In this paper, we introduce mixed Pareto quasi-optimization problems and show some sufficient conditions on the existence of their solutions. As special cases, we obtain several results for the mixed Pareto quasi-equilibrium problem and also system of two Pareto quasioptimization problems. | ON THE EXISTENCE OF SOLUTIONS TO MIXED PARETO QUASI-OPTIMIZATION PROBLEMS Nguyen Thi Quynh Anh 0 University of Infomation Techlonogy & Communication, Thai Nguyen University Abstract. In this paper, we introduce mixed Pareto quasi-optimization problems and show some sufficient conditions on the existence of their solutions. As special cases, we obtain several results for the mixed Pareto quasi-equilibrium problem and also system of two Pareto quasioptimization problems. Key Words. Mixed Pareto quasi-optimization problems, C-convex, C-quasiconvex-like mappings, C- continuous mappings, diagonally C2 -convex, diagonally C2 -quasi-convex-like mappings. 1 Introduction Throughout this paper, unless otherwise specify, we denote by X, Y, Y1 , Y2 , Z real locally convex Hausdorff topological vector spaces. Assume that D ⊂ X, K ⊂ Z are nonempty subsets. and Ci ⊆ Yi , i = 1, 2, are convex closed cones. 2A denotes the collection of all subsets in the set A. Given multivalued mappings S : D × K → 2D , T : D × K → 2K ; P : D → 2D , Q : K × D → 2K and single-valued mappings F1 : K × K × D → Y1 , F2 : K × D × D → Y2 , we consider the following problem: Mixed Pareto quasi-optimization problems Find (¯ x, y¯) ∈ D × K x ¯ ∈ S(¯ x, y¯); y¯ ∈ T (¯ x, y¯) such that there are no v ∈ T (¯ x, y¯), v 6= y¯, t ∈ P (¯ x), y ∈ Q(¯ x, t), t 6= x ¯ with F1 (¯ y , y¯, x ¯) C1 F1 (¯ y , v, x ¯); F2 (y, x ¯, x ¯) C2 F2 (y, x ¯, t). Where a C b means that a − b ∈ C. The multivalued mapping Q(x, .) : D → 2K , GrQ(¯ x, .) = {(t, y) ∈ D × K|y ∈ Q(¯ x, t)} Setting (GrQ(¯ x, .)) ∩ (P (¯ x) × K) = A × B, the above problem becomes to find (¯ x, y¯) ∈ D × K such that x ¯ ∈ S(¯ x, y¯); y¯ ∈ T (¯ x, y¯); F1 (¯ y , y¯, x ¯) ∈ P M in(F1 (¯ y , T (¯ x, y¯), x ¯)|C1 ); F2 (y, x ¯, x ¯) ∈ P M in(F2 (A, x ¯, B)|C2 ), with P M in(A|C) = {x ∈ A| there are no y ∈ A, y 6= x such that x C y} is the set of Pareto efficient points of A to C . The purpose of this paper is to study the existence for solutions of .
