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Positive periodic solutions for a nonlinear differential system with two parameters
Chengbo Zhai,Ruixiong Fan
Keywords:positive periodic solutions; differential system; existence and uniqueness
      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.