Oscillatory Behavior of Third-Order Nonlinear Difference Equations with a Nonlinear-Nonpositive Neutral Term

  • Said R. GraceEmail author


We shall present some new oscillation criteria for third-order nonlinear difference equations with a nonlinear-nonpositive neutral term of the form:
$$\begin{aligned} \Delta \left( \left( a(t)\left( \Delta ^{2}\left( x(t)-p(t)x^{\alpha }(t-k) \right) \right) ^{\gamma } \right) +q(t)x^{\beta }(t-m+1)=0,\right. \end{aligned}$$
with positive coefficients via comparison with first-order equations whose oscillatory behavior are known, or via comparison with second-order inequalities with solutions having certain properties. The obtained results are new, improve and correlate many of the known oscillation criteria appeared in the literature even for the case of Eq. (1.1) with \(\hbox {p (t)} = 0\). Examples are given to illustrate the main results.


Oscillation third order neutral difference equation nonlinear- nonpositive neutral term 

Mathematics Subject Classification

34N05 39A10 



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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Engineering Mathematics Faculty of EngineeringCairo UniversityGizaEgypt

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