Abstract. Alexseev's formula generalizes the variation of constants formula and permits the study of a nonlinear perturbation of a system with certain stability properties. In recent years M. Pinto introduced the notion of h-stability. S.K. Choi et al. investigated h-stability for the nonlinear differential systems using the notion of t ∞ -similarity. Applying these two notions, we study bounds for solutions of the perturbed differential systems.

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In this paper, we investigate bounds for solutions of nonlinear perturbed differential systems.

We characterize the h-stability in variation for nonlinear difference systems via n -similarity and Lyapunov functions. Furthermore, using Lyapunov's method and ϱ comparison principle, we obtain some results related to stability for the perturbations of nonlinear difference systems. ᮊ

The present paper is concerned with the notions of Lipschitz and asymptotic stability for perturbed nonlinear differential system knowing the corresponding stability of nonlinear differential system. We investigate Lipschitz and asymtotic stability for perturbed nonlinear differential systems. The main tool used is integral inequalities of the Bihari-type, in special some consequences of an extension of Bihari's result to Pinto and Pachpatte, and all that sort of things.