Appearance of oscillations under the effect of an oscillatory force (Q1328424)

This paper concerns semilinear differential equations of the type \[ x''(t)= h(t)f(x(t))+ g(t),\quad t\in \mathbb{R}_ += [0,\infty),\tag{1} \] where \(f\in C(\mathbb{R},\mathbb{R})\), \(g\in C(\mathbb{R}_ +,\mathbb{R})\), \(h\in C(\mathbb{R}_ +,\mathbb{R}_ +)f\) is nondecreasing, and \(xf(x)> 0\) for \(x\neq 0\). If \(g\) is nonoscillatory, one theorem states that (1) has no oscillatory solutions. If \(g\) is oscillatory, sufficient conditions on \(h\) are found for every bounded solution of (1) to be oscillatory. Specializations to linear equations (1) are included. Results for \(h(t)\leq 0\) were found earlier by \textit{J. S. W. Wong} [SIAM J. Math. Anal. 19, 667-675 (1988; Zbl 0655.34023)].