Coexistence of unbounded solutions and periodic solutions of Liénard equations with asymmetric nonlinearities at resonance

In this paper, we deal with the existence of unbounded orbits of the mapping {θ1 = θ 2nπ 1/ρμ(θ) o(ρ-1),ρ1=ρ c-μ′(θ) o(1), ρ→∞,where n is a positive integer, c is a constant and μ(θ) is a 2π-periodic function. We prove that if c ＞ 0 and μ(θ) ≠ 0, θ∈ 0, 2π], then every orbit of the given mapping goes to infinity in the future for ρ large enough; if c ＜ 0 and μ(θ) ≠ 0, θ∈ 0, 2π], then every orbit of the given mapping goes to infinity in the past for ρ large enough. By using this result, we prove that the equation x″ f(x)x′ ax -bx- φ(x) =p(t) has unbounded solutions provided that a, b satisfy 1/√a 1/√b = 2/n and F(x)(= ∫x0 f(s)ds),and φ(x) satisfies some limit conditions. At the same time, we obtain the existence of 2π-periodic solutions of this equation.

Coexistence of unbounded solutions and periodic solutions of Liénard equations with asymmetric nonlinearities at resonance

In this paper, we deal with the existence of unbounded orbits of the mapping , where n is a positive integer, c is a constant and μ(θ) is a 2π-periodic function. We prove that if c > 0 and μ(θ) ≠ 0, θ, ∈ 0, 2ΰ], then every orbit of the given mapping goes to infinity in the future for ρ large enough; if c < 0 and μ(θ) ≠ 0, θ ∈ 0, 2π], then every orbit of the given mapping goes to infinity in the past for ρ large enough. By using this result, we prove that the equation x″+f(x)x′+ax ⁺−bx ⁻+ϕ(x)=p(t) has unbounded solutions provided that a, b satisfy and ϕ(x) satisfies some limit conditions. At the same time, we obtain the existence of 2π-periodic solutions of this equation. This work was supported by the National Natural Science Foundation of China (Grant No. 10471099), the Fund of Beijing Education Committee (Grant No. KM200410028003) and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, Ministry of Education of China