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Oscillatory and asymptotic properties of solutions of even order neutral difference equations. (English) Zbl 1194.39009
The authors consider the following difference equation
\[ \Delta^m(x_n-px_{n-\tau})=q_nx_{n-\sigma} + h_nx_{n+\eta},\tag{1} \] where \(n\) runs over \({\mathbb N}\), \(\{m,\tau\}\subset{\mathbb N}\), \(\sigma\in{\mathbb N}_0\), \(p\in(0,1)\), \(\{g_n,\,h_n\}\subset(0,+\infty)\) for all the \(n\in{\mathbb N}\).
First I must make the following two remarks.
Remark 1. The authors give the following definitions. I quote: “A solution \((x_n)\) of the equation (1) is called oscillatory, if for every \(n_0\geq1\) there exists \(n\geq n_0\) such that \(x_nx_{n+1}\leq 0\). Otherwise the solution is called nonoscillatory.”
According to these definitions the trivial (zero) solution of the specified above equation (1) is oscillatory.
Remark 2. As an enemy of nonstandard notations, I must remark also that, if \(\{m,n\}\subset{\mathbb N}_0\) then \(n^{\underline{ m}}/m!\) is equal to the binomial coefficient \(\binom nm.\)
The authors deduce their results from the corresponding results of the behavior of solutions of the inequality
\[ \Delta^m(x_n-px_{n-\tau})\geq q_nx_{n-\sigma} + h_nx_{n+\eta},\tag{2} \] They obtain the following result.
Theorem 4. Let all the following conditions (a), (b), (c) be fulfilled:
(a) if \(m>2,\) then
\[ \sum_{n=1}^{+\infty} n^{m-3}h_n=+\infty,\tag{3} \]
(b) There exists \(k\in{\mathbb N}_0,\) such that \(\eta\geq m+k\tau\) and
\[ \limsup_{n\to+\infty}\left(\sum_{i=0}^k p^i\right) \left(\sum_{j=0}^k \binom{n+\eta-k\tau-j-1}{m-1}h_j\right)>1, \tag{4} \]
(c) \[ \limsup_{n\to+\infty} \left(\sum_{j=0}^k \binom{j-n+\sigma+m-1}{m-1}\right)>1-p. \tag{5} \] Then each solution of the inequality (2) is neither eventually positive nor eventually negative.
This result is deduced by the authors from two other Theorems.
Theorem A. (Theorem 2 + Remark 1 at the end of the article).
Let both conditions (a) and (b) are fulfilled. If a solution of the inequality
\[ \Delta^m(x_n-px_{n-\tau})\geq h_nx_{n+\eta},\tag{6} \] is either eventually positive or eventually negative, then it is bounded and if \(m>2\) it tends to \(0,\) when \(n\) tends to \(+\infty\); moreover if \(m=2,\) condition (b) holds and
\[ \sum_{n=1}^{+\infty} nh_n=+\infty, \tag{7} \] then this solution tends to \(0,\) when \(n\) tends to \(+\infty.\)
Theorem B. (Theorem 1; this result was obtained by the authors earlier). Let the condition (c) be fulfilled. If a solution of the inequality
\[ \Delta^m(x_n-px_{n-\tau})\geq g_nx_{n-\sigma}, \tag{8} \] is either eventually positive or it is eventually negative, then it is unbounded.
The authors give two examples which illustrate their results.
Reviewer’s remarks. Remark 3. (I made two remarks above.) Formulation of Lemma 2 is wrong.
Counterexample 1. Let \(m=1,\)
\[ z_{2k-1}=z_{2k}=k,\quad\Delta z_{2k-1}=0,\quad\Delta z_{2k}=1, \] where \(k\in{\mathbb N}.\) Then \(z(n)>0,\,\Delta z_{n}\geq0\) and not eventually zero, but \[ z_{2k-1}\Delta z_{2k-1}=(-1)^{m-l}z_{2k-1}\Delta z_{2k-1}=0. \]
Counterexample 2. Let \(m=2,\)
\[ z_{2k-1}=\frac{\pi^2}3-2\sum_{i=1}^k\frac1{k^2},\quad 2k z_{2k}=z_{2k-1}-\frac1{((k+1)^2} \] where \(k\in{\mathbb N}.\) Then
\[ \Delta z_{2k-1}=\Delta z_{2k}=-\frac1{(1+k)^2},\quad\Delta^2 z_{2k-1}=0, \]
\[ \Delta^2 z_{2k}=\frac1{(1+k)^2}-\frac1{(2+k)^2}>0, \] \(z(n)>0,\,\Delta^2 z_{n}\geq0\) and not eventually zero, but
\[ z_{2k-1}\Delta^2 z_{2k-1}=\Delta(-1)^{m-l}z_{2k-1}\Delta z_{2k-1}=0. \] The mistake is recoverable. It is necessary to replace in the formulation of the Lemma the words “\(\Delta^m z_n\) of constant sign with \(\Delta^m z_n\) not eventually zero” by the words “\(\Delta^m z_n\) is either eventually positive or eventually negative”. Exactly under the last conditions this Lemma is applied in the paper.
The goal of the following remarks about the style is to make reading of this paper more easier.
Remark 4. The authors spend 6 lines to explain how they deduce the inequality
\[ \Delta^l z_n\geq z_{n+\eta} \sum_{j=n}^{+\infty}\binom{j-n+m-l-1}{m-l-1}h_j. \] Instead of that it would be better if they simply write the inverse (backwards) interpolation formula (or, as they say now, discrete Taylor formula):
\[ y_n=(-1)^s\left(\sum_{i=n}^N\binom{i-n+s-1}{s-1} \Delta^sy_j\right)+\sum_{k=0}^{s-1}\binom{N-n}k(-1)^k\Delta^ky_{N+s-k} \] with \(y_n=\Delta^lz_n\) and \(s=m-l,\) and all would be clear without further explanation.
Analogously, making use of direct interpolation formula,
\[ z_{b}=\sum_{j=a}^{b-s}n\binom{b-1-i}{s-1} + \sum_{k=0}^{s-1}\binom{b-s-a}k\Delta^kz_a \] with \(s=m\,,a=n,\,b=n+\eta-k\tau,\,\eta\geq k\tau+m,\) one gets the condition (b), which excludes the case \(l=m\) in Theorem A. Finally, making use of backwards interpolation formula one obtains the equality
\[ z_{n-\sigma}=\sum_{j=n-\sigma}^n\binom{m-1+j-n+\sigma}{m-1}+ \sum_{k=0}^{m-1}\binom{m+\sigma}k(-1)^k\Delta^kz_{n+m-k} \]
and then get the condition (c), which exclude the case \(l=0\) in the Theorem B. So, in my opinion, applying of Lemma 1 instead of the discrete Taylor formula makes the paper longer and less intelligible.
Conclusion: The positive content of the paper give me a reason to say that it is interesting.

39A21 Oscillation theory for difference equations
39A10 Additive difference equations
39A22 Growth, boundedness, comparison of solutions to difference equations
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