The method of lower and upper solutions and the stability of periodic oscillations
This paper is devoted to the method of lower and upper solutions to solve the nonlinear periodic problem (1) \(\ddot x+g(t,x)=0\), \(x(0)= x(T)\), \(\dot x(0)= \dot x(T)\), where \(g\in \mathbb{C}^{0,4} (\mathbb{R}/T\mathbb{Z}\times \mathbb{R})\). A function \(\alpha\in \mathbb{C}^2(\mathbb{R}/T\mathbb{Z})\) is said to be a lower solution of (1), if \(\ddot\alpha(t)+ g(t,\alpha(t))\geq 0\) \(\forall\,t\in \mathbb{R}\). An upper solution \(\beta\) is defined in a similar way by reversing the above inequality. C. de Coster and P. Habets showed that if \(\alpha(t)\leq \beta(t)\) \(\forall\,t\in\mathbb{R}\), then (1) has a solution laying between \(\alpha\) and \(\beta\). \textit{E. N. Dancer} and \textit{R. Ortega} [J. Dyn. Differ. Equ. 6, 631--637 (1994; Zbl 0811.34018)] have proved that, when \(\alpha\) and \(\beta\) are strict lower and upper solutions and the region \(\alpha\leq \beta\) contains a unique solution of (1), then this solution is unstable. The main result of this paper gives sufficient conditions for the existence of a unique solution \(\varphi\in [\beta,\alpha]\) that is stable. Significant examples are considered.
scientific article; zbMATH DE number 1836640
scientific article