Sharp conditions for the oscillation of delay difference equations

Suppose that { p n } is a nonnegative sequence of real numbers 
and let k be a positive integer. We prove that 
 \frac{{k^k }} {{\left( {k + 1} \right)^{k + 1} }} \]" id="E3" xmlns:mml="http://www.w3.org/1998/Math/MathML"> lim n → ∞ inf  [ 1 k ∑ i = n − k n − 1 p i ] > k k ( k + 1 ) k + 1 
is a sufficient condition for the oscillation of all solutions of the 
delay difference equation 
 A n + 1 − A n + p n A n − k = 0 ,    n = 0 , 1 , 2 , … . 
This result is sharp in that the lower bound k k / ( k + 1 ) k + 1 in 
the condition cannot be improved. Some results on difference 
inequalities and the existence of positive solutions are also presented.


INTRODUCTION AND PRELIMINARIES
Recently there has been some activity concerning the oscillation of all solutions of the delay difference equation An+l An + pnAn-O, n-O, 1,2,...

(i)
where {Pn} is a sequence of nonnegative real numbers and k is a positive integer. See, for example, [1]- [3] and the references cited therein. Throughout this paper, the sequence {Pn} is supposed to be defined for n > 0.
By a solution of Eq. (1) we mean a sequence {An} which is defined for n >_ -k and which satisfies Eq. (1) for n >_ 0. A solution {A,} of Eq. (1) is said to be oscillatory if the terms An of the sequence are not eventually positive or eventually negative. Otherwise, the solution is called nonoscillalory.
Our aim in Section 2 is to establish the following result.
For a proof of this result see [3]. If lim infpn > )k+l' n--*oo (k+ 1 then it follows from [2] that every solution of Eq. (1) oscillates. Clearly (2) is a substantial improvement over (5), replacing the p, of (5) by the arithmetic mean of the terms p,_,..., p,-x in (2). . See [4]. We should also remark here that it is the proof of this latter theorem (see Theorem 2.1.1 in [4]) which we used as our guide in arriving at the statement and the proof of Theorem 1. One should notice that condition (2) can be written in the form and that n-x ( kl)+l limn_ooinf Pi > k +' ).
--. In Section 3 we present some results about difference inequalities. In particular we prove that, under appropriate hotheses, if the difference inequality z+x-z+p,zn-0, n=0,1,2,... h a positive solution so does Eq. (1). Finally, we utilize this result to give a "sharp" sufficient condition for the existence of a positive solution of Eq. (1).
Next observe that because of (9), for n sufficiently large, i=n-k where M k/3 > 0. Choose m such that > This is possible because from (9),/3 > a. Then for n sufficiently large, say for n > no, (13) is satisfied for the specific m which was chosen in (15), also (9) and (14) hold, and {A,, } is decreasing for n > no. Now in view of (14) and for n > no + k, there exists an integer n* with n-k < n* < n such This contradicts (15) and so the proof of the theorem is complete.

DIFFERENCE INEQUALITIES
A slight modification in the proof of Theorem 1 leads to the following result about the difference inequalities,  Bn _ Z PiXi-k <_ X,n for n > 0 and so T is well-defined and T" S S. If It follows by induction that < A r+l < A r < < A < A . Clearly, x, > 0 for n > -k and, by Theorem 3, it suffices to show that {xn} is a solution of the difference inequality (18). To this end, in view of (27) we have, for n >_ 0, 1--1n-I H 1--pi