Oscillation results via comparison theorems for fourth-order delay three-terms difference equations

In this paper we establish sufficient conditions for the oscillation of all solutions of equation Δ4x(n)+p(n)Δx(n+1)+q(n)x(n-τ)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \varDelta ^4x(n)+p(n)\varDelta x(n+1)+q(n)x(n-\tau )=0 \end{aligned}$$\end{document}via comparison with some first order delay difference equations whose oscillatory characters are known. The presented criterion is easily verifiable. Examples are also given to illustrate the main result.


Introduction
In this paper we assume that p and q are sequences of positive real numbers, τ ∈ N = {0, 1, 2, . . . , }. We consider the linear fourth-order difference equation of the form Δ 4 x(n) + p(n)Δx(n + 1) + q(n)x(n − τ ) = 0, where n ∈ N n 0 = {n 0 , n 0 + 1, . . . } and n 0 ∈ N, Δ is the forward difference operator defined by Δx(n) = x(n + 1) − x(n). By a solution of equation (E) we understand a sequence x of real numbers, that is defined for n ≥ n 0 − τ and satisfies the equation for sufficiently large n. A solution is nonoscillatory if it is eventually positive or eventually negative. Otherwise, the solution is said to be oscillatory. We call equation (E) oscillatory if all its solutions are oscillatory. The background of difference equations and discrete oscillation theory can be found in the monographs of Agarwal [1] and Agarwal, Bohner, Grace and O'Regan [2]. In recent years, the study of oscillatory and asymptotic behavior of solutions of second order difference equations has received great attention. Compared to this, the study of higher-order difference equations receives considerably less attention in literature. Some recent results for the oscillation of third-order difference equations can be found in [3,6,7,9,12,13,24,25], and for fourthorder in [4,10,16,18,23]. In particular, in [7] and [13], the authors studied oscillation and asymptotic properties of solutions of the three-term difference equation by assuming the coefficient of the damping term is nonnegative. In [12], the difference equation Δ 3 x(n) + p(n)Δx(n + 1) + q(n)x(n − τ ) = 0 with positive coefficient p was investigated. In these papers the comparison theorems with suitable first-order difference equations were used. It seemed to be natural to continue the research in the case of higher-order equations especially since fourth-order differential and difference equations often occur as models in mathematical biology, economics and engineering (for example see [4,5,8,26]).
In [15], the authors investigated the oscillatory behavior of solutions of the fourth order difference equations with damping Δ 4 u(n) + p(n)Δu(n + 1) + q(n)u(s(n)) = 0, under the assumption that the auxiliary third order difference equation The purpose of this paper is to study the oscillation of equation (E). Since for three-terms equations it is difficult to introduce the classification of nonoscillatory solutions, first we prove two lemmas that allow us to rewrite equation (E) in equivalent form as a two-terms equation with quasidifferences. Then, in our main result we give sufficient conditions for the oscillation of all solutions of equation (E) in terms of the two associated first order delay difference equations. Applying the appropriate criteria to these delay difference equations allows us to obtain new, easily verifiable oscillation results for equation (E) (see Theorem 2.10). It is worth noticing, that in this criterion the explicit form of solutions of the auxiliary equations is not needed. Numerical methods are frequently used in the investigation of the properties of the solutions of differential and difference equations. However, such methods are difficult to apply when investigating the oscillation of solutions. Our results may be helpful in the interpretation of numerical solutions, see Examples 3.1 and 3.2.
Let us recall some results that will be used in the sequel.
has an eventually positive solution if and only if the difference equation has an eventually positive solution.

Lemma 1.2 Consider the equation
where a, b, c are sequences of positive real numbers and f : and u f (n, u) > 0 for all u = 0, n ∈ N hold. Let x be an eventually positive solution of (1.2). Then exactly one of the following statements holds for all sufficiently large n:

Main results
We start this section with two lemmas, which allow us to rewrite equation (E) in equivalent binomial form in terms of solutions of two auxiliary linear equations. Because the proofs of these lemmas are technically complicated, they are presented in Sect. 4.

Lemma 2.1 If z is an eventually positive solution of equation
then the equality holds for any sequence x and for large n.
has an eventually positive solution v, then the equality holds for any sequence x and for large n .
Applying (2.4) to equation (E) we get The question arises whether equations (2.1) and (2.3) have eventually positive solutions. The lemma below presents a possible criterion. Since Δ 3 z(n) < 0 for n ≥ n 1 , we get Δ 2 z(n) > 0 for n ≥ n 1 Then, by (2.6), we have Based on Trench [27], we say that a linear difference operator where r j are eventually positive real sequences. The sequences L i x are called quasidifferences of x.
The quasidifferences in equation (E ) have the form Note, that if z and v are solutions of the equations (2.1) and (2.3) respectively, and both converge to positive constants, then and then the linear operator in (E ) is in canonical form, which means equation (E ) is in canonical form. Hence, by Lemma 1.3, we get the following classification of the nonoscillatory solutions of (E ).

Lemma 2.4 Assume that condition (2.5) is satisfied. Let x be an eventually positive solution of equation (E ).
Then there exists n 2 ≥ n 0 such that for all n ≥ n 2 and either In summary, we get the following remark.

