On the existence and stability of approximate solutions of perturbed vector equilibrium problems

Abstract

In this paper we consider several concepts of approximate minima of a set in normed vector spaces and we provide some results concerning the stability of these minima under perturbation of the underlying set with a sequence of sets converging in the sense of Painlevé-Kuratowski to the initial set. Then, we introduce the concept of approximate solution for equilibrium problem governed by set-valued maps and we study the stability of these solutions. The particular case of linear continuous operators is considered as well.

Painlevé-Kuratowski Convergences of the solution sets to perturbed generalized systems

“…It appears that the only relevant paper is [7], where Durea considered the vector equilibrium problems with the perturbations of the multifunction and obtained the Painlevé-Kuratowski upper convergence of the solution sets. Since the perturbed vector equilibrium problem with a sequence of mappings converging is different from the parametric vector equilibrium problem with the parameter perturbed in a space of parameters, it is important to study the Painlevé-Kuratowski Convergence of the sequence of the solution sets.…”

Existence conditions for generalized vector variational inequalities

“…In the latter problem (the weak variant of the first one) the set Y \ − int K is closed and this fact is essential for proving existence results of solutions for this problem: see, for instance, Durea (2007), among others. On the other hand, int K = ∅ also ensures the possibility to apply powerful nonsmooth separation techniques in order to get various kinds of results.…”

Stability of Efficient Solutions for Semi-infinite Vector Optimization Problems

“…Now, we recall the well known notion of set-convergence, namely Painlevé-Kuratowski set-convergence. A sequence of sets {B n ⊂ X : n ∈ N} is said to converge in the sense (see also [8,22]) of Painlevé-…”

“…There are some stability results for vector optimization problems and related issues with a sequence of sets converging in the sense of Painlevé-Kuratowski. Examples of fresh literatures include, for vector optimization problems, we can see Attouch and Riahi [2], Huang [12], Lucchetti and Miglierina [18], Lalitha and Chatterjee [14]; for vector equilibrium problem, we can refer to Durea [8], Fang and Li [10], Zhao et al [23], Peng and Yang [21], etc. However, to the best of our knowledge, the Painlevé-Kuratowski stability of efficient solutions set for semi-infinite vector optimization problems has not been found.…”