Hölder Continuity of Solutions to Parametric Generalized Vector Quasiequilibrium Problems

and Applied Analysis 3 0 in any metric linear spaces. Let X,Λ,M, Y be metric linear spaces. Let Y ∗ be the topological dual space of Y . Let C ⊂ Y be a pointed, closed, and convex cone with intC/ ∅, where intC denotes the interior of C. Let C∗ : {f ∈ Y ∗ : f y ≥ 0, for all y ∈ C} be the dual cone of C. Since intC/ ∅, the dual cone C∗ of C has a weak∗ compact base. Letting e ∈ intC be given, then B∗ e : {ξ ∈ C∗ : ‖ξ‖ 1} is a weak∗ compact base of C∗. Let N λ0 ⊂ Λ and N μ0 ⊂ M be neighborhoods of considered points λ0 and μ0, respectively. LetK : X ×Λ ⇒ X be a set-valued mapping, and let F : X ×X ×M ⇒ Y be a setvalued mapping. For each λ ∈ N λ0 and μ ∈ N μ0 , consider the following parameterized generalized vector quasiequilibrium problem of finding x0 ∈ K x0, λ such that F ( x0, y, μ ) ⊂ Y \ − intC, ∀y ∈ K x0, λ . PGVQEP For each λ ∈ N λ0 and μ ∈ N μ0 , let E λ : {x ∈ X | x ∈ K x, λ }. 2.1 Let S λ, μ be the solution set of PGVQEP , that is, S ( λ, μ ) : { x ∈ E λ | Fx, y, μ ⊂ Y \ − intC, ∀y ∈ K x, λ . 2.2 For each ξ ∈ C∗ \ {0}, each λ ∈ N λ0 and μ ∈ N μ0 , let Sξ λ, μ denote the set of ξ-solution set to PGVQEP , that is, Sξ ( λ, μ ) : { x ∈ E λ : inf z∈F x,y,μ f z ≥ 0, ∀y ∈ K x, λ } . 2.3 Special Case i When K x, λ K λ , that is, K does not depend on x, the PGVQEP reduces to the parametric generalized vector equilibrium problem PGVEP considered by Li et al. 23 . ii If F : X × X × M → R, the PGVQEP collapses to the quasiequilibrium problem QEP considered by Anh and Khanh 26 . iii If K x, λ K λ and F is a vector-valued mapping, that is, F : X × X × M → Y , the PGVQEP reduce to the parametric Ky Fan inequality PKI considered by S. J. Li and X. B. Li 25 . Now we recall some basic definitions and their properties which are needed in this paper. Definition 2.1 classical notion . A set-valued mapping G : M ⇒ X is said to be · α-Hölder continuous at μ0 if there is a neighborhood U μ0 of μ0 such that, for all μ1, μ1 ∈ U μ0 , G ( μ1 ) ⊆ Gμ2 ) B ( 0, d ( μ1, μ2 )) , 2.4 where ≥ 0 and α > 0. 4 Abstract and Applied Analysis Definition 2.2. A set-valued mapping G : X × Λ ⇒ Y is said to be 1 · α1, 2 · α2 -Hölder continuous at x0, λ0 if and only if there exists neighborhoods N x0 of x0 and N μ0 of μ0 such that, for all x1, x2 ∈ N x0 , for all λ1, λ2 ∈ N λ0 , G x1, λ1 ⊆ G x2, λ2 1d1 x1, x2 2d2 λ1, λ2 B 0, 1 , 2.5 where 1, 2 ≥ 0 and α1, α2 > 0. Definition 2.3 see 25 . A set-valued mapping G : M ⇒ Y is said to be · α -Hölder continuous with respect to e ∈ intC at μ0 if and only if there exists neighborhoods N μ0 of μ0 such that, for all μ1, μ2 ∈ N μ0 , G ( μ1 ) ⊆ Gμ2 ) d ( μ1, μ2 ) −e, e , 2.6 where ≥ 0, α > 0 and −e, e {x : x ∈ e − C, x ∈ −e C}. Definition 2.4. Let F : X×X×Λ ⇒ Y be a set-valuedmapping with nonempty values; F x, ·, μ is called C-like convex on A λ if and only if for any x1, x2 ∈ X and any t ∈ 0, 1 , there exists x3 ∈ X such that tF x, x1, λ 1 − t F x, x2, λ ⊂ F x, x3, λ C. 2.7 Remark 2.5. If for each μ ∈ N μ0 and each x ∈ E N λ0 , F x, ·, μ is C-like convex on E N λ0 , then F x, E N λ0 , μ C is a convex set.