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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.

MSC:

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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