On the Existence of Solutions of Symmetric Vector Equilibrium Problems via Nonlinear Scalarization

The connectedness of weakly and strongly efficient solution sets of nonconvex vector equilibrium problems

“…In the theorem above, we use the pseudo triangle inequality property to establish the existence conditions for weakly efficient solutions of the vector equilibrium problems without assuming any convexity conditions. Hence, our result is different from many existing ones in the literature (see, e.g., [6,7,8,25]). Moreover, by weakening the triangle inequality condition, Theorem 4.1 also improves the main results obtained in [2,3,4].…”

“…Thus, the assumptions of Theorem 4.1 hold true, and so WEff(X , f ) is nonempty and compact (in the fact that WEff(X , f ) = {−1}). However, f is neither concave in the first component nor convex in the second component on X , the results of [6,7,8,25] do not work. Furthermore, f does not hold the triangle inequality condition defined by (4.1), due to…”

A Note on the Generalized Nonlinear Vector Variational-Like Inequality Problem

“…In [15], the concepts of transfer closed and intersectionally closed are used along with an assumption milder than C-pseudomonotonicity on the set-valued mappings. On the other hand, relaxed monotonicity and relaxed Lipschitz's continuity are used in [13] and generalized C-quasi-convexity is used in [17] for the setvalued mappings.…”

“…In the same year, Tavakoli et al studied the C-pseudomonotone property for the set-valued mappings in order to solve a generalized variational inequality problems [15]. On the other hand, vector equilibrium problems for the set-valued mappings were studied by Farajzadeh et al and Chen et al during this period [16,17]. This wide range of literature is a clear indication of the importance that variational inequality problems have gained in the recent years.…”

Stability on parametric strong symmetric quasi-equilibrium problems via nonlinear scalarization

“…Thus the study of the problem (SQEP) is interesting and meaningful. In recent years, the research on symmetric equilibrium problem mainly aims to the following two aspects: a) the existence of the solutions (see, e.g., [14,15,16,17,18] and the references therein); b) the stability of solutions, including well-posedness, connectedness, and Berge-semicontinuity (see, e.g., [6,19,20,21,22] and the reference therein). It is noticeable that there are few results concerning with the stability of the solution mapping to (SQEP) with the parameter perturbation.…”