Asymptotic convergence of the solutions of a discrete equation with several delays

摘要

A discrete equationΔy(n)=∑i=1sβi(n)[y(n-ji)-y(n-ki)]with integer delays ki and ji, ki > ji ⩾ 0 is considered. It is assumed that βi:Zn0-k∞→[0,∞),n0∈Z,k=max{k1,k2,…,ks},Zp∞≔{p,p+1,…},p∈Z,∑i=1sβi(n)>0,s∈Z1∞ is a fixed integer and n∈Zn0∞. Criteria of asymptotic convergence of solutions are expressed in terms of inequalities for the functions βi, i = 1, … , s. Some general properties of solutions are derived as well. It is, e.g., proved that, for the asymptotical convergence of all solutions, the existence of a strictly monotone and asymptotically convergent solution is sufficient. A crucial role in the analysis of convergence is played by an auxiliary inequality derived from the form of a given discrete equation.