On oscillation of solutions of a nonautonomous quasilinear second-order equation

Abstract

Sufficient conditions are obtained for the initial values of nontrivial oscillating (for $t = ω$) solutions of the nonautonomous quasilinear equation $y'' \pm \lambda (t)y = F(t,y,y'),$, where $t ∈ Δ = [a, ω[,-∞ < a < ω ≤ + ∞, λ(t) > 0, λ(t) ∈ C_Δ^{(1)},$ $|F((t,x,y))| ≤ L(t)(|x|+|y|)^{1+α}, L(t) ≥ -0, α ∈ [0,+∞[,$ $F: Δ × R^2 →R, F ∈ C_{Δ × R^2}, R$ is the set of real numbers, and $R^2$ is the two-dimensional real Euclidean space.