Perturbations of Nonlinear Systems of Differential Equations , Ill

1. The effect of a perturbation on the solutions of a linear system of differential equations can be studied by means of the variation of constants formula. The nonlinear variation of constants formula of Alekseev [l] has been used to obtain various results on the effect of a perturbation on the solutions of a nonlinear system; see, for example [2], [3], [5]. In most of these results, the trivial solution of the unperturbed system is assumed to be asymptotically stable. In this paper, we wish to study perturbations of systems which are stable but not necessarily asymptotically stable. If the unperturbed system is linear, then the assumption of uniform stability suffices to yield quite general results. If the unperturbed system is not linear, uniform stability of the trivial solution does not imply any useful perturbation theorems. Thus, it is necessary to consider more restricted types of stability. One type which enables one to consider integrable perturbations is integral stability, introduced by Vrkoc [7]. Here, we consider another type of stability, which has been mentioned previously in [4] and [5], called uniform stability in variation. This is still more restrictive than integral stability, but sometimes easier to verify and amenable to a broader class of perturbations. For a linear