Bounded perturbations with multiple delays of forced harmonic oscillators at resonance. (English) Zbl 0758.34056

The problem of existence of \(2\pi\)-periodic solutions for the two delay differential equations \(x''(t)+x(t)+h_ 1(x(t-r))+h_ 2(x(t-s))=p(t)\) and \(x''(t)+x(t)+h(x(t-r))+g(x'(t-s)=p(t)\) is investigated. The results which are proved by using a slight modification of an abstract result due to R. K. Nagle and Z. Sinkala [Differential Equations: Stability and Control, Proc. Int. Conf., Colorado Springs/CO (USA) 1989, Lect. Notes Pure Appl. Math. 127, 401-408 (1990; Zbl 0711.34053)] extend some known results obtained for bounded perturbations of forced harmonic oscillators at resonance and for similar problems where the perturbations involve the derivative of the solution.


34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
34C25 Periodic solutions to ordinary differential equations
34D10 Perturbations of ordinary differential equations
34K10 Boundary value problems for functional-differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
70K30 Nonlinear resonances for nonlinear problems in mechanics


Zbl 0711.34053