Vol 24, No 127 (2019)

We consider functional-differential equation x(gt = f t; xht , t ∈ 0; 1 , where function f satisfies the Caratheodory conditions, but not necessarily guarantee the boundedness of the respective superposition operator from the space of the essentially bounded functions into the space of integrable functions. As a result, we cannot apply the standard analysis methods (in particular the fixed point theorems) to the integral equivalent of the respective Cauchy problem. Instead, to study the solvability of such integral equation we use the approach based not on the fixed point theorems but on the results received in [A.V. Arutyunov, E.S. Zhukovskiy, S.E. Zhukovskiy. Coincidence points principle for mappings in partially ordered spaces // Topology and its Applications, 2015, v. 179, № 1, 13-33] on the coincidence points of mappings in partially ordered spaces. As a result, we receive the conditions on the existence and estimates of the solutions of the Cauchy problem for the corresponding functional-differential equation similar to the well-known Chaplygin theorem. The main assumptions in the proof of this result are the non-decreasing function f(t; ·) and the existence of two absolutely continuous functions v, w, that for almost each t ∈ [0; 1] satisfy the inequalities vgt ≥ f t; vht , wgt ≤ f t;wht . The main result is illustrated by an example.

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.

Operator functions eA , sin B , cos B of the operator argument from the Banach algebra of bounded linear operators acting from E to E are considered in the Banach space E . For trigonometric operator functions sin B , cos B , formulas for the sine and cosine of the sum of the arguments are derived that are similar to the scalar case. In the proof of these formulas, the composition of ranges with operator terms in the form of Cauchy is used. The basic operator trigonometric identity is given. For a complex operator exponential function eZ of an operator argument Z from the Banach algebra of complex operators, using the formulas for the cosine and sine of the sum, the main property of the exponential function is proved. Operator functions eAt , sin Bt , cos Bt , eZt of a real argument t∈(-∞;∞) are considered. The facts stated for the operator functions of the operator argument are transferred to these functions. In particular, the group property of the operator exponent eZt is given. The rule of differentiation of the function eZt is indicated. It is noted that the operator functions of the real argument t listed above are used in constructing a general solution of a linear n th order differential equation with constant bounded operator coefficients in a Banach space.