Unbounded solutions for asymmetric oscillations in the degenerate resonant case (Q2062053)

In this paper, the asymmetric equation \[ x'' + ax^+ - bx^- = f(t) \] is considered, where \(x^+=\max\{x,0\}\), \(x^-=\max\{-x,0\}\), and \(a\) and \(b\) are two different positive constants satisfying \((1/\sqrt{a})+(1/\sqrt{b})=2n/m\) with \(m\) and \(n\) relatively prime. Moreover, \(f(t)\) is a continuous \(2\pi\) periodic function and \[ \Phi_f(\theta) = \int_0^{2\pi} C\Bigl(\frac{n}{m}\theta + nt \Bigr)f(nt)de, \quad \theta \in {\mathbb R} \] has some degenerate zeros, where \(C(t)\) is a solution of the initial value problem \[ x'' + ax^+ - bx^- = 0, \quad x(0)=1, \ x'(0)=0. \] Especially, the existence of unbounded solutions is proved for the case when \(a=4\), \(b=1\) and \(f(t)=\pm(1/45)+\cos4t\).