Oscillation of neutral second order half-linear differential equations without commutativity in deviating arguments

Abstract

Abstract In this paper we derive oscillation criteria for the second order half-linear neutral differential equation [ r ( t ) Φ ( z ′ ( t ) ) ] ′ + c ( t ) Φ ( x ( σ ( t ) ) ) = 0 , z ( t ) = x ( t ) + b ( t ) x ( τ ( t ) ) , $$ \displaystyle \Bigl[r(t)\Phi(z'(t))\Bigr]'+c(t)\Phi(x(\sigma(t)))=0, \quad z(t)=x(t)+b(t)x(\tau(t)), $$ where Φ(t) = |t| p−2 t, p ≥ 2, is a power type nonlinearity. We improve recent results published in the literature by obtaining better oscillation constants and removing the usual condition σ(τ(t)) = τ(σ(t)). Two methods (comparison method and Riccati equation method) are used.