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Oscillation of a class of the fourthorder nonlinear difference equations
Advances in Difference Equations volume 2012, Article number: 99 (2012)
Abstract
In this article, a class of fourthorder difference equations with quasidifferences and deviating argument is considered. We state a new oscillation theorem for the sublinear case and we complete the existing results in the literature. Our approach is based on considering Equation (1) as a system of the fourdimensional difference system and on the cyclic permutation of the coefficients in the difference equations.
Introduction
In this article, we consider a class of fourthorder nonlinear difference equations of the form
where α, β, γ, λ are the ratios of odd positive integers, \tau \in \mathbb{Z} is a deviating argument and \{{a}_{n}\}, \{{b}_{n}\}, \{{c}_{n}\}, \{{d}_{n}\} are positive real sequences defined for n\in {\mathbb{N}}_{0}=\{{n}_{0},{n}_{0}+1,\dots \}, {n}_{0} is a positive integer, and Δ is the forward difference operator defined by \mathrm{\Delta}{x}_{n}={x}_{n+1}{x}_{n}.
By a solution of Equation (1) we mean a real sequence \{{x}_{n}\} satisfying Equation (1) for n\in {\mathbb{N}}_{0}. A nontrivial solution \{{x}_{n}\} of (1) is said to be nonoscillatory if it is either eventually positive or eventually negative, and it is otherwise oscillatory. Equation (1) is said to be oscillatory if all its solutions are oscillatory.
In the last few years, great attention has been paid to the study of fourthorder nonlinear difference equations, see [1–10] and references therein.
If {a}_{n}={c}_{n}=1\alpha =\gamma =1 and \tau =3, then (1) takes the form
The oscillatory and asymptotic properties of solutions of (2) have been investigated in [4, 11, 12] under the conditions
while articles [2, 10, 13] deal with cases where at least one of these series is convergent (see also the references therein).
Equation (1) is a special case of nonlinear fourthorder equation with deviating argument investigated in the recent articles [1, 2]. In [1], necessary and sufficient conditions for the oscillation of all bounded solutions of (1) (the socalled Boscillation) have been given. In [2], oscillation criteria for (1) have been established using the analysis of nonoscillatory solutions and by comparison with certain first and secondorder difference equations.
Equation (1) with \tau =2 can be seen as a coupled system of two secondorder difference equations of the form
Indeed, eliminating z from the first equation, this system can be rewritten as
System (3) is a special case of more general coupled systems. Those oscillatory properties have been investigated in [5].
Our approach here is to consider (1) as a fourdimensional system. If
then Equation (1) can be written as the nonlinear system
where
Obviously, if (x,y,z,w) is a solution of system (S) and one of its components is of one sign, then all its components are of one sign.
System (S) can be viewed as a discrete analogue of the fourdimensional differential system investigated by Kusano et al.[14], and by Chanturia [15]. In that article, the oscillation of the ndimensional differential systems was investigated in terms of Property A (which reads for equations of even order as the oscillation of all solutions) and Property B. Observe that in [16] we have used this approach to study Property B for system (S) assuming {D}_{n}<0.
Motivated by these articles, we study the oscillatory properties of solutions of (1). First we show the influence of the deviating argument τ on the existence of quickly oscillatory solutions and we describe the so called cyclic permutation for (1). Our main goal is to state a new oscillation theorem for Equation (1) in the sublinear case \lambda <\alpha \beta \gamma and to extend the existing oscillation results in the literature in case where the difference operator in (1) is in the canonical form, i.e., when
Our results are based on the conditions for the nonexistence of nonoscillatory solutions and on the change of summation for double series. Due to our approach considering (1) as a fourdimensional system, we extend for any \tau \in \mathbb{Z} some results of [2] stated for a delay \tau \le 0. Using cyclic permutation we show how it is possible to extend oscillation criteria to the case when one of the series in (H) is convergent.
Existence of quickly oscillatory solutions
Prototypes of oscillatory solutions of (1) are solutions of the form
Such solutions are called quickly oscillatory and the following result can be seen as a necessary condition for their existence.
Theorem 1 Equation (1) with τ even has no quickly oscillatory solutions.
