Oscillation of nonlinear functional equations (Q1965345)

Let \(\mathbb{R}_+\) denote the set of nonnegative real numbers and \(I\) be an unbounded subset of \(\mathbb{R}_+\). Further, let \(p, q :I \to \mathbb{R}_+\) and \(g:I \to I\) be three given functions. The authors study the functional equation \[ F(x(g(t))) = p(t)x(t) + q(t)f(x(g^2 (t))) \tag{1} \] where \(F, f: \mathbb{R} \to \mathbb{R}\) are continuous and increasing functions such that \(uF(u) > 0\) and \(uf(u) > 0 \) for all real numbers \(u \neq 0\). A solution \(x : \mathbb{R} \to \mathbb{R}\) to (1) is called oscillatory if there exists a sequence \(\{ t_n \}_{n=1}^{\infty}\) in \(I\) such that \[ \lim_{n \to \infty} t_n = \infty \qquad \text{and} \qquad x(t_n) x(t_{n+1}) \leq 0 \] for \(n=1,2,3, \cdots \) and nonoscillatory otherwise. In this paper, the authors give conditions under which all solutions of the functional equation (1) are ocillatory.