Existence Results for Generalized Vector Equilibrium Problems with Multivalued Mappings via KKM Theory

1 Department of Mathematics, School of Science, Razi University, P.O. Box 67149-67346, Kermanshah, Iran 2 Center for Research in Mathematics and Physics, School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran 3 Department of Mathematics, Faculty of Basic Sciences, University of Shahrekord, P.O. Box 88186-34141, Shahrekord, Iran 4 Department of Mathematics, College of Arts, Social Sciences, & Celtic Studies, National University of Ireland, Galway, Ireland


Introduction and preliminaries
Throughout this paper, unless otherwise specified, we always let X and Y be real Hausdorff topological vector spaces, K ⊆ X a nonempty convex set, C : K → 2 Y with pointed closed cone convex values we recall that a subset A of Y is convex cone and pointed whenever A A ⊆ Y, tA ⊆ A, for t ≥ 0, and A ∩ −A {0}, resp., where 2 Y denotes all the subsets of Y. Denote by L X, Y the set of all continuous linear mappings from X into Y.For any given l ∈ L X, Y , x ∈ X, let l, x denote the value of l at x. Let T : K → L X, Y and g : K → K be two mappings.Finally, let F : K × K → 2 Y be a set-valued mapping.We need the following definitions and results in the sequel.Definition 1.1.Let F : K × K → 2 Y be a set-valued mapping.One says that F is i strongly C-pseudomonotone if, for any given x and y ∈ K, / ⊆ − int C x ⇒ F y, x ⊆ −C y . 1.1 ii C-pseudomonotone if, for any given x and y ∈ K, ,Y be a set-valued mapping and let g : K → K be a mapping.If we define F x, y Tx, y − g x , for each x, y ∈ K × K, then strongly C-pseudomonotonicity and C-pseudomonotonicity reduce to the strongly Cpseudomonotonicity and C-pseudomonotonicity, of T with respect to g, respectively, introduced in 1 .
Remark 1.4.One can see that ii is equivalent to the following statement: G is l.s.c. at x ∈ X if for each closed set C ⊆ Y, any net {x α } ⊆ K, x α converges to x, and G x α ⊆ C, for all α imply that G x ⊆ C. Lemma 1.5 see 2 .Let X and Y be two topological spaces.Suppose that G : X → 2 Y is a setvalued mapping.Then the following statements are true.
b Let, for any x ∈ X, G x be compact.If G is u.s.c. on X then for any net {x α } ⊂ X such that x α → x and for every y α ∈ G x α , there exist y ∈ G x and a subnet {y β } of {y α } such that y β → y.
For the converse of b in Lemma 1.5, we refer the reader to 3 .
Definition 1.6.Let X be a topological vector space and Y a topological space.A set-valued mapping G : X → 2 Y is called upper hemicontinuous if the restriction of G on straight lines is upper semicontinuous.
Definition 1.7.One says that the mapping G : K × K → 2 Y is C-upper sign continuous if, for all x, y ∈ K, the following implication holds: If we define G x, y {f x, y }, for all x, y ∈ K and C x 0, ∞ , then Definition 1.7 reduces to the upper sign continuous introduced by Bianchi and Pini in 4 .The upper sign continuity notion was first introduced by Hadjisavvas 5 for a single-valued mapping in the framework of variational inequality problems.

Main results
In this section, we consider the following generalized vector equilibrium problems for short, GVEPs in the topological vector space setting: Clearly, a solution of GVEP 2 is also a solution of problem GVEP 1 .We need the following lemma in the sequel.
Then for any given y ∈ K, the following are equivalent: Proof.I ⇒ II is obvious from the definition of C-pseudomonotonicity of F. Suppose that II holds.For each z ∈ K, put z t y t z − y , where t ∈ 0, 1 and y ∈ K as above.By II , we have which contradicts ii note the first inclusion follows from iv , the second inclusion follows from 2.1 and 2.2 , and the third follows from the relation Therefore, for all t ∈ 0, 1 , the set F z t , z ∩ C z t is nonempty.Thus, by iii there is a u ∈ F y, z ∩ C y .Hence, since C y ∩ −C y \ {0} ∅, we get u/ ∈ −C y \ {0} .Consequently, F y, z / ⊆ − C y \ {0}.This completes the proof.
Remark 2.2.If the set-valued mapping C : K → 2 Y has closed graph and for each fixed z ∈ K the mapping x → F x, z is upper hemicontinuous with nonempty compact values, then condition iii in Lemma 2.1 holds.To see this, let x and y be arbitrary elements of K and u t ∈ F z t , y ∩ C z t / ∅, where z t 1 − t x ty, t ∈ 0, 1 .By Lemma 1.5 b , there exists a subnet of u t without loss of generality u t and u ∈ F x, y such that u t → u, where t → 0. Now, since C : K → 2 Y has closed graph note u t → u and z t → x as t → 0 and u t ∈ C z t , we have u ∈ C x .Hence, u ∈ F x, y ∩ C x and so F x, y ∩ C x / ∅.This shows that F is C-upper sign continuous.Therefore, Lemma 2.1 improves Lemma 2.3 in 1 .
By a similar argument as in Lemma 2.1 and using Remark 2.2, we can deduce the following result.