Proof Let {x}_{n}={(1)}^{n}{p}_{n} be a quickly oscillatory solution of (1). Then
From the first equation of system (S) we have
where {q}_{n}={(\frac{{p}_{n+1}}{{C}_{n}}+\frac{{p}_{n}}{{C}_{n}})}^{\gamma}>0. From the second equation of (S) we obtain
where {r}_{n}={(\frac{{q}_{n+1}}{{B}_{n}}+\frac{{q}_{n}}{{B}_{n}})}^{\beta}>0. Repeating argument, we get from the third equation of (S)
where {s}_{n}={(\frac{{r}_{n+1}}{{A}_{n}}+\frac{{r}_{n}}{{A}_{n}})}^{\alpha}>0. Consequently, from here and from the fourth equation we have
which gives a conclusion. □
By the method used in the proof of Theorem 1 we can easily construct equations possessing a quickly oscillatory solution.
Example 1 Consider the equation
where τ is an odd positive integer. This equation has a quickly oscillatory solution {x}_{n}={(1)}^{n}{2}^{n}. Indeed, {p}_{n}={2}^{n}, {q}_{n}={2}^{n}3, {r}_{n}={2}^{n\beta}{3}^{2\beta}, {s}_{n}={2}^{n\beta}{3}^{2\beta}({2}^{\beta}+1) and the value of {d}_{n} follows from the relation {d}_{n}=({s}_{n+1}+{s}_{n})/{p}_{n+\tau}^{\lambda}.
If \beta \ge \lambda, then Equation (6) has all solutions oscillatory (see, e.g., Proposition 1 below). However, if \beta <\lambda, then by [4], Theorems 3.5, 3.6] Equation (6) has also nonoscillatory solutions.
Cyclic permutation
In this section, we describe the leftordered cyclic permutation of coefficients in (1).
Lemma 1 The following statements are equivalent:

(i)
x is a solution of (1).

(ii)
y=\{{y}_{n}\}, where {y}_{n}={c}_{n}{(\mathrm{\Delta}{x}_{n})}^{\gamma}, is a solution of
\mathrm{\Delta}\left(\frac{1}{{d}_{n}^{1/\lambda}}{\left(\mathrm{\Delta}{a}_{n}{\left(\mathrm{\Delta}{b}_{n}{(\mathrm{\Delta}{y}_{n})}^{\beta}\right)}^{\alpha}\right)}^{1/\lambda}\right)+\frac{1}{{c}_{n+\tau}^{1/\gamma}}{y}_{n+\tau}^{1/\gamma}=0.(R1) 
(iii)
z=\{{z}_{n}\}, where {z}_{n}={b}_{n}{(\mathrm{\Delta}{y}_{n})}^{\beta}, is a solution of
\mathrm{\Delta}\left({c}_{n+\tau}{\left(\mathrm{\Delta}\frac{1}{{d}_{n}^{1/\lambda}}{\left(\mathrm{\Delta}{a}_{n}{(\mathrm{\Delta}{z}_{n})}^{\alpha}\right)}^{1/\lambda}\right)}^{\gamma}\right)+\frac{1}{{b}_{n+\tau}^{1/\beta}}{z}_{n+\tau}^{1/\beta}=0.(R2) 
(iv)
w=\{{w}_{n}\}, where {w}_{n}={a}_{n}{(\mathrm{\Delta}{z}_{n})}^{\alpha} is a solution of
\mathrm{\Delta}\left({b}_{n+\tau}{\left(\mathrm{\Delta}{c}_{n+\tau}{\left(\mathrm{\Delta}\frac{1}{{d}_{n}}{(\mathrm{\Delta}{w}_{n})}^{1/\lambda}\right)}^{\gamma}\right)}^{\beta}\right)+\frac{1}{{a}_{n+\tau}^{1/\alpha}}{w}_{n+\tau}^{1/\alpha}=0.(R3)
Proof First we prove that (i) is equivalent to (ii). If we express x from the last equation in (S) we obtain
Thus, from here and the first equation in (S) we have
which yields Equation (R1). To prove that (i) is equivalent to (iii) we use the same process. Using (5) and (7) we have
Substituting this into
and using the second equation of (S) we get Equation (R2).
To prove that (i) is equivalent to (iv) we proceed as before, expressing Δz in terms of w from the third equation of (S) and from (5) and comparing both expressions. □
Theorem 2 Equation (1) is oscillatory if and only if Equation (R_{ i }) is oscillatory fori\in \{1,2,3\}.
