Oscillations of difference equations with non-monotone retarded arguments

Keyword: Difference equation Non-monotone argument Retarded argument Oscillatory solutions Nonoscillatory solutions Consider the first-order retarded difference equation DxðnÞ þ pðnÞx sðnÞ ð Þ 1⁄4 0; n 2 N0 where ðpðnÞÞnP0 is a sequence of nonnegative real numbers, and ðsðnÞÞnP0 is a sequence of integers such that sðnÞ 6 n 1, n P 0; and limn!1 sðnÞ 1⁄4 1. Under the assumption that the retarded argument is non-monotone, a new oscillation criterion, involving lim inf, is established. An example illustrates the case when the result of the paper implies oscillation while previously known results fail. 2015 Elsevier Inc. All rights reserved.