Positive Periodic Solutions of Coupled Singular Rayleigh Systems
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Positive Periodic Solutions of Coupled Singular Rayleigh Systems Fanchao Kong1 · Feng Liang1 · Juan J. Nieto2 Received: 16 May 2020 / Accepted: 13 September 2020 © Springer Nature Switzerland AG 2020
Abstract This paper mainly aims to investigate the positive periodic solutions for coupled singular Rayleigh systems. In order to establish the coupled structure, the basic framework of graph theory is employed. By means of Lyapunov method, inequality techniques and a classical consequence of Mawhin’s continuation theorem, some sufficient criterion for the positive periodic solutions has been provided. After that, taken the effects of the delays into account and without imposing more conditions, we further study the positive periodic solutions for a kind of coupled singular Rayleigh system with delays. Here not only the structure is more general than the existing works but the conditions imposed are concise. Consequently, compared with the previous results on the singular systems and coupled systems, the results we established are more generalized and some previous ones can been complemented and improved. Finally, the effectiveness of the established results are validated via an numerical example. Keywords Coupled Rayleigh system · Graph theory · Lyapunov method · Continuation theorem · Singularity
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Fanchao Kong [email protected] Feng Liang [email protected] Juan J. Nieto [email protected]
1
School of Mathematics and Statistics, Anhui Normal University, Wuhu 241000, Anhui, People’s Republic of China
2
Instituto de Matemáticas, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain 0123456789().: V,-vol
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1 Introduction During the past several decades, the following Rayleigh system has been widely considered: x (t) + f (t, x (t)) + g(t, x(t)) = e(t),
(1.1)
where f ; g : R × R → R and e : R → R are continuous functions. The dynamic behaviors of system (1.1) have been an active research topic due to their extensive applications in physics, mechanics, engineering technique, and other areas, see [8,14, 23] and the references therein. It has been shown that these successful applications are greatly dependent on the existence of periodic solutions for system (1.1). Moreover, the periodicity analysis of system (1.1) has been a subject of intense activities, and many results have been reported. See, to name a few, [13,15,27]. Singular equations appear in a great deal of physical models, see, e.g., [1,21,22,24, 28], differential equations with singularities arise naturally in the study of the motion of particles under the influence of gravitational or electrostatic forces. For example, in paper [21], the singularity models in which the restoring force caused by a compressed perfect gas; in paper [22,24,28], the singular term can be regarded as a generalized Lennard–Jones potential or Van der Waals force and it is widely found in molecular dynamics to model the interaction between atomic particles. In recent years, different kinds of singu
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