Nonsingular Integro-Differential Boundary Value Problem Not Solved With Respect to the Derivative

Abstract

The study of the differential-algebraic boundary value problems was established in the papers of K. Weierstrass, M.M. Lusin and F.R. Gantmacher. Works of S. Campbell, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, M.O. Perestyuk, V.P. Yakovets, O.A. Boi- chuk, A. Ilchmann and T. Reis are devoted to the systematic study of differential-algebraic boundary value problems. At the same time, the study of differential-algebraic boundary-value problems is closely related to the study of linear boundary-value problems for ordinary di- fferential equations, initiated in the works of A. Poincare, A.M. Lyapunov, M.M. Krylov, N.N. Bogolyubov, I.G. Malkin, A.D. Myshkis, E.A. Grebenikov, Yu.A. Ryabov, Yu.A. Mitropolsky, I.T. Kiguradze, A.M. Samoilenko, M.O. Perestyuk and O.A. Boichuk. The study of the linear differential-algebraic boundary value problems is connected with numerous applications of corresponding mathematical models in the theory of nonlinear osci- llations, mechanics, biology, radio engineering, the theory of the motion stability. Thus, the actual problem is the transfer of the results obtained in the articles and monographs of S. Campbell, A.M. Samoilenko and O.A. Boichuk on the linear boundary value problems for the integro-differential boundary value problem not solved with respect to the derivative, in parti- cular, finding the necessary and sufficient conditions of the existence of the desired solutions of the linear integro-differential boundary value problem not solved with respect to the derivative. In this article we found the conditions of the existence and constructive scheme for finding the solutions of the linear Noetherian integro-differential boundary value problem not solved with respect to the derivative. The proposed scheme of the research of the nonlinear Noetherian integro-differential boundary value problem not solved with respect to the derivative in the critical case in this article can be transferred to the seminonlinear integro-differential boundary value problem not solved with respect to the derivative.

Nonlinear Integrodifferential Boundary-Value Problems Unsolvable with Respect to the Derivative

“…In the posed problem, we generalize the investigations of linear integrodifferential boundary-value problem [2,3] to the case of the nonlinear boundary-value problem (1), (2) with rectangular matrix of the derivative [4][5][6].…”

“…Theorem . Suppose that, for the generating boundary-value problem (3) with a rectangular matrix (or a square degenerate matrix of constant rank) A.t /; the conditions of the theorem in [6, p. 15] are satisfied. Under these conditions with degeneration of order p; in the critical case P Q ⇤ 6 D0; for any fixed continuous vector function…”

“…Corollary . Suppose that, for the generating boundary-value problem (3) with a rectangular matrix (or a degenerate square matrix of constant rank) A.t /; the conditions of the theorem in [6, p. 15] 1), (2) unsolved with respect to the derivative is determined by the vector L c satisfying Eq. ( 7).…”