Oscillation of difference equations with delay. (English) Zbl 0722.39003

Differential equations and applications, Proc. Int. Conf., Columbus/OH (USA) 1988, Vol. I, 257-263 (1989).
[For the entire collection see Zbl 0707.00014.]
A solution \(\{y_ n\}\) of a second order nonlinear difference equation of the form \((1)\quad \Delta^ 2y_{n-1}+q_ ny^{\alpha}_ n=0,\quad n=1,2,...\) (where \(\alpha_ 0\) is the quotient of odd positive integers and \(\Delta y_ n=y_{n+1}-y_ n)\) is said to be oscillatory if for every \(N\geq 0\) there exists \(n\geq N\) such that \(y_ ny_{n+1}\leq 0.\)
In Section 2 the authors consider a version of (1) with discrete delay, namely \((2)\quad \Delta^ 2y_{n-1}+q_ ny^{\alpha}_{\sigma_ n}=0,\quad n=1,2,...,\) where \(\sigma_ n\) is a positive integer with \(\sigma_ n<n\), and establish that under some conditions the solutions are oscillatory.
In the third section they consider the oscillation of a two-dimensional system of difference equations \((3)\quad \Delta y_ n=p_ nz_ n,\quad \Delta z_ n=-f(n,y_{n-m}),\quad n=1,2,...\) and obtain some sufficient conditions for oscillation of (3).


39A10 Additive difference equations
39A12 Discrete version of topics in analysis


Zbl 0707.00014