Calculus Rules For Directional Versions Of Normal Cone, Coderivative And Sequential Normal Compactness In Variational Analysis

This dissertation mainly devoted to generalized differential theory in variational anal-ysis, including establishing calculus rules of directional versions of normal cones, coderiva-tives and subdifferentials in general Banach space, especially in Asplund space, and then developing calculus of the directional sequential normal compactness in infinite dimen-sional space.The whole dissertation is divided into five chapters. The first chapter, introduction part, mainly introduces the research background of variational analysis and the research status and progress of the generalized differential theory, which involves the normal cones of sets, the coderivatives of set-valued mappings and the sequential normal compactness in infinite dimensional spaces. The second chapter provides preliminary knowledge for the study of generalized differential theory, including a review of basic definitions and results on the normals cone of sets, the coderivative of set-valued mappings structure, the subdifferentials of generalized real-valued function. Combine with the tangential ap-proximation in the original space, we define directional versions of differentiability and Lipschitz property of mappings between Banach spaces and then discuss their relation-ships, which generalizes the well-known differentiability and Lipschitz property in Banach space.The main contents of this paper include following three aspects:First, it mainly studies the directional Mordukhovich normal cone of sets. First, we introduce the directional Mordukhovich normal cone of sets. Second, we present the basic properties of directional Mordukhovich normal cone. In particular, we studies the characterization of the directional Mordukhovich normal cone of sets with separated structure. These results enrich the connotation of generalized differential theory, broaden the scope of application of these structures.Second, it concerns the directional versions of various coderivatives of set-valued map-pings and the directional Mordukhovich subdifferential of single-valued mapping, where the direction in the original space are embedded. First, we consider various kinds of direc-tional coderivative structure of set-valued mappings and then present the corresponding basic properties. Second, we establish some basic calculus of the directional coderiva-tives of set-valued mappings between Banach spaces, including sum rule and composition rule. Third, we introduce directional versions of Mordukhovich subdifferential and sin-gular subdifferentials present their basic properties and structural characterization, and then establish their complete calculus rule. Finally, as applications, we mainly study the characterization and the upper bound estimation of the directional Mordukhovich subdifferentials of marginal functions.Third, it is devoted to the calculus rules of directional forms of sequential normal compactness and the coderivative of set-valued mappings. Firstly, we consider some basic calculus rules of directional forms of sequential normal compactness of set-valued mappings and sets in Banach spaces. Secondly, we establish complete calculus rules for directional coderivative of set-valued maps in Asplund space. Finally, we present more complete calculus rules for directional sequential normal compactness of set-valued mappings in Asplund space. These results are the cornerstone of the whole theory of generalized differential, which is vital for applications in infinite dimensional space.