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**Existence and uniqueness of periodic solutions of mixed monotone functional differential equations.**
*(English)*
Zbl 1187.34090

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 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 \(A(x^*,x^*)=x^*\).

\[ y'(t)=-a(t)y(t)+f_1(t,y(t-\tau(t)))+f_2(t,y(t-\tau(t))), \tag{1} \]

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 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 \(A(x^*,x^*)=x^*\).

Reviewer: Adriana Buică (Cluj-Napoca)

### Keywords:

first order periodic functional-differential equation; periodic solution; existence and uniqueness; fixed point theorem; mixed monotone operators; ordered Banach spaces
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\textit{S. Kang} and \textit{S. S. Cheng}, Abstr. Appl. Anal. 2009, Article ID 162891, 13 p. (2009; Zbl 1187.34090)

### References:

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