## 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
Full Text: