Regular boundary-value problems in a band

Abstract

It has been established in the fifties that the classes of uniqueness of solution of Cauchy problems for linear differential equations (and systems) with constant coefficients and the classes of correct solvability of such problems are determined by entirely different characteristics of the equation: for the conditions of the uniqueness of the solution only the order of the equation responsible [i], while the classes of the correct solvability of the problem depends on the algebraic properties of the corresponding differential expression [2]. It turns out that in a series of other problems the situation is different: the uniqueness of the solution of the problem in the class of bounded (smooth) functions implies its correct solvability in this class. Such problem will be said to be regular. The purpose of this paper is the investigation of regularity conditions for two types of problems: Cauchy problems in a strip for a linear differential equation with loads along the time coordinate and a nonlocal multipoint problem in a strip for a linear differential equation with constant coefficients.