On the Cauchy Problem for a Semilinear Functional Differential Inclusion of the Fractional Order with Impulse Response and Infinite Delayina Banach Space

In this paper, applying the theory of topological degree of condensing multi-valued mappings, we prove the existence of solution and the compactness of the set of solutions of the Cauchy problem for a semilinear functional differential inclusion of fractional order with infinite delay and impulse responses in a Banach space. The article consists of an introduction and three sections. In the introduction the urgency of this problem, outlines the background and provides links to articles and monographs in which the reader can find the applications of the theory of functional differential equations and inclusions of fractional order. In the second section we describe the formulation of the problem, we introduce the space, which addresses this problem and give a criterion for the relative compactness of the set in the input space. The third section consists of four sub-items, which provide preliminary information. In the first subparagraph the concept of fractional derivative and fractional primitive is given. Second paragraph provides the necessary information from the theory of multi-valued mappings. The third sub-paragraph is devoted to information from the theory of measurable multifunctions. In the fourth paragraph we formulate a modified phase space entered by Hale and Kato. In the last section we formulate conditions that we impose on the elements included in the original inclusion and on the basis of auxiliary statements prove our main result.

fractional derivative, differential inclusions, Cauchy problem, measure of noncompactness, fixed point, condensing multimap, the impulsive characteristics