Periodic and unbounded motions in asymmetric oscillators at resonance

We consider the existence of periodic and unbounded solutions for the asymmetric oscillatorx″+ax+−bx−+g(x)=p(t),x″+ax+−bx−+g(x)=p(t), where x+=max⁡{x,0}x+=max⁡{x,0}, x−=max⁡{−x,0}x−=max⁡{−x,0}, a and b   are two positive constants, p(t)p(t) is a 2π  -periodic smooth function and g(x)g(x) satisfies lim|x|→+∞⁡x−1g(x)=0lim|x|→+∞⁡x−1g(x)=0. We have proved previously that the boundedness of all the solutions and the existence of unbounded solutions have a close relation to the interaction of some well-defined functions Φp(θ)Φp(θ) and Λ(h)Λ(h). In this paper, we consider some critical cases and obtain some new sufficient conditions for the existence of 2π  -periodic and unbounded solutions. In particular, the obtained periodic solutions are new and cannot be deduced from Landesman–Lazer's and Fabry–Fonda's existence conditions. In contrast with many existing results, the function g(x)g(x) may be unbounded or oscillatory without asymptotic limits.