Connectedness of Solution Sets for Weak Vector Variational Inequalities on Unbounded Closed Convex Sets
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چکیده
and Applied Analysis 3 (ii) The scalar C-pseudomonotonicity in Definition 2 is weaker than C-pseudomonotonicity in Definition 1(ii). In fact, for any ξ ∈ C∗ \ {0}, x, y ∈ X, if ⟨ξ(u∗), y − x⟩ ≥ 0, then we have ⟨u∗, y − x⟩ ∉ − intC. Then, it follows from the Cpseudomonotonicity of T that ⟨V∗, y − x⟩ ∈ C, which implies that ⟨ξ(V∗), y − x⟩ ≥ 0. Definition 5. The topological space E is said to be connected if there do not exist nonempty open subsets V i of E, i = 1, 2, such that V 1 ∪ V 2 = X and V 1 ∩ V 2 = 0. Moreover, E is said to be path connected if for each pair of points x and y in E, there exists a continuous mapping φ : [0, 1] → E such that φ(0) = x and φ(1) = y. Definition 6. Let E, F be two topological spaces. A set-valued mapping G : E → 2F is said to be (i) closed if graphG = {(x, y) ∈ E × F : y ∈ G(x)} is closed in E × F, (ii) upper semicontinuous, if for every x ∈ E and every open set V satisfying G(x) ⊂ V, there exists a neighborhood U of x such that G(U) ⊂ V. Lemma 7 follows directly fromTheorem 3.1 of [14]. Lemma 7. Let K be a closed convex subset ofX and T : K → 2L(X,Y) scalar pseudomonotone and upper semicontinuous with nonempty compact convex values. If for any ξ ∈ C, S ξ (T,K) is nonempty and compact, then Sw(T,K) is nonempty and compact. Remark 8. It is pointed out in [14] that ifK ∞ ∩(T(K)) w∘ = {0}, then S ξ (T,K) is nonempty and compact for any ξ ∈ C, and so Sw(T,K) is nonempty and compact. FromTheorem 2 of [15], we have the following lemma. Lemma 9. LetK be a closed convex subset ofX and T : K → 2 L(X,Y) scalar pseudomonotone and upper semicontinuous with nonempty compact convex values. Then, for any ξ ∈ C, S ξ (T,K) is nonempty and compact if and only if K ∞ ∩ [ξ(T(K))] − = {0}. Lemma 10 (see [16]). Let E, F be two topological spaces. If the set-valued mapping G : E → 2F is closed and F is compact, then G is upper semicontinuous. Lemma 11 (see [17]). Let E, F be two topological spaces. Assume that E is connected and the set-valued mapping G : E → 2 F is upper semicontinuous. If for every x ∈ E, the set G(x) is nonempty and connected, then the set G(X) is connected. Lemma 11 follows immediately from the definition of path connectedness. Lemma 12. Let E, F be two topological spaces. Assume that E is path connected and the mapping A : E → F is continuous. Then, the set A(E) is path connected. Lemma 13. Let Y be a normed space with its dual space of Y . Let {y β } a net in Y and {y∗ β } be a net in Y. Suppose that y β converges to y in the norm topology of Y and y∗ β weak∗converges to y. Then, ⟨y∗ β , y β ⟩ → ⟨y∗, y⟩. Proof. Indeed, we have
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