Periodic solutions for dissipative-repulsive systems
Following results are obtained in this paper: (1) If \(x'=f(t,x)\) is dissipative-repulsive, then it admits an \(\omega\)-periodic solution, where \(f: \mathbb{R}\times \mathbb{R}^n\to \mathbb{R}^n\) is continuous, locally Lipschitzian in \(x\) and \(f(t+\omega, x)=f(t,x)\) for every \(t\) and some \(\omega>0\). (2) If \(x'= F(t,x_t)\) is dissipative-repulsive, then it admits an \(\omega\)-periodic solution, where \(F:\mathbb{R}\times C\to\mathbb{R}^n\) is continuous, locally Lipschitzian in \(\varphi\in C\), \(F(t+\omega, \varphi)=F(t,\varphi)\) for all \((t,\varphi)\) and some \(\omega>0\) and \(F\) takes any bounded sets in \(C\) into bounded sets in \(\mathbb{R}^n\); here, \(C=C([-r,0], \mathbb{R}^n)\), \(r>0\). It is further shown that dissipative-repulsive systems admit equilibria. An example is given to illustrate some results.
scientific article; zbMATH DE number 1558403
scientific article