Existence Results for Systems of Quasi-Variational Relations

Abstract

The existence of solutions for a system of variational relations, in a general form, is studied using a fixed point result for contractions in metric spaces. As a particular case, sufficient conditions for the existence of solutions of a system of quasi-equilibrium problems are given.

On interval-valued vector variational-like inequalities and vector optimization problems with generalized approximate invexity via convexificators

“…Introduction. Giannessi [5] first proposed the idea of vector variational inequality in 1980,containing broader applicability in optimization, adaptive control, finance, and stability problems, see, for example, [3,8,11] and the references cited therein. In optimization theory, nonsmooth phenomenon frequently occur, which has prompted the development of several subdifferential and generalized directional derivative notions.…”

“…From (8) and using the fact that ∂ * * (−Γ)(ω) = −∂ * * (Γ)(ω), we have ξ L k , η(ω, υ) = −ξ L k , η(υ, ω) ≤ −e k ∥η(ω, υ)∥, for every ξ L k ∈ ∂ * * Γ L k (ω), ξ U k , η(ω, υ) = −ξ U k , η(υ, ω) ≤ −e k ∥η(ω, υ)∥, for every ξ U k ∈ ∂ * * Γ U k (ω). Thus there exist ρ > 0 such that ξ L , η(ω, υ) p ≦ −e∥η(ω, υ)∥, for every…”