Existence and uniqueness of periodic solutions of mixed monotone functional differential equations (Q1035050)

The authors investigate the existence and uniqueness of periodic solutions for \[ y'(t)=-a(t)y(t)+f_1(t,y(t-\tau(t)))+f_2(t,y(t-\tau(t))), \tag{1} \] where \(a\) and \(\tau\) are continuous and \(T\)-periodic functions, \(f_1,f_2\in C(\mathbb{R}^2,\mathbb{R})\) are \(T\)-periodic with respect to their first variable \(t\), and \(a(t)>0\) for all \(t\in\mathbb{R}\). It is assumed also that \(f_1\) is increasing with respect to its second variable, while \(f_2\) is decreasing with respect to its second variable. Another basic assumption is the existence of an ordered pair of lower and upper ``quasisolutions''. The main tool for proving the results is a fixed point theorem in ordered Banach spaces for mixed monotone operators. We remind that an operator \(A:E\times E\to E\), where \(E\) is an ordered Banach space, is called \textit{mixed monotone} whenever \(A(x_1,y_1)\leq A(x_2,y_2)\) for any \(x_2,x_2,y_1,y_2\in E\) that satisfy \(x_1\leq x_2\) and \(y_2\leq y_1\). Also, \(x^*\in E\) is called a fixed point of \(A\) if \(A(x^*,x^*)=x^*\).