Perturbations of asymptotically stable differential systems.

Abstract

We consider systems of differential equations of the form (1) x’ = f(t,x), for t ∊ [a,∞), x in some domain D ⊂ IRn and f ∈ c1 ([a,∞) × D) (a a fixed real number). We assume that the solution x(t,to,xo) of (1) defined for t ≥ a satisfies | x(t,to, xo) | ≤ c| xo |h(t)h(to)-1 (t ≥ to ≥ a) for xo small enough, for some constant c > 0 and h a continuous positive function defined in [a,∞). We give conditions for the perturbed system y’ = f(t,y) + g(t,y) (g ∈ c([a,∞) x D)) to have the same type of stability as (1). 1980 Mathematical subject classification. 34 C 11; 34 D 10. © 1984, Walter de Gruyter. All rights reserved.