Boundedness properties and perturbing Lyapunov functions

Abstract

It is well-known [ 1,4] that in investigating uniform boundedness of solutions of a given differential system by means of Lyapunov functions, it is enough to impose conditions on the complement of a compact set in 1w”, where as in the case of nonuniform boundedness properties, the proofs demand that the assumptions hold everywhere in KY. In [2], using the idea of perturbing Lyapunov functions, the nonuniform boundedness property is discussed without assuming conditions everywhere, as in the case of uniform boundedness. Recently [ 31 uniform boundedness criteria are studied by using two Lyapunov functions together with the theory of differential inequalities to obtain sufficient conditions under weaker assumptions. In this paper, we investigate nonuniform boundedness properties employing the method of perturbing Lyapunov functions and the theory of differential inequalities. Our results include several existing results as special cases as well as those in [2, 33.