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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 \(p,q:\mathbb R\to\mathbb R^+\) 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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