Existence, uniqueness, boundedness and stability of periodic solutions of a certain second-order nonlinear differential equation with damping and resonance effects

In this paper, some qualitative behaviors of solutions for certain second-order nonlinear differential equation with damping and resonance effects are considered. By employing Lyapunov’s direct method, a complete Lyapunov function was used to investigate the stability of the system. Krasnoselskii’s...

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Bibliographic Details
Published inBiometrical letters Vol. 60; no. 2; pp. 109 - 124
Main Authors Eze, Everestus Obinwanne, Obasi, Uchenna Emmanuel, Ezugorie, Godwin, Hannah, Enyiduru Ekwomchi
Format Journal Article
LanguageEnglish
Published Sciendo 01.12.2023
Subjects
Duffing’s equation
Krasnoselskii’s fixed point theorem
Lyapunov function
Periodic solutions
Stability
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Summary:In this paper, some qualitative behaviors of solutions for certain second-order nonlinear differential equation with damping and resonance effects are considered. By employing Lyapunov’s direct method, a complete Lyapunov function was used to investigate the stability of the system. Krasnoselskii’s fixed point theorem was used to establish sufficient conditions that guaranteed the existence and boundedness of a unique solution. The results show that the equilibrium point was asymptotically stable. Furthermore, a test for periodicity was conducted using the Bendixson criterion, and the results showed that the solution of the second-order nonlinear differential equation is aperiodic, which extends some results from the literature.
ISSN:2199-577X
2199-577X
DOI:10.2478/bile-2023-0008