POSITIVE PERIODIC SOLUTIONS FOR A NONLINEAR DIFFERENTIAL SYSTEM WITH TWO PARAMETERS

In this article, we investigate a nonlinear system of differential equations with two parameters $ \left\{ {\begin{array}{*{20}{l}} \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} \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.