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Periodic solutions for a second order nonlinear functional differential equation. (English) Zbl 1151.34056
Define $$L(x(t)):=x''(t)+p(t)x'(t)+q(t)x(t),$$ where $p,q:\Bbb R\to\Bbb R^+$ are continuous $T$-periodic functions with positive average, and $T>0$. The main result of this paper establishes sufficient conditions to ensure the existence of at least one $T$-periodic solution for the second order delay-differential equation $$L(x(t))=r(t)x'(t-\tau(t))+f(t,x(t),x(t-\tau(t))).\tag{1}$$ Here, $r,\tau:\Bbb R\to\Bbb R$ are continuous and $T$-periodic, and the continuous function $f(t,x,y)$ is $T$-periodic in $t$ for all $(x,y)\in\Bbb R^2$. The Green function for the periodic problem associated to the ordinary differential equation $L(x(t))=\phi(t)$ is used to define a suitable abstract operator whose fixed points are the periodic solutions of (1). Then, a fixed point theorem due to Krasnosel’skii is applied to get the desired existence result. Under an additional condition, this operator is shown to be a contraction, and therefore the $T$-periodic solution is unique.

MSC:
 34K13 Periodic solutions of functional differential equations 34B27 Green functions
Full Text:
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