On the construction of asymptotic solutions of two-point boundary-value problems for degenerate singularly perturbed systems of differential equations

Abstract

We consider the problem of the existence of a solution of a two-point boundary-value problem for degenerate singularly perturbed linear systems of differential equations. We obtain asymptotic formulas for this solution.Consider the boundary-value problemwhere x.t; "/ is the required n-dimensional vector, t 2 OE0I 1; " 2 .0I " 0 / is a small parameter, h 2 N; A.t; "/ and B.t / are n n matrices, d is a p-dimensional column vector, f .t; "/ is an n-dimensional column vector, and M and N are p n matrices with constant elements. Assume that the following conditions are satisfied:(i) on the segment OE0I 1; the matrix A.t; "/ and vector f .t; "/ admit uniform asymptotic expansions in powers of the small parameter "; i.e.,(ii) the coefficients A k .t / and f k .t / of expansions (3) and the matrix B.t/ are infinitely differentiable on the segment OE0I 1I(iii) det B.t / D 0 8t 2 OE0I 1:Boundary-value problems of the type (1), (2) were investigated in [1][2][3][4][5] in the case where B.t/ D E; where E is the identity matrix. In [1,2], for the construction of asymptotic solutions of these problems, the method of regularization was used. In [3,4], the method of boundary functions was used for this purpose. In [5], the asymptotic behavior of a solution of a boundary-value problem was described using the idea of reduction of a linear system to an almost diagonal form. In [6], for the solution of this problem, the system was reduced to a system with a singular matrix coefficient of the derivative. In [7], a method was proposed for the asymptotic Nizhyn University, Nizhyn, Ukraine.

Molecular decomposition and a class of Fourier multipliers for bi-parameter modulation spaces

“…The aim of this section is to prove Theorem 1.3. We follow the idea of Kobayshi and Sawano [33]. For our purpose, we will need the following lemma, which is a bi-parameter analogue of the Theorem in […”

“…Modulation spaces, introduced by Feichtinger [15] in the early 1980s, were originally used to measure smoothness of functions and to analyze local properties of frequency space. During the last decades, modulation spaces have been studied systematically; see, for instance, [2,16,17,19,20,33,35,49] and the references therein. These spaces have turned out to be useful in the study of pseudo-differential operators, partial differential equations, signal analysis and quantum mechanics.…”

“…One main objective of the present paper is to derive a molecular decomposition for the bi-parameter modulation spaces BM s p,q (R n × R m ). We point out that various definitions of molecules of (single-parameter) modulation spaces has been introduced in [1,21,33]. Our definition of molecules for bi-parameter modulation spaces is inspired by [33].…”

The boundary-value problem for the linear degenerated singularly perturbed system of differential equations of the second order

“…Крайова задача для лiнiйної системи диференцiальних рiвнянь першого порядку дослiджувалася автором в роботах [1][2][3][4]. При цьому було використано метод асимптотичного iнтегрування вироджених сингулярно збурених лiнiйних систем, розроблений в [5].…”