Periodic solutions of linear degenerate systems of ordinary differential equations of the second order

Abstract

We propose sufficient conditions for the existence of a periodic solution of a system of linear ordinary differential equations of the second order with a degenerate symmetric matrix in the coefficient of the second-order derivative in the case of an arbitrary periodic inhomogeneity.1. We investigate the system of differential equationswhere the square matrices A; B; and C and the n-dimensional vector f are periodic functions with period 2 ; A 0 .t / Á A.t /; the prime means the transposition of a matrix, the dot denotes differentiation with respect to the independent variable t; the matrix A.t / is degenerate (moreover, it is possible that rank A.t/ 6 Á const/; and " is a small parameter.We consider the problem of the existence of a smooth periodic solution of system (1) for an arbitrary inhomogeneity f .t /:There are many works (see, e.g., [1-6]) devoted to the development of methods for the construction of solutions of degenerate differential systems and the investigation of their qualitative behavior. System (1) was studied in [1] under certain assumptions, one of which is the constancy of the rank of the matrix A.t/: In [2], sufficient conditions for the existence of a periodic solution of the scalar equation (1) were obtained on the basis of investigation of the degenerate Riccati equation. In the present paper, we generalize these results.Let C r .T 1 / be the space of vector or matrix functions that take real values and are periodic with period 2 and continuous together with all derivatives up to the order r inclusive. Let H r .T 1 / denote the space of functions square integrable on T 1 D OE0I 2 together with all generalized derivatives up to the order r inclusive. Assume that . ; / r is the scalar product in H r .T 1 /;h ; i is the scalar product in R n ;jˆ.t /j r D max t 2T 1 ; 0Ä Är kˆ. / .t/k; and k k is the Euclidean vector norm or a consistent matrix norm.