О разрешимости каузальных функциональных включений с бесконечным запаздыванием

Abstract

In the present article we develop the results of works devoted to the study of problems for functional differential equations and inclusions with causal operators, in case of infinite delay. In the introduction of the article we substantiates the relevance of the research topic and provides links to relevant works A. N. Tikhonov, C. Corduneanu, A. I. Bulgakov, E. S. Zhukovskii, V. V. Obukhovskii and P. Zecca. In section two we present the necessary information from the theory of condensing multivalued maps and measures of noncompactness, also introduced the concept of a multivalued causal operator with infinite delay and illustrated it by examples. In the next section we formulate the Cauchy problem for functional inclusion, containing the composition of multivalued and single-valued causal operators; we study the properties of the multiopera-tor whose fixed points are solutions of the problem. In particular, sufficient conditions under which this multioperator is condensing on the respective measures of noncompacness. On this basis, in section four we prove local and global results and continuous dependence of the solution set on initial data. Next the case of inclusions with lower semicontinuous causal multioperators is considered. In the last section we generalize some results for semilinear differential inclusions and Volterra integrodifferential inclusions with infinite delay.