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Journal of Applied Mathematics and Simulation
Volume 2, Issue 2, Pages 101-111
http://dx.doi.org/10.1155/S1048953389000080

Sharp conditions for the oscillation of delay difference equations

1Department of Mathematics, The University of Rhode Island, Kingston, RI 02881, USA
2Department of Mathematics, University of Ioannina, P.O. Box 1186, Ioannina 45110, Greece

Copyright © 1989 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Suppose that {pn} is a nonnegative sequence of real numbers and let k be a positive integer. We prove that limninf [1ki=nkn1pi]>kk(k+1)k+1 is a sufficient condition for the oscillation of all solutions of the delay difference equation An+1An+pnAnk=0,   n=0,1,2,. This result is sharp in that the lower bound kk/(k+1)k+1 in the condition cannot be improved. Some results on difference inequalities and the existence of positive solutions are also presented.