A sharp oscillation criterion for a difference equation with constant delay

where {p(n)}∞n=0 is a nonnegative sequence of reals and k is a natural number, oscillate if lim infn→∞ ∑n–1 i=n–k p(i) > ( k k+1 ) k+1. In the case that ∑n–1 i=n–k p(i) is slowly varying at infinity, it is proved that the above result can be essentially improved by replacing the above condition with lim supn→∞ ∑n–1 i=n–k p(i) > ( k k+1 ) k+1. An example illustrating the applicability and importance of the result is presented.


Introduction and preliminaries
Consider the first-order linear difference equation where denotes the forward difference operator, i.e., x(n) = x(n + 1)x(n), {p(n)} ∞ n=0 is a nonnegative sequence of reals, and k is a natural number. By a solution of the difference equation (1) we mean a sequence of real numbers {x(n)} ∞ n=-k which satisfies Eq. (1) for all n ≥ 0. A solution {x(n)} ∞ n=-k of the difference equation (1) is said to be oscillatory if the terms of the sequence {x(n)} ∞ n=-k are neither eventually positive nor eventually negative. Otherwise, the solution {x(n)} ∞ n=-k is said to be nonoscillatory. In the last few decades, the oscillatory behavior of the solutions to difference equations has been extensively studied. See, for example, [3-9, 13, 14, 16, 18, 20, 21, 23-30] and the references cited therein. For the general theory of difference equations, the reader is referred to the monographs [1,2,13,19]. In 1981, Domshlak [8] considered the case where k = 1. In 1989, Erbe and Zhang [9] proved that all solutions of Eq. (1) oscillate if β := lim inf n→∞ p(n) > 0 and lim sup n→∞ p(n) > 1β (2) or lim inf while Ladas, Philos, and Sficas [18] improved condition (3) as follows: Note that this condition is sharp in the sense that the fraction on the right-hand side cannot be improved, since when p(n) is a constant, say p(n) = p, then this condition reduces to which is a necessary and sufficient condition [21] for the oscillation of all solutions to Eq. (1). Moreover, concerning the constant k k (k+1) k+1 in (3) it should be emphasized that, as it is shown in [9], if then Eq. (1) has a nonoscillatory solution. In 1990, Ladas [16] conjectured that Eq. (1) has a nonoscillatory solution if for all large n.
However, this conjecture is not correct, and a counter-example was given in 1994 by Yu, Zhang, and Wang [30]. Moreover, in 1999 Tang and Yu [28] showed that Eq. (1) has a nonoscillatory solution if the so-called "corrected Ladas conjecture" is satisfied. In 2017, Karpuz [14] improved the above result by replacing condition (8) with which is a weaker condition. In the case that the above mentioned conditions (4) and (5) are not satisfied, in 2004 Stavroulakis [25] and in 2006 Chatzarakis and Stavroulakis [6] derived the following sufficient oscillation conditions for Eq. (1): where α is as in (5) and satisfies 0 < α ≤ ( k k+1 ) k+1 . Also Chen and Yu [7] derived the following oscillation condition: In 2001, Shen and Stavroulakis [23] established several oscillation conditions which, in the case of the difference equation with k = 1 reduce to the sufficient oscillation condition which cannot be improved in the sense that the lower bound 1/4 cannot be replaced with a smaller number. Indeed, by condition (7), we see that Eq. (14) has a nonoscillatory solution if sup p(n) < 1/4. Note, however, that even in the critical state where lim n→∞ p(n) = 1/4, Eq. (14) can be either oscillatory or nonoscillatory. For example, if p(n) = 1 4 + c n 2 , then Eq. (14) will be oscillatory in case c > 1/4 and nonoscillatory in case c < 1/4 (the Kneserlike theorem [8]).
Recall that the difference equation (1) is the discrete analogue of the delay differential equation where p : [t 0 , ∞) → R + is a nonnegative continuous function andτ is a positive constant.
In 1972, Ladas, Lakshmikantham, and Papadakis [17] proved that all solutions of Eq. (17) oscillate if while in 1982 Koplatadze and Canturija [15] established the following result: If which is a necessary and sufficient condition [13] for all solutions of the delay differential equation e as k → ∞, and therefore conditions (5) and (6) can be interpreted as the discrete analogue of (19) and (20), respectively.
Very recently, Garab et al. [11] essentially improved the above condition (19) by replacing it with under the additional assumptions that p is a bounded and uniformly continuous function such that a := lim inf Our aim in this paper is to extend this result to the discrete case. To this end, assume that the sequence A(n) = n-1 i=n-k p(i) is slowly varying at infinity (see [11]), i.e., for every λ ∈ N, lim n→∞ [A(n + λ) -A(n)] = 0. It should be mentioned that the idea how to obtain sharp oscillation conditions in the continuous time case by considering slowly varying coefficients originated from Pituk [22]. Moreover, the continuous-time result has been recently generalized in [10] and in [12] for variable delay and several variable delays, respectively. The next work of the present authors is to study discrete analogues of these problems.

