Stonyakin F. Compact properties for mappings and their applications to Bochner integral in locally convex spaces.

In the thesis the classical Radon-Nikodym problem for Bochner integral is solved on the base of new compact properties for mappings taking values in locally convex spaces (LCS). Object of researches: mappings taking values in separable LCS. Purpose of researches: to construct of developed theory for new compact properties LCS-valued mappings: compact subdifferential, compact variation, strong compact variation, strong compact absolute continuity and to solve Radon-Nikodym problem for Bochner integral on the base of these properties. Methods of researches: nonsmooth analysis, infinite-dimensional differential calculus, measure and integration theory, operator theory. Theoretical and practical results: conditions of representability for mappings as indefinite Bochner integrals are obtained in terms of compact subdifferential and compact variation, a validity of Radon-Nikodym property limit form is proved for each Frechet space. Novelty: all results are new. Degree of introduction, sphere of the use: results have theoretical character, can be used to research of some problems in operator theory and mathematical analysis in LCS.