Unbounded solutions for asymmetric oscillations in the degenerate resonant case

Abstract

In this paper, we are concerned with the existence of unbounded solutions of the asymmetric equationx″+ax+−bx−=f(t), where x+=max⁡{x,0}, x−=max⁡{−x,0}, a and b are two different positive constants satisfying 1/a+1/b=2n/m with m and n relatively prime, f(t) is a continuous 2π periodic function, and the discriminative functionΦf(θ)=∫02πC(nmθ+nt)f(nt)dt,θ∈R has some degenerate zeros, where C(t) is the solution ofx″+ax+−bx−=0 with the initial conditions C(0)=1,C′(0)=0. Especially, when a=4,b=1,f(t)=±1/45+cos⁡4t, the existence of unbounded solutions will be proved.