A nonoscillation theorem for half-linear differential equations with periodic coefficients

Abstract

The half-linear differential equation (φ{symbol}p (x′))′ + a (t) φ{symbol}p (x′) + b (t) φ{symbol}p (x) = 0 is considered under the assumption that the coefficient a(t) and an indefinite integral B(t) of b(t) are periodic functions with period T > 0. It is proved that {(p - 1) φ{symbol}p* (B (t)) - a (t)} B (t) ≤ 0 (0 ≤ t ≤ T) is sufficient for all nontrivial solutions to be nonoscillatory. Here, p > 1 and φ{symbol}q (y) = | y |q - 2 y for q = p or q = p* = p/(p - 1). The proof is given by means of Riccati technique. The condition is shown to be sharp. Sufficient conditions are also presented for all nontrivial solutions are oscillatory in the linear case p = 2. Some examples and simulations are included to illustrate our results. © 2007 Elsevier Inc. All rights reserved.