Lower convergence of approximate solutions to vector quasi-variational problems

In this article, various types of approximate solutions for vector quasivariational problems in Banach spaces are introduced. Motivated by [M.B. Lignola, J. Morgan, On convergence results for weak efficiency in vector optimization problems with equilibrium constraints, J. Optim. Theor. Appl. 133 (2007), pp. 117–121] and in line with the results obtained in optimization, game theory and scalar variational inequalities, our aim is to investigate lower convergence properties (in the sense of Painleve´– Kuratowski) for such approximate solution sets in the presence of perturbations on the data. Sufficient conditions are obtained for the lower convergence of ‘strict approximate’ solution sets but counterexamples show that, in general, the other types of solutions do not lower converge. Moreover, we prove that any exact solution to the limit problem can be obtained as the limit of a sequence of approximate solutions to the perturbed problems.