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This paper is mainly devoted to applications of modern tools of variational analysis and generalized differentiation to the metric regularity of parametric generalized inequality systems in infinite-dimensional spaces. The basic tools of our analysis involve the Mordukhovich normal coderivatives of set-valued mappings, the limiting subgradient estimate for the marginal functions, and the Ekeland variational principle. Using these tools, we establish new sufficient conditions for the metric regularity of parametric generalized inequality systems. Our results extend the corresponding results in [23] which established some pointbased sufficient conditions for the metric regularity in the Robinson’s sense of implicit multifunctions in finite-dimensional setting.

We study the controllability problem for a system governed by a degenerate semilinear functional differential inclusion in a Banach space with infinite delay. Notice that we are not assuming that the generalized semigroup generated by the linear part of inclusion is compact. Instead we suppose that the multivalued nonlinearity satisfies the regularity condition expressed in terms of the Hausdorff measure of noncompactness. It allows to obtain the general controllability principle in the terms of the topological degree theory for condensing multivalued operators. Two realizations of this principle are considered.