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Abstract
We study the existence of periodic solutions of Liénard equation with a deviating argumentx′′+f(x)x'+n2x+g(x(t-τ))=p(t),wheref,g,p:R→Rare continuous andpis2π-periodic,0≤τ<2πis a constant, andnis a positive integer. Assume that the limitslimx→±∞g(x)=g(±∞)andlimx→±∞F(x)=F(±∞)exist and are finite, whereF(x)=∫0xf(u)du. We prove that the given equation has at least one2π-periodic solution provided that one of the following conditions holds:2cos(nτ)[g(+∞)-g(-∞)]≠∫02πp(t)sin(θ+nt)dt, for allθ∈[0,2π],2ncos(nτ)[F(+∞)-F(-∞)]≠∫02πp(t)sin(θ+nt)dt, for allθ∈[0,2π],2[g(+∞)-g(-∞)]-2nsin(nτ)[F(+∞)-F(-∞)]≠∫02πp(t)sin(θ+nt)dt, for allθ∈[0,2π],2n[F(+∞)-F(-∞)]-2sin(nτ)[g(+∞)-g(-∞)]≠∫02πp(t)sin(θ+nt)dt, for allθ∈[0,2π].
Highlights
We are concerned with the existence of periodic solutions of Lienard equation with a deviating argument as follows: x + f (x) x + n2x + g (x (t − τ)) = p (t), (1)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.In recent years, the periodic problem of Lienard equations with a deviating argument has been widely studied because of its background in applied sciences.In the case when f(x) ≡ 0, for all x ∈ R and τ = 0, (1) becomes x + n2x + g (x) = p (t) . (2)Assume that limits (h1) limx → ±∞ g(x) = g(±∞)exist and are finite
We study the existence of periodic solutions of Lienard equation with a deviating argument x + f(x)x + n2x + g(x(t − τ)) =
We introduce a continuation theorem which will be used to prove the existence of periodic solutions of (1)
Summary
We are concerned with the existence of periodic solutions of Lienard equation with a deviating argument as follows: x + f (x) x + n2x + g (x (t − τ)) = p (t) ,. Lazer and Leach [9] proved that (2) has at least one 2π-periodic solution provided that the following condition holds:. Besides (h1), the limits (h2) limx → ±∞ F(x) = F(±∞) exist and are finite, where F(x) = ∫0x f(u)du It was proved in [10] that the following equation: x + f (x) x + n2x + g (x (t − τ)) = p (t). Abstract and Applied Analysis has at least 2π-periodic solution provided that one of the following conditions holds:. By using the continuation theorem [11], we prove the following result.
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