Electron. J. Diff. Eqns., Vol. 2001(2001), No. 10, pp. 1-15.

Oscillation criteria for delay difference equations

Jianhua Shen & I. P. Stavroulakis

This paper is concerned with the oscillation of all solutions of the delay difference equation
$$  x_{n+1}-x_n+p_nx_{n-k}=0, 
 \quad n=0,1,2,\dots   $$
where $\{p_n\}$ is a sequence of nonnegative real numbers and $k$ is a positive integer. Some new oscillation conditions are established. These conditions concern the case when none of the well-known oscillation conditions
$$  \limsup_{n\to \infty}\sum_{i=0}^kp_{n-i} greater than 1 \quad{\rm and}\quad
  \liminf_{n\to \infty}\frac{1}{k}\sum_{i=1}^kp_{n-i} greater than
is satisfied.

Submitted January 9, 2001. Published January 23, 2001.
Math Subject Classifications: 39A10.
Key Words: Oscillation, non-oscillation, delay difference equation.

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Jianhua Shen
Department of Mathematics
Hunan Normal University
Changsha, Hunan 410081, China
e-mail: jhsh@public.cs.hn.cn
Ioannis P. Stavroulakis
Department of Mathematics
University of Ioannina
451 10 Ioannina, Greece
e-mail: ipstav@cc.uoi.gr

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