Oscillatory behavior of higher order nonlinear neutral delay dynamic equations with positive and negative coefficients - II

Abstract

In this paper, we derive some sufficient conditions for the oscillatory and asymptotic behavior of solution of the higher order nonlinear Neutral Delay Dynamic Equations (NDDEs) of the form (r(t)(y(t) + p(t)y(α(t))) ∆n ) ∆2 + q(t)G(y(β(t))) − h(t)H(y(γ(t))) = 0 (H) and (r(t)(y(t) + p(t)y(α(t))) ∆n ) ∆2 + q(t)G(y(β(t))) − h(t)H(y(γ(t))) = f(t) (NH) for t ∈ [t 0 , ∞) T , t 0 ( > 0) ∈ T, where T is a time scale with sup T = ∞, and n ∈ N, are studied under the assumption Zt0∞ (σ(t)) n − 1∆ t < ∞ (H 1 ) r(t) for the various ranges of p(t). In addition, sufficient conditions are obtained for the existence of bounded positive solutions of the equation (NH) by using Krasnosel’skii’s fixed point theorem. The results in this paper extended and generalizes the results of ([10],[11]). Examples are included to illustrate the validation of the results.