Linear Differential Equations with Almost Periodic Coefficients

Abstract

Introduction. We shall be interested in linear systems of the form x' A (t)x + B (t), where A (t) is an nXn matrix, B (t) is an n vector, and their entries are almost periodic functions (either real or complex valued). Historically, the approach to this type of problem has been to assume that the solutions of the system x' = A (t) x have certain properties and then obtain the existence of an almost periodic solution of x' = A (t)x + B (t). The limitations of such a procedure are apparent. While the results obtained in this paper are restricted to rather special types of almost periodic matrices, they do possess the advantage that the restrictions are placed directly upon the coefficients. For the case B-=0, we shall concern ourselves with a modified form of the represen-tation problem considered by Cameron [4]. Our results divide into two main types, the first type being when A (t) is a superdiagonal matrix. An example (example A) is obtained which illustrates the difficulties involved and suggests the restrictions imposed in Theorem 1. The second type of matrix A(t) considered is that in which the frequencies of the aij(t) are all positive and bounded away from zero. The results in this case are an extension of earlier results due to Wintner and Putnam [12]. For the case B = 0, we consider the existence of almost periodic solutions. The main result is found in Theorem 3. We shall employ the notation used by Besicovitich [1], writing a. p. for almost periodic and denoting the unique association of an a. p. function with its Fourier series by -.