Abstract
Two kinds of parametric set-valued vector quasi-equilibrium problems are introduced. The existence of solutions to these problems is studied. The upper and lower semicontinuities of their solution maps with respect to the parameters are investigated.
Highlights
Introduction and PreliminariesEquilibrium problems are a class of general problems that contains many other problems, such as optimization problems, variational inequality problems, saddle point problems, and complementarity problems, as special cases
By the lower semicontinuity of F, there exists fβ ∈ F xβ, yβ, zβ for each β such that fβ → f0, which together with the closedness of W · and 3.2 implies that f0 ∈ Y \ − int C x0
By using a similar reasoning as in part one of the proof of Theorem 3.1, we can conclude that there exists a net { xβ, yβ, zβ } such that xβ, yβ, zβ → x0, y0, z0 and
Summary
Equilibrium problems are a class of general problems that contains many other problems, such as optimization problems, variational inequality problems, saddle point problems, and complementarity problems, as special cases. Let K : A × M → 2X, T : A × Λ → 2Y , F : A × X × Y → 2Y , and C : A → 2Y be set-valued mappings such that A ∩ K x, μ / ∅ for all x ∈ A and μ ∈ M and C x be a closed convex pointed cone of Y with int C x / ∅ for each x ∈ A. F is said to be Y \ − int C quasi convex-like of type 2 with respect to T see 1 if for any nonempty finite subset {y1, . A set-valued mapping G : B → 2B is said to be a KKM mapping if for each nonempty finite subset {x1, .
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