Unbounded solutions in asymmetric oscillations (Q1764951)

The aim of the paper is to analyze the existence of unbounded solutions for the nonlinear asymmetric oscillator \[ x''+\alpha x^+-\beta x^-= f(t),\tag{1} \] where \(\alpha\), \(\beta\) are positive constants satisfying \({1\over\sqrt{\alpha}}+{1\over \sqrt{\beta}}= {2\over\omega}\) for some \(\omega\in \mathbb{R}^+\setminus\mathbb{Q}\), \(f(t)\) is \(2\pi\)-periodic and bounded and \(x^{\pm}= \max\{\pm x,0\}\). Some sufficient conditions for the existence of unbounded solutions of (1) are presented.