## 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 $$A(x^*,x^*)=x^*$$.

### MSC:

 34K13 Periodic solutions to functional-differential equations 47H10 Fixed-point theorems
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### References:

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