Abstract
Painleve-Kuratowski convergence of the solution sets is investigated for the perturbed set-valued weak vector variational inequalities with a sequence of mappings converging continuously. The closedness and Painleve-Kuratowski upper convergence of the solution sets are obtained. We also obtain Painleve-Kuratowski upper convergence when the sequence of mappings converges graphically. By virtue of a sequence of gap functions and a key assumption, Painleve-Kuratowski lower convergence of the solution sets is established. Some examples are given for the illustration of our results.
Highlights
Since the concept of vector variational inequality VVI was introduced by Giannessi 1 in 1980, many important results on various kinds of vector variational inequality problems have been established, such as existence of solutions, relations with vector optimization, stability of solution set maps, gap function, and duality theories see, e.g., 2–8 and the references cited therein
The stability analysis of the solution set maps for the parametric VVI problem is of considerable interest amongst researchers in the area
Some results on the semicontinuity of the solution set maps for the parametric VVI problem with the parameter perturbed in the space of parameters are available in the literature
Summary
Since the concept of vector variational inequality VVI was introduced by Giannessi 1 in 1980, many important results on various kinds of vector variational inequality problems have been established, such as existence of solutions, relations with vector optimization, stability of solution set maps, gap function, and duality theories see, e.g., 2–8 and the references cited therein. Almost no stability results are available for the perturbed VVI problem with the sequence of mappings converging continuously or graphically It appears that the only relevant paper is 10 , where Lignola and Morgan considered generalized variational inequality in a reflexive Banach space with a sequence of operators converging continuously and graphically and obtained the convergence of the solution sets under an assumption of pseudomonotonicity. We should establish Painleve-Kuratowski upper and lower convergences of the solution sets of the perturbed set-valued weak variational inequity SWVVI with a sequence of converging mappings in a Banach space.
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