Boundedness in asymmetric oscillations (Q1283105)

The equation \(x''+ ax^+- bx^-= f(t)\) is studied where \(f(t)\) is a \(2\pi\)-periodic function, \(a\) and \(b\) are positive constants such that \(a\neq b\). It is shown that under some assumptions, all solutions to the equation are bounded, more precisely, if \(x(t)\) is a solution to the equation, then it is defined for all \(t\in \mathbb{R}\), and \(\sup_{t\in\mathbb{R}}(| x(t)|+| x'(t)|)< \infty\).