On the boundedness of solutions of perturbed linear systems

Abstract

Y’ = A (0 Y + W), W,) where x, y, g, h are n-vectors, A(t) is a continuous n X n matrix for t > 0, g(t, y) is continuous for 12 0, y E R”, and h(t) is continuous for t > 0. Strauss and Yorke [8,9] have studied the perturbing uniform asymptotically stable systems, and Furuno and Hara [4] have shown some more detailed results. Bernfeld [ 1 ] and Lovelady [6] have studied the perturbing uniformly bounded and uniformly ultimately bounded systems. On the other hand Coppel [2,3] has studied the boundedness of solutions of (PL,) from the point of view of the dichotomy theory. Lovelady [5] referred to the connections between the perturbation problem and the dichotomy theory. Here we shall give some further results on the boundedness of solutions of perturbed linear systems. This paper is much influenced by Strauss and Yorke [9]. The purpose of this paper is to prove theorems on the perturbation from (L) to (PL) and (PL,) of uniform boundedness (Theorem 3.1), uniform boundedness and ultimate boundedness (Theorem 4.1), and uniform boundedness and uniform ultimate boundedness (Theorem 5.1). Let G, be the class of functions g(t, y) such that I] g(t, r)l] 0 and II y]] > R, where j: y(t) dt r,, > 0 and