Remark 2.5
If condition (2.5) holds, then the three-term difference equation (E) can be rewritten as a two-terms equation of the form (E ), which is in the canonical form.
Our goal is to present an easily verifiable oscillation criterion for equation (E). In our investigation we utilize the form (E ). Using the comparison theorem we will deduce the oscillation of (E) from the oscillation of certain first-order difference equations whose properties are well-explored.
Let z be a solution to equation (2.1) which tends to a positive constant and let v be a solution to equation (2.3) which tends to a positive constant. Then there exist positive constants z * , we define sequences K 1 , K 2 by where n 4 = max{n 2 , n 3 } + τ + 3 and n 2 as in Lemma 2.4. It is easy to see that condition Proof Assume to the contrary that there exists an eventually positive solution x to (E). Let x be such a solution. Notice that x is also an eventually positive solution to (E ). Let z be a positive decreasing solution of (2.1). According to Lemma 2.4 we consider two cases. Case I. Assume conditions (2.7) and (2.8) hold. Summation of the both sides of (E ) from s ≥ n 4 to infinity, leads to equality From the above, by properties of the sequence Summing the above inequality from n to infinity, we obtain Again by (2.8), By properties of the sequence L 2 x, we have Combining the above two inequalities, we obtain Hence, using (2.8) we get (2.13) Based on the sequence L 1 x properties, we have for n > n 4 . Therefore (2.14) Combining (2.13) with (2.14), we get for n ≥ n 4 + τ . Using the definition of K 1 , this leads to inequality By assumption, the sequence x is an eventually positive solution of inequality (2.15). By virtue of Lemma 1.1, it is also an eventually positive solution of equation (2.11). This contradicts that all solutions of (2.11) are oscillatory. Case II. Set u(n) : = L 3 x(n). Let conditions (2.7) and (2.8) hold. Thus the the sequence L 2 x is positive for n ≥ n 4 . Hence, the following estimation holds From the properties of the sequence L 3 x, we have .
Combining the above two inequalities, we get v(m) Summing both sides of the above inequality from n 4 to k − 1, we obtain Since the sequence L 1 x is positive for n ≥ n 4 , we have On the other hand, again by (2.8), we get (s + 1) .
By summation of both sides of the above inequality from n 4 to n − 1, we obtain . .
Since the sequence L 3 x is decreasing, . .
Using the definition of u, equation (E ) takes the form We get Hence, using (2.8) we get Finally, using the definition of K 2 , we obtain Since u is an eventually positive solution of the above inequality, by Lemma 1.1 it is also an eventually positive solution of equation (2.12). In that way there is contradiction with the assumption that all solutions of equation (2.12) are oscillatory.
Applying well-known oscillation criteria for first-order delay difference equations to equations (2.11) and (2.12) (see [11] and [17]), we obtain the following oscillation criteria for equation (E). If τ > 0, then we can also use the following criterion.
Note, that in the above results the explicit form of the eventually positive solutions of (2.1) and (2.3) are needed. But, it is well known, that it is difficult to find the explicit form of solutions of second and third order linear difference equations with variable coefficients. Therefore, we present a criterion in which the assumptions depend only on the coefficients p and q of the equation (E). First we prove a simple lemma.
Combining Lemma 2.9 and Corollary 2.7 we get the following oscillation criterion.

Examples
By using computer algebra systems there is always possible to find recursively an approximate solution of the considered equation, but sometimes it is very difficult to determine whether the approximate solution is oscillatory or not. In many cases, it is rather easy to use our last criterion to verify that the considered equation is oscillatory. Such cases are shown in the following examples.

Example 3.1 Consider the following equation of type (E):
Δ 4 x(n)+ 12 n 2 (n + 1) ln 2 (n + 1) Δx(n + 1)+ (0.01 + sin 2 n) 10 √ n n 3 x(n −1) = 0, (3.1) where n ≥ 2, p(n) = 12 n 2 (n + 1) ln 2 (n + 1) , q(n) = (0.01 + sin 2 n) 10 √ n n 3 , τ = 1. Using the iterative scheme we solve (3.1) with the following initial conditions Except for the initial conditions all computed values of x are rounded to the nearest integer number. As we see in Table 1, the obtained terms of this solution are positive until the 88th term, then negative. The next sign change of terms is on 346th term, next on 900th term and so on. Since the length of intervals with one sign are increasing, the oscillations of the solutions are not easily visible. On the other hand, it is easy to verified that ∞ n=1 n 2 12 n 2 (n + 1) ln 2 (n + 1) < ∞, ∞ n=1 n sin 2 n n 3 < ∞,   For a numerical solution of (3.2), we set a = 10, b = 5, τ = 2, α = 5, β = 2, 5. The recurrence formula for the equation takes the form x(n + 4) = 4x(n + 3)−6x(n + 2) + 4x(n + 1)−x(n)− 10 n 5 x(n + 2)−x(n + 1) − 5 n 2.5 x(n − 2). Taking the initial values we get a solution x whose trajectory from n = 1 to n = 150 is shown in Fig. 1. To show the oscillatory nature of this solution, we present its graph divided into four parts with increasingly larger scales.
Proof of Lemma 2. 2 We start from the right hand side of (2.4): 1 v(n + 1) Δ v(n)v(n + 1) Δ v(n)v(n + 1) z(n + 1) 1 v(n + 1) Δ z(n)z(n + 1)Δ Δx(n) z(n) article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.