Openness stability and implicit multifunction theorems: Applications to variational systems

Abstract

In this paper we aim to present two general results regarding, on one hand, the openness stability of set-valued maps and, on the other hand, the metric regularity behavior of the implicit multifunction related to a generalized variational system. Then, these results are applied in order to obtain, in a natural way, and in a widely studied case, several relations between the metric regularity moduli of the field maps defining the variational system and the solution map. Our approach allows us to complete and extend several very recent results in literature.Keywords: set-valued mappings · linear openness · metric regularity · Lipschitz-like property · implicit multifunctions Mathematics Subject Classification (2010): 90C30 · 49J53 · 54C60 X, Y, P be Banach spaces, H : X × P ⇒ Y be a multifunction and define the implicit set-valued map S : P ⇒ X byThen, in this second part of our work we find how metric regularity and Lipschitz properties of H and S are related under certain assumptions (Theorem 3.6). This second main result is in fact a general implicit multifunction theorem. After fixing these two main tools we are able to present in a natural way the situation where H is a sum of two set-valued maps F, G of the formNote that this case is more general than the situation considered in [1] by the presence of the set-valued map F instead of a single-valued map. Moreover, this is the most general situation one can consider because it is not possible to get good results concerning the Lipschitz properties when G depends on the parameter p, in virtue of [1, Remark 3.6. (iii)]. However, note that in ([10]) the authors deal with a sort of metric regularity of the solution map associated to the sum of two parametric set-valued maps. In our framework, when we put at work together the two main results, we are in the position to indicate is a smooth manner the relations between the regularity moduli of S, G and F. We hope that our main results and their combination will bring more light on the previous results on this topic. We would like to mention that, in comparison with [1] and [2], we get here only results in which the assumptions are on F and G and the conclusion concerns S. The converse situation considered in the quoted works (from S to G) is not presented here because it has too many similarities with the corresponding results of Artacho and Mordukhovich. Any interested reader could find the arguments to obtain such results in our framework, but with Artacho and Mordukhovich tools.The paper is organized as follows. In the next section we present the notations, the concepts and the basic facts we use in the sequel. The third section contains the main results of the paper we have presented in few words above. The last section investigates the widely studied form of the parametric variational systems we can find in literature. We show here how the main results concerning the stability of the linear openness of set-valued maps and the implicit multifunctions could be combined in order to get quite easily the estimation ...

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