An Integrodifferential Equation
T. A. Burton · Proceedings of the American Mathematical Society · 1980
The vector equation x'(t) = A(i)x(t) + ('C(t, s)D(x(s))x(s) ds + F(t) is considered in which A is not necessarily a stable matrix, but A(f) + G{t, t)D(0) is stable where G is an antiderivative of C with respect to t. Stability and boundedness results are then obtained. We also point out that boundedness results of Levin for the scalar equation u'(t) = - f'0 a(t-s)g(u(s)) ds can be extended to a vector system x'(i) = - /{, H(t, s)x{s) ds. 1. Introduction. We consider the equation x'(t) = A(t)x(t) + f'c(t, s)D(x(s))x(s) ds + F(t) (1) •'o in which A, C, and D are n X n matrices, while x and F are n-vectors. In particular, A and F are continuous for 0 < t < oo, C is continuous for 0 < s < t < oo and D is defined in a neighborhood of zero and continuous at x = 0. Let G(t, s) be an n X n matrix with