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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: + 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-τ(t))+f(t,x(t),x(t-τ(t)))·(1)

Here, r,τ: are continuous and T-periodic, and the continuous function f(t,x,y) is T-periodic in t for all (x,y) 2 .

The Green function for the periodic problem associated to the ordinary differential equation L(x(t))=φ(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:
34K13Periodic solutions of functional differential equations
34B27Green functions