NEW CONDITIONS FOR K- LIKE PROPERTIES OF ASYMPTOTICALLY STABLE SOLUTIONS FOR WEAKLY PERTURBED

For a certain class of nonlinear differential equations, this work describes one broad method to the investigation of stable – like features of solutions for a weakly nonlinear system with tiny perturbing motions. The properties of the analysed equations include K- Stability, K- Unstable, and Asymptotically K- Stables. The method is based on a generalisation of the direct Lyapunov method mixed with nonlinear mechanics' asymptotic and averaging methods. In some cases, this strategy allows researchers to investigate classes of systems with a tiny parameter while making new, broader assumptions about the features of the researched systems' solutions. As a consequence of the research, the following conclusions were reached: K- stability for the system [1.3], K- unstable for the system [1.1], and Asymptotically K- stable for the system [1.1]. In the scenario where more than two reasons of the researched system attributes were established, my technique and results improved on [4.15] in the literature.

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