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 $$p,q:\mathbb R\to\mathbb R^+$$ 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, $$r,\tau:\mathbb R\to\mathbb R$$ are continuous and $$T$$-periodic, and the continuous function $$f(t,x,y)$$ is $$T$$-periodic in $$t$$ for all $$(x,y)\in\mathbb R^2$$.
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.
Reviewer: Eduardo Liz (Vigo)

MSC:

 34K13 Periodic solutions to functional-differential equations 34B27 Green’s functions for ordinary differential equations
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References:

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