Lazer-Leach Type Conditions on Periodic Solutions of Liénard Equation with a Deviating Argument at Resonance

Abstract EN

We study the existence of periodic solutions of Liénard equation with a deviating argument x′′+f(x)x'+n2x+g(x(t-τ))=p(t), where f,g,p:R→R are continuous and p is 2π-periodic, 0≤τ<2π is a constant, and n is a positive integer.

Assume that the limits limx→±∞g(x)=g(±∞) and limx→±∞F(x)=F(±∞) exist and are finite, where F(x)=∫0xf(u)du.

We prove that the given equation has at least one 2π-periodic solution provided that one of the following conditions holds: 2cos(nτ)[g(+∞)-g(-∞)]≠∫02πp(t)sin(θ+nt)dt, for all θ∈[0,2π],2ncos(nτ)[F(+∞)-F(-∞)]≠∫02πp(t)sin(θ+nt)dt, for all θ∈[0,2π],2[g(+∞)-g(-∞)]-2nsin(nτ)[F(+∞)-F(-∞)]≠∫02πp(t)sin(θ+nt)dt, for all θ∈[0,2π],2n[F(+∞)-F(-∞)]-2sin(nτ)[g(+∞)-g(-∞)]≠∫02πp(t)sin(θ+nt)dt, for all θ∈[0,2π].