FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2006, VOLUME 12, NUMBER 5, PAGES 29-47

Bounded solutions of families of systems of differential equations and their approximation

D. S. Dzhumabaev

Abstract

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In the paper, we consider the problem of finding a bounded solution of a one-parametric family of systems of ordinary differential equations. Using the parametrization method, we prove necessary and sufficient conditions for the existence of a unique solution of the problem considered that is bounded on the whole axis in terms of a two-sided infinite block-band matrix composed with respect to integrals over intervals of length $h\; >\; 0$ the matrix of the system of differential equations. Also, we construct a family of two-point boundary-value problems on a finite interval that approximate the problem of finding the bounded solution and finds an interconnection between the correct solvability of the initial and approximating problems.

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Last modified: February 21, 2007