Linear differential-algebraic boundary value problem with singular pulse influence

Abstract

The study of differential-algebraic boundary value problems was initiated in the works of K. Weierstrass, N. N. Luzin and F. R. Gantmacher. Systematic study of differential-algebraic boundary value problems is devoted to the work of S. Campbell, Yu. E. Boyarintsev, V. F. Chistyakov, A. M. Samoilenko, M. O. Perestyuk, V. P. Yakovets, O. A. Boichuk, A. Ilchmann and T. Reis. The study of the differential-algebraic boundary value problems is associated with numerous applications of such problems in the theory of nonlinear oscillations, in mechanics, biology, radio engineering, theory of control, theory of motion stability. At the same time, the study of differential algebraic boundary value problems is closely related to the study of pulse boundary value problems for differential equations, initiated M. O. Bogolybov, A. D. Myshkis, A. M. Samoilenko, M. O. Perestyk and O. A. Boichuk. Consequently, the actual problem is the transfer of the results obtained in the articles by S. Campbell, A. M. Samoilenko, M. O. Perestyuk and O. A. Boichuk on a pulse linear boundary value problems for differential-algebraic equations, in particular finding the necessary and sufficient conditions for the existence of the desired solutions, and also the construction of the Green’s operator of the Cauchy problem and the generalized Green operator of a pulse linear boundary value problem for a differential-algebraic equation. In this article we found the conditions of the existence and constructive scheme for finding the solutions of the linear Noetherian differential-algebraic boundary value problem for a differential-algebraic equation with singular impulse action. The proposed scheme of the research of the linear differential-algebraic boundary value problem for a differential-algebraic equation with impulse action in the critical case in this article can be transferred to the linear differential-algebraic boundary value problem for a differential-algebraic equation with singular impulse action. The above scheme of the analysis of the seminonlinear differential-algebraic boundary value problems with impulse action generalizes the results of S. Campbell, A. M. Samoilenko, M. O. Perestyuk and O. A. Boichuk and can be used for proving the solvability and constructing solutions of weakly nonlinear boundary value problems with singular impulse action in the critical and noncritical cases.

Solvability conditions for nonlinear matrix equations

“…To find approximations for the solutions of nonlinear matrix equations in the case of unknown square matrix, Newton's method is applicable [8,10]. To find approximations for the solutions of nonlinear matrix equations in the case of unknown rectangular matrix, we apply the Newton-Kantorovich method [12]. Let f (z) denote a vector function…”