| Volume 10, Number 6, 2020, Pages 2659-2668 DOI：10.11948/20200014 |
| Positive periodic solutions for a nonlinear differential system with two parameters |
| Ruixiong Fan,Chengbo Zhai |
| Keywords:Positive periodic solutions, differential system, existence and uniqueness. |
| Abstract: |
| In this article, we investigate a nonlinear system of differential equations with two parameters
$$\left\{ \begin{array}{l} x""(t)=a(t)x(t)-\lambda f(t, x(t), y(t)),\y""(t)=-b(t)y(t)+\mu g(t, x(t), y(t)),\end{array}\right.$$
where $a,b \in C(\textbf{R},\textbf{R}_+)$ are $\omega-$periodic for some period $\omega > 0$, $a,b \not\equiv 0$, $f,g \in C(\textbf{R} \times \textbf{R}_+ \times \textbf{R}_+ ,\textbf{R}_+)$ are $\omega-$periodic functions in $t$, $\lambda$ and $\mu$ are positive parameters. Based upon a new fixed point theorem, we establish sufficient conditions for the existence and uniqueness of positive periodic solutions to this system for any fixed $\lambda,\mu>0$. Finally, we give a simple example to illustrate our main result. |
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