Rocky Mountain Journal of Mathematics

Nontrivial periodic solutions of second order singular damped dynamical systems

Jifeng Chu, Shengjun Li, and Hailong Zhu

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Abstract

Assuming that the linear equation $x'' + h(t)x' + a(t)x = 0$ has a positive Green's function, we study the existence of nontrivial periodic solutions of second order damped dynamical systems \[ x'' + h(t)x' + a(t)x = f(t, x) + e(t), \] where $h$, $a\in \C(\!(\R/T\Z),\R)$, $e\! =\! (e_1,\ldots, e_N\!)^T\!\! \in \C(\!(\R/T\Z),\R^N\!)$, $N \ge 1$, and the nonlinearity $f = (f_1,\ldots, f_N)^T\in\C((\R=T\Z)\times\R^N\setminus\{0\},\R^N)$ has a repulsive singularity at the origin. We consider a very general singularity and do not need any kind of strong force condition. The proof is based on a nonlinear alternative principle of Leray-Schauder. Recent results in the literature are generalized and improved.

Article information

Source
Rocky Mountain J. Math., Volume 45, Number 2 (2015), 457-474.

Dates
First available in Project Euclid: 13 June 2015

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1434208483

Digital Object Identifier
doi:10.1216/RMJ-2015-45-2-457

Mathematical Reviews number (MathSciNet)
MR3356624

Zentralblatt MATH identifier
1327.34068

Subjects
Primary: 34C25: Periodic solutions 34D20: Stability

Keywords
Nontrivial periodic solutions singular damped dynamical systems Leray- Schauder alternative principle

Citation

Chu, Jifeng; Li, Shengjun; Zhu, Hailong. Nontrivial periodic solutions of second order singular damped dynamical systems. Rocky Mountain J. Math. 45 (2015), no. 2, 457--474. doi:10.1216/RMJ-2015-45-2-457. https://projecteuclid.org/euclid.rmjm/1434208483


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