Main result
Our main result is the following theorem.

Assume also that A(n) = n-1 i=n-k p(i) is slowly varying at infinity and
Then all solutions of Eq. (1) oscillate.
To prove this theorem, we need the following lemma.

Lemma 2.1 ([6]) Assume that x(n) is an eventually positive solution of Eq.
(1) and that A(n) = n-1 i=n-k p(i) ≥ P > 0 for large n. Then x(nk) for large n.
Now we give a proof of Theorem 2.1.
Proof of Theorem 2.1 Suppose, for the sake of contradiction, that the assumptions of Theorem 2.1 hold and Eq. (1) has a nonoscillatory solution x(n). Without loss of generality, we may assume that x(n) is eventually positive (otherwise take the solution -x(n)) and since x(n) = x(n + 1)x(n) = -p(n)x(nk) ≤ 0, the solution x(n) is nonincreasing. By the virtue of Lemma 2.1 there exists a positive number K such that and by the monotonicity of x(n), we get In view of (22), there exists a strictly increasing sequence δ(n) n∈N of natural numbers such that Introduce the sequence Observe that y(n) (m) ≤ 1. Furthermore, since x(n) is an eventually positive solution of Eq. (1), we have and hence y(n) (m+1)y(n) (m) + p δ(n) + m y(n) (m-k) = 0.
As it is shown above, for each m ∈ N, y(n) (m) is a bounded sequence, and since p is bounded, q(n) (m) is also bounded. In view of the above, by the Bolzano-Weierstrass theorem and Cantor's diagonal argument, it follows that there exists a strictly increasing sequence s(n) n∈N of natural numbers such that, for all m ∈ N, By virtue of (27), for every m, n ∈ N, we get y(s(n)) (m+1)y(s(n)) (m) + q(s(n)) (m) y(s(n)) (m-k) = 0, and taking n → ∞, we have That is, the positive sequence y(m) is a (nonoscillatory) solution of Eq. (29). Now, taking into account (25) and the fact that A(n) is slowly varying at infinity, for m ∈ N, we have That is, m-1 i=m-k q(i) is equal to the constant B and therefore By virtue of (5), the sequence y(m) (being a solution of Eq. (29)) is oscillatory, a contradiction. The proof is complete.
Remark 2.2 Note that in the special case that k = 1, i.e., in the case of the difference equation x(n) + p(n)x(n -1) = 0, condition (22) reduces to lim sup n→∞ p(n) > 1/4, that is, to condition (16) which is sharp and cannot be further improved. Observe, however, that condition (16) is derived from condition (15) only in the case that α = 1/4. In general, that is, without the assumption that A(n) is slowly varying, lim sup n→∞ A(n) > 1/4 does not seem to imply oscillation of all solutions, which points out that our main result improves oscillation condition (15) even in the case of k = 1.

Example
In this section an example, which illustrates the significance of our result, is presented. It is to be pointed out that in this example all conditions of Theorem 2.1 are satisfied and therefore all solutions oscillate while none of the conditions presented in Sect. 1 is satisfied.