Subdominant positive solutions of the discrete equationΔu(k+n)=−p(k)u(k)

Abstract

A delayed discrete equationΔu(k+n)=−p(k)u(k)with positive coefficientpis considered. Sufficient conditions with respect topare formulated in order to guarantee the existence of positive solutions ifk→∞. As a tool of the proof of corresponding result, the method described in the author's previous papers is used. Except for the fact of the existence of positive solutions, their upper estimation is given. The analysis shows that every positive solution of the indicated family of positive solutions tends to zero (ifk→∞) with the speednot smaller than the speed characterized by the functionk·(n/(n+1))k. A comparison with the known results is given and some open questions are discussed.