On Perturbing Lyapunov Functions
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It is known [2,3] that in proving uniform boundedness of a differential system by means of Lyapunov functions, it is sufficient to impose conditions in the complement of a compact set in [see pdf for notation], whereas, in the case of equiboundedness, the proofs demand that the assumptions hold everywhere in [see pdf for notation]. We wish to present, in this paper, a new idea which permits us to discuss nonuniform properties of solutions of differential equations under weaker assumptions. Our results will show that the equiboundedness can be proved without assuming conditions everywhere in [see pdf for notation] (as in the case of uniform boundedness), provided we appropriately perturb the Lyapunov functions. Our results also imply that in those situations when the Lyapunov function found does not satisfy all the desired conditions, it is fruitful to perturb that Lyapunov function rather than discard it. We also discuss the corresponding situation relative to equistability. We feel that the idea of perturbing Lyapunov functions introduced in this paper is a useful and important tool in the study of nonuniform properties of solutions as well as the preservation of those properties under constantly acting perturbations and therefore deserves further investigation.