Oscillation criteria for quasi-linear neutral delay dynamic equations on time scale

Abstract

Abstract In this paper, we consider the oscillation for the second-order quasi-linear neutral dynamic equation ( r ( t ) | Z Δ ( t ) | α − 1 Z Δ ( t ) ) Δ + q ( t ) | x ( δ ( t ) ) | β − 1 x ( δ ( t ) ) = 0. $${(r(t)|{Z^\Delta }(t{)|^{\alpha - 1}}{Z^\Delta }(t))^\Delta } + q(t)|x(\delta (t{))|^{\beta - 1}}x(\delta (t)) = 0.$$ on time scale 𝕋, where Z(t) = x(t) + p(t)x(τ(t)), α,β > 0 are constants. We establish some new oscillation criteria and give sufficient conditions to insure that all solutions of quasi-linear neutral dynamic equation are oscillatory on time scale 𝕋. The new oscillation criteria are presented that improve some known results in the literature.