Oscillation Conditions for Difference Equations with a Monotone or Nonmonotone Argument

Abstract

Consider the first-order delay difference equation with a constant argument Δ ( ) + ( ) ( − ) = 0, = 0, 1, 2, . . . , and the delay difference equation with a variable argument Δ ( ) + ( ) ( ( )) = 0, = 0, 1, 2, . . . , where ( ) is a sequence of nonnegative real numbers, is a positive integer, Δ ( ) = ( + 1) − ( ), and ( ) is a sequence of integers such that ( ) ≤ − 1 for all ≥ 0 and lim →∞ ( ) = ∞. A survey on the oscillation of all solutions to these equations is presented. Examples illustrating the results are given.