Розв’язки слабозбурених крайових задач та умови їх виникнення (випадок k=-1)

Анотація

SOLUTIONS OF WEAKLY PERTURBED BOUNDARY PROBLEMS AND TERMS OF OCCURRENCE OF SOLUTIONS (CASE )

R. Ovchar

A mathematical description of many problems of natural science is reduced to boundary value problems for systems of ordinary differential equations with impulse action, the order of which in general does not coincide with the number of m boundary conditions. The boundary-value problems with impulse action were studied earlier in the assumption of  by such well-known scientists as A. Mishkis,    A. Samoilenko, M. Perestyuk, M. Ronto, A. Anokhin,   S. Gurgula,  A. Halanay,     D. Wexler, with the most fully studied periodic boundary value problems for systems of ordinary differential equations with impulse action in a noncritical case                (A. Samoilenko, M. Perestyuk), as well as periodic boundary value problems in a critical case (O. Boychuk, M. Perestyuk, A. Samoilenko). In the article we consider the most general, incomplete, undefined and redefined boundary value problems with impulse action, in which boundary conditions are given by a linear or weakly nonlinear vector function, whose number m components, in the general case, do not coincide with the order of n differential systems.

The purpose of the paper is to find the necessary and sufficient conditions for the existence of solutions of linear and weakly nonlinear boundary value problems for systems of ordinary differential equations with impulse action, as well as the construction of convergent iterative algorithms for finding solutions to such problems.

In the article the effective methods of perturbation theory, asymptotic methods of nonlinear mechanics, are developed in the works of  M. Krylov, M. Bogolyubov,     Y. Mitropolsky and A. Samoilenko, the apparatus of generalized Green’s operators of linear semiautonomous boundary value problems, which was studied by O. Boychuk.

The article deals with the following basic provisions that determine the scientific novelty of the results of the article:

-         obtained effective coefficient conditions for the occurrence of solutions of linear inhomogeneous pulsed boundary value problems with small perturbations in the case when generating a boundary value problem with impulse action does not have solutions for arbitrary right parts;

-         an equation is obtained for generating weakly nonlinear amplitudes of critical boundary value problems with impulse action, which determines the necessary condition for the existence of solutions to such problems;

-         obtained effective sufficient conditions for the existence of solutions of weakly nonlinear critical and noncritical boundary value problems with impulse action; convergent iterative algorithms for their construction are proposed;

-         the general scheme of research of linear and weakly nonlinear boundary value problems with impulsive action is proposed.

The article is theoretical, generalizes and deepens the previously known results from linear and weakly nonlinear periodic boundary value problems in the general case when boundary conditions are given by a weakly linear vector functionality, and the number of  boundary conditions does not coincide with the order of the differential system. The practical sense of the paper is due to the fact that the question of the existence and construction of solutions of boundary value problems with impulse action occupies an important place in the qualitative theory of differential equations; wide application of the theory of boundary value problems with impulse action in various physical nature and functional assignments of technical problems; constructivism of the proposed coefficient conditions for the existence of solutions and algorithms for their construction.

inimum (maximum) will be topologically equivalent to a cone. It is clear that in the case when the graph of a continuous function z = f (x, y) is a cone over a curve that bounds a starlike region, then z = f (x, y) is topologically equivalent to a smooth function z = x²+y².