Lemma 2.3. Suppose that
i for each fixed z ∈ K, the mapping x → F x, z is upper semicontinuous with compact values; ii F is strongly C-pseudomonotone; iii F x, x / ⊆ − int C x , for each x ∈ K; v for each fixed x ∈ K, the mapping z → F x, z is convex.
Then for any given y ∈ K, the following are equivalent: Remark 2.4.Let T : K → 2 L X,Y be a set-valued mapping.If we define F x, y Tx, y − x , where x, y ∈ K, then Lemma 2.3 reduces to Lemma 3 of Yin and Xu 6 .Lemma 2.5.Under the assumptions of Lemma 2.1, the solution set of (GVEP 2 ) is convex.
Proof.Let x 1 and x 2 be solutions of GVEP 2 .By Lemma 2.1, we have From this and condition iv of Lemma 2.1, for all t ∈ 0, 1 , we deduce that for all z ∈ K. Hence, from Lemma 2.1, we get This means that 1 − t x 1 tx 2 is a solution of GVEP 2 .The proof is complete.
Similarly, we can prove the following lemma.
Lemma 2.6.Under the assumptions of Lemma 2.3, the solution set of (GVEP 1 ) is convex.
Definition 2.8.Let K 0 be a nonempty subset of K.A set-valued mapping Γ : K 0 → 2 K is said to be a KKM map if co A ⊆ x∈A Γ x , for every finite subset A of K 0 , where co denotes the convex hull.
Lemma 2.9 Fan-KKM lemma 7 .Let K be a nonempty subset of a topological vector space X and Γ : K → 2 X be a KKM mapping with closed values.Assume that there exists a nonempty compact convex subset Lemma 2.10 see 8 .Let K be a convex subset of a metrizable topological vector space X and F : K → 2 K be a compact upper semicontinuous set-valued mapping with nonempty closed convex values.Then F has a fixed point in K.
Theorem 2.11.Let all the assumptions of Lemma 2.1 hold and for each fixed x ∈ K, the mapping y → F x, y is lower semicontinuous, where y ∈ K.If there exist a nonempty compact subset B of K and a nonempty convex compact subset D of K such that, for each x ∈ K \ B there exists y ∈ D such that F y, x / ⊆ − C y , then the solution set of problem (GVEP 2 ) is nonempty and compact in K.
We claim that Γ is a KKM mapping.If not, there exist y 1 , y 2 , . . ., y n ∈ K and t i > 0, which is a contradiction to condition ii of Lemma 2.1.Therefore, Γ is a KKM mapping and so Γ is also a KKM mapping note, Γ y ⊆ Γ y , for all y ∈ K .By Remark 1.4, the values of Γ are closed in K note, for each fixed x ∈ K, the mapping y → F x, y is lower semicontinuous and by our assumption, we obtain that y∈D Γ y is a closed subset of the compact set B and hence y∈D Γ y is compact in K. Therefore, Γ fulfils all the assumptions of Lemma 2.9 and so y∈K Γ y / ∅.This means that there exists z ∈ K such that F y, z ⊆ −C z , ∀y ∈ K.

2.12
Now, it follows from Lemma 2.1 that and hence z is a solution of the problem GVEP 2 .This proves that the solution set of GVEP 2 is nonempty.By Lemma 2.1, the solution set of GVEP 2 equals y∈K Γ y and so it is a compact set in K note, in the above that the set y∈K Γ y is a closed subset of the compact set B .The proof is complete.
As an application of Theorem 2.11, we derive the existence result for a solution of the following problem which consists of finding a u ∈ K such that where A : K × K → 2 L X,Y and g : K → K.This problem was considered by Fang and Huang 1 in reflexive Banach spaces setting for a set-valued mapping which is demi-C-pseudomonotone.Theorem 2.12.Let X be metriziable topological vector space, K nonempty convex subset of X, A : K × K → 2 L X,Y , and g : K → K be two mappings.Assume that i for each fixed w ∈ K, the mapping u, v → A w, u , v − g u is C-pseudomonotone and C-upper sign continuous; ii A w, u , u − g u ∩ C u / ∅, for each w, u ; iii for each fixed v ∈ K, the mapping w, u → A w, u , u−g v is lower semicontinuous; iv for each finite dimensional subspace M of X with K M K ∩ M / ∅, there exist compact subset B M and compact convex subset D M of K M such that ∀ w, z ∈ K M × K M \ B M , ∃u ∈ D M such that A w, u , z − g u / ⊆ − C u .
Then there exists u ∈ K such that A u, u , v − g u / ⊆ − C u \ {0}, ∀v ∈ K.

2.15
and so since v was an arbitrary element of K, then 2.21 is true, for all v ∈ K.This completes the proof of claim.From 2.21 and Lemma 2.1, we have A u, u , v − g u / ⊆ − C u \ {0}, ∀v ∈ K, 2.23 and so the proof of the theorem is complete.