Asymptotic behavior of solutions of certain higher order nonlinear difference equation

摘要

In this paper, we consider the higher order difference equationy(k+n)+p1(k)y(k+n-1)+p2(k)y(k+n-2)+⋯+pn(k)y(k)(1)=fk,y(k),y(k+1),…,y(k+n-1),∑s=k0k-1g(k,s,y(s),…,y(s+n-1)),k∈N(k0)={k0,k0+1,k0+2,…}, k0∈{1,2,…}. With the aid of a discrete inequality, we obtain some sufficient conditions which guarantee that for every solution y(k) of (1) satisfies the equation as k→∞,y(k)=∑i=1n(δi+o(1))zi(k),where δi, i=1,2,…,n are constants, {zi(k)}i=1n are any independent solutions of the equationz(k+n)+p1(k)z(k+n-1)+p2(k)z(k+n-2)+⋯+pn(k)z(k)=0,k∈N(k0).