NEW CONDITIONS FOR K- LIKE PROPERTIES OF ASYMPTOTICALLY STABLE SOLUTIONS FOR WEAKLY PERTURBED SYSTEMS FOR A CERTAIN CLASS OF NONLINEAR DIFFERENTIALS EQUATIONS

This paper gives the description of one general approach to the study of stable – like properties of solutions for a weakly nonlinear system with small perturbing motions for a certain class of nonlinear differential equations. The properties are, K- Stability, K – Unstable, and Asymptotically K – Stables of the given studied equations. The approach is based on the generalization of the direct Lyapunov method combined with the asymptotic and averaging method of nonlinear mechanics. In some instances, such an approach makes it possible to study classes of systems with a small parameter under new wider assumptions on the properties of solutions of the studied systems .Hence, the study established the following results; K- stability for the system [1.3], K- unstable for the system [1.1] and Asymptotically K- stable for the system, [1.1]. My approach and results of this study improved on [4.15] in the literature to the case where more than two arguments of the studied system properties were established.