The method of lower and upper solutions and the stability of periodic oscillations. (English) Zbl 1043.34044

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 \(\beta\). E. N. Dancer and 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.


34C25 Periodic solutions to ordinary differential equations
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion


Zbl 0811.34018
Full Text: DOI


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