Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)8249Research ArticleOn the Cauchy Problem for a Semilinear Functional Differential Inclusion of the Fractional Order with Impulse Response and Infinite Delayina Banach SpacePetrosyanG GDepartment of Higher Mathematicsgarikpetrosyan@yandex.ruVoronezh State Pedagogical University15012014152208092016Copyright © 2014,2014In 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 derivativedifferential inclusionsCauchy problemmeasure of noncompactnessfixed pointcondensing multimapthe impulsive characteristicsфункционально-дифференциальное включениедробная производнаязадача Кошибесконечное запаздываниеимпульсная характеристикамера некомпактностинеподвижная точкауплотняющее мультиотображение