Asymptotic Behavior of Solutions to Difference Equations of Neutral Type

Abstract

AbstractWe present sufficient conditions for the existence of a solution x to an equation $$\begin{aligned} \Delta ^m(x_n-u_nx_{n-k})=a_nf(x_{n-\tau })+b_n, \end{aligned}$$ Δ m ( x n - u n x n - k ) = a n f ( x n - τ ) + b n , which is “close” to a given solution y to the linear homogeneous equation of neutral type $$\Delta ^m(y_n-\lambda y_{n-k})=0$$ Δ m ( y n - λ y n - k ) = 0 , where $$\lambda $$ λ is the limit of the sequence u. Closeness of solutions to above equations is understood as $$x_n-y_n=\textrm{o}(\omega _n)$$ x n - y n = o ( ω n ) , where $$\omega $$ ω is a given nonincreasing sequence with positive values. Moreover, we establish under which conditions for a given solution x to $$\Delta ^m(x_n-u_nx_{n-k})=a_nf(x_{n-\tau })+b_n$$ Δ m ( x n - u n x n - k ) = a n f ( x n - τ ) + b n and a given nonincreasing sequence with positive values $$\omega $$ ω there exists a polynomial sequence $$\varphi $$ φ of degree less than m such that $$x_n=\varphi (n)+\textrm{o}(\omega _n)$$ x n = φ ( n ) + o ( ω n ) . Presented conditions strongly depend on $$\lambda $$ λ .