P m−1 ⊆ P m for m = 1, 2, • • • .We deduce that Thus, [19, p.66] and hypothesis (b) guarantee that Then, Eqs. ( 1) and ( 2) guarantees that Now, hypothesis (b) satisfying that M w (and so P w ) is also weakly compact.This yields that H(P w ) ⊂ P w and therefore It is clear that R is nonempty set (using hypothesis (d)).Since H(P w ) is relatively weakly compact, we deduce that R is relatively weakly compact.
On the other hand, because R ⊂ S, we have B(R) ⊂ B(S) and by using hypothesis (a) shows B(R) is relatively weakly compact.
According to the steps in the above, (B + H)(R) is also relatively weakly compact.Applying Theorem 2.2, we conclude there exists x ∈ R (and x ∈ S) such that x ∈ (B + H)(x) which concludes the proof of Theorem.
Let S be a subset of the space X and let H : S → K cl,cv (S) be a w.s.u.s.c.multivalued map, in the following theorem, we replace the hypothesis (b) of Theorem 3.1 with a less restrictive hypothesis; a countable set M ⊆ Q, there exist z 0 ∈ S such that M w = co {z 0 }∪H(M ) implies that M is also relatively weakly compact.Theorem 3.2 Suppose that B : S → K cl,cv (S) and H : M → K cl,cv (S) be two w.s.u.s.c.multivalued maps satisfying the conditions (a), (c), (d) and (e) of Theorem 3.1.In addition assume that.
(A) There exists z 0 ∈ S such that satisfying M is relatively weakly compact, (B) H(F w ) ⊆ H(F ) w for each subset F of S are satisfied.Then there exists x ∈ S such that x ∈ H(x) + B(x).
Proof Let P m defines as the same steps in Theorem 3.1, we have that P m is relatively weakly compact for all m = 0, 1, • • • .Assume that M m , P and M be as the last theorem, we obtain P (= P w ) = co {z 0 } ∪ H(P ) and P w = M w . ( Now, condition (B) implies that thus, Then, Eqs. ( 3) and ( 4) imply that is relatively weakly compact.Next, put R = B(S) ∩ H(M w ).Note that, applying hypothesis (d), we get R is nonempty and (B + H)(R) is relatively weakly compact.Now arguing the same steps in the end of the proof of Theorem 3.1, we deduce that there exists x ∈ R (and so x ∈ S) such that x ∈ (B + H)(x) which conclude the proof.Remark 3.1 The conditions (b), (A) and (B) in Theorems 3.1 and 3.2 introduced by Donal O'Regan [16], to prove a fixed set results for w.s.u.s.c.multivalued maps, our objective of this work , we use these conditions to show that the perturbation B(M ) is also relatively weakly compact in Banach space X with S ⊂ X.
Let S be a subset of the space X and let µ be the measure of weak noncompactness on X. Assume that B : S → K cl,cv (S) is a w.s.u.s.c.multivalued map.F is said to be countably µcondensing if µ B(S) < ∞ and µ(B(M )) < µ(M ) for all countable bounded subset M of S with µ(M ) > 0.
Definition 3.1 Assume S be a subset convex of X and let B : S → K(X) be a given map.We say that F ⊂ S is a Mönch-set for B if there exists z 0 ∈ S such that F w = co {z 0 } ∪ B(F ) and there exists a countable set M w ⊂ F w with F w = M w .
Corollary 3.1 Assume hat S is a closed convex, nonempty subset of X, z 0 ∈ S and B : S → K cl,cv (S).Let H : S → K cl,cv (S) be two w.s.u.s.c.multivalued maps satisfying the assumptions of Theorem 3.2.If H is countably µ-condensing, then there exists z ∈ S such that z ∈ H(z) + B(z).
Proof We start to prove that H satisfies assumption (A) of Theorem 3.2.We fix y 0 in S and we assume that M w be a Mönch-set for the map H contained also in S, that is, M w = co {y 0 } ∪ H(M ) , we have where F w is also a countable subset of M w .Since F w ⊂ co({y 0 } ∪ H(M )), for each point of F w , we can be written as a finite combination of points belonging to the M is relatively weakly compact, we set {z 0 } ∪ H(M ).Further, there exists a countable set O ⊂ M w such that We claim that the measure of F is equal zero, that is µ(F w ) = 0. in fact, the use of ( 6), H is countably µcondensing and also the properties of µ, we have If µ(O) = 0, then the proof of the corollary is finished.Otherwise, combining ( 5) and ( 7) we get which is a contradiction.Thus, µ(F w ) = 0 which gives the prove of our claim.Consequently, the use of ( 5) improve that M w is weakly compact.Keeping in mind that M w = co({z 0 } ∪ H(M )), we infer that H(M w ) ⊂ M w and further H(M w ) is also relatively weakly compact.
Put R = B(S) ∩ H(M w ).By the assumption (d), we have that R is nonempty and so (B + H)(R) is relatively weakly compact.Now arguing as the same steps in the end of the proof of Theorem 3.1, we deduce that there exists z ∈ R (and also then z ∈ S) such that z ∈ (B + H)(z).

Conclusion
We conclude that we have in this paper some contrubtion for the new results of fixed points theory setting in general classes Mönch's type for multivalued mappings by using the generation of Himmelberg theorem that is given in [2] and we develop these results by using the measure of weak noncompactness on X when the perturbation is countably µcondensing.