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Zbl 1151.34056
Wang, Youyu; Lian, Hairong; Ge, Weigao
Periodic solutions for a second order nonlinear functional differential equation. (English)
[J] Appl. Math. Lett. 20, No. 1, 110-115 (2007). ISSN 0893-9659

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 $T>0$. \par 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 $(x,y)\in\Bbb R^2$. \par 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.
[Eduardo Liz (Vigo)]

MSC 2000:: *34K13 Periodic solutions of functional differential equations
34B27 Green functions

Keywords: Delay-differential equations; periodic solution; Green function; Fixed point theory

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