Existence of rotating-periodic solutions for nonlinear systems via upper and lower solutions. (English) Zbl 1385.34032

In this paper, the authors study the following system \[ x'= f(t, x),\quad '=\frac{d}{dt}\tag{S1} \] where \(f: \mathbb{R}^1\times \mathbb{R}^n\rightarrow\mathbb{R}^n\) is continuous and satisfies the assumption \[ f(t+T,x)=Qf(t,Q^{-1}x)\text{ for all }t\in\mathbb{R}^1,\quad x\in \mathbb{R}^n, \] where \(Q\in O(n)\), i.e. \(Q\) is an orthogonal matrix. By using Brouwer’s fixed point theorem, they present a Massera-type criterion on affine-periodic solutions. Combining Massera’s criterion with the topological degree theory, they prove the existence of affine-periodic solutions for systems (S1). Moreover, some applications are given.


34C25 Periodic solutions to ordinary differential equations
34C14 Symmetries, invariants of ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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