Oscillation of First-order Neutral Difference Equation

Abstract

<p class="MsoNormal" style="margin: 0cm 0cm 4pt; line-height: 12pt; mso-line-height-rule: exactly; mso-outline-level: 1;"><span style="font-size: 10pt;" lang="EN-US"><span style="font-family: Times New Roman;">The oscillation of the first order neutral difference equation</span></span></p><p class="MsoNormal" style="margin: 0cm 0cm 4pt; line-height: 12pt; mso-line-height-rule: exactly;"><span style="font-size: 10pt;" lang="EN-US"><span style="position: relative; top: 5pt; mso-text-raise: -5.0pt;"></span></span></p><p class="MsoNormal" style="margin: 0cm 0cm 4pt; line-height: 12pt; mso-line-height-rule: exactly;"><span style="font-size: 10pt;" lang="EN-US"><span style="font-family: Times New Roman;">is studied in this paper,where</span><span style="position: relative; top: 5pt; mso-text-raise: -5.0pt;"><span style="font-family: Times New Roman;"> </span></span><span style="font-family: Times New Roman;">and </span><span style="position: relative; top: 5pt; mso-text-raise: -5.0pt;"></span><span style="font-family: Times New Roman;">,</span><span style="position: relative; top: 3pt; mso-text-raise: -3.0pt;"><span style="font-family: Times New Roman;"> </span></span><span style="font-family: Times New Roman;">,</span><span style="position: relative; top: 3pt; mso-text-raise: -3.0pt;"><span style="font-family: Times New Roman;"> </span></span><span style="font-family: Times New Roman;">are positive constants. The sufficient conditions for oscillation of the equation are obtained by suitable inequality and characteristic equation.</span></span></p

Necessary and sufficient conditions for oscillations of first order neutral delay difference equations with constant coefficients

“…In [1,10], several sufficient conditions involving p,τ , q and σ have been obtained under which all solutions of (1) oscillate. In the present paper we can very easily derive sufficient conditions in terms of p,τ , q and σ by comparing element of the set ∧(z) in each case.…”

“…In [10], Xiaohui Gong et al proved that if −1 < p < 0, τ − σ > 1, then q − τ τ −σ−1 (1 − p 1 τ ) > 0 is a sufficient conditions for oscillation of Eq. (1).…”

Necessary and sufficient conditions for oscillation of first order neutral delay difference equations

“…Xiaohui Gong et al [15] proved that all the solutions of the equation (1) are oscillatory if 0 < p < 1, τ − σ > 1…”