Trichotomy of Non-oscillatory Solutions for Nonlinear First-order Neutral Difference Equation with Generalized Difference Operators

Abstract

In this study, we examine the trichotomy of non-oscillatory solution for the nonlinear first-order neutral difference equation triangle a (xn-a((n+1)) x_(n-1) )+triangle(a) (x(n-1)/b(n-1) )+q(n) f(x_(n-tau) )=0,n is an element of N-max{1,tau} , where triangle(a) x(n)=x(n+1)-ax(n),tau is an element of N,a is an element of R+ with triangle(a)m Delta=triangle((m-1))(a) (triangle(a) ), a is a general term of exponential sequence, (q(n)) is real valued sequences. ; -<with (-)->+infinity as ->+infinity;under the assumption Sigma(infinity)(=0)1/of the primary findings is demonstrated by examples.