Remark 1 By Theorem 2 Equation (2) is oscillatory if and only if the equation
is oscillatory. Observe that the difference operator in this equation is in the canonical form if {\sum}_{n={n}_{0}}^{\mathrm{\infty}}{d}_{n}=\mathrm{\infty}.
Remark 2 The cyclic permutation for the coupled system (3) means that equations in (3) are considered in the opposite order. Hence, (x,z) is a solution of (3) if and only if (u,v)=(z,x) is a solution of the system
which is again system of the form (3). Oscillation results of [5] for (3) assume
that is (4) is not in the canonical form. Hence, to compare results of [5] and our oscillation criteria we have to apply results of [5] to (8), see Remark 6.
Nonoscillatory solutions
Throughout this and the next sections, we use the convention
The aim of this section is to study nonoscillatory solutions of (1). If (S) has a solution (x,y,z,w), then (x,y,z,w) is a solution of (S), too. Hence, when studying the nonexistence conditions for nonoscillatory solutions, we can consider solutions such that {x}_{n}>0 for large n.
We start with the classification of nonoscillatory solutions of (S). This has been presented in [2] without the proof, so we formulate this statement including the proof.
Lemma 2 Assume (H). Then any solution(x,y,z,w)of system (S) such that{x}_{n}>0for large n is one of the following types:
Type (a) {x}_{n}>0, {y}_{n}>0, {z}_{n}>0, {w}_{n}>0for all large n,
Type (b) {x}_{n}>0, {y}_{n}>0, {z}_{n}<0, {w}_{n}>0for all large n.
Proof Let (x,y,z,w) be a nonoscillatory solution of (S). Assume that there exists a solution such that {y}_{n}>0, {z}_{n}<0, {w}_{n}<0 for large n. Since \mathrm{\Delta}{z}_{n}<0, there exists k>0 such that {z}_{n}\le k for large n. Using the summation of the second equation of system (S) we get
Passing n\to \mathrm{\infty}, we get lim{y}_{n}=\mathrm{\infty}, which is a contradiction.
Let there exist a solution so that {y}_{n}<0, {z}_{n}>0, {w}_{n}>0 for large n. Since z is positive increasing there exists k>0 so that {z}_{n}\ge k for large n. Summation of the second equation of system (S) leads to lim{y}_{n}=+\mathrm{\infty}, which is a contradiction with the fact {y}_{n}<0.
Let there exist a solution so that {y}_{n}<0, {z}_{n}<0 for large n. Since y is negative decreasing there exists k>0 so that {y}_{n}\le k for large n. By summation of the first equation of system (S) and passing n\to \mathrm{\infty}, we get a contradiction.
The case when {z}_{n}>0 and {w}_{n}<0 for large n can be treated by the similar way by summation of the third equation of (S). □
Proposition 1 Assume (H) and
Then Equation (1) is oscillatory.
Proof In view of Lemma 2 we can assume without loss of generality that {x}_{n}>0, {y}_{n}>0 and {w}_{n}>0. Hence exists k>0 and {n}_{0}>1 such that {x}_{n}\ge k for n\ge {n}_{0}. By summation of the fourth equation of system (S) we find that (9) leads to a contradiction with the positiveness of {w}_{n}. □
Hence, under assumptions (H), if (1) has a nonoscillatory solution, then
We say that a solution x of (1) is of type (a) (type (b)) if the corresponding solution (x,y,z,w) of system (S) is of type (a) (type (b)).
In the next, we give sufficient conditions for the nonexistence of both types of nonoscillatory solutions of (1). To this goal, the following lemma will be used.
Lemma 3 Letk\in (0,1)and\{{w}_{n}\}be a sequence such that{w}_{n}>0and\mathrm{\Delta}{w}_{n}<0. Then
Proof We have
Summing this from N to ∞
□
The nonexistence of solutions of type (a) is ensured by the following conditions.
Lemma 4 Equation (1) has no solution of type (a) if any of the following conditions hold: (i)
(ii)

(iii)
\lambda <\alpha \beta \gamma and
\sum _{n={n}_{0}}^{\mathrm{\infty}}{d}_{n}{\left(\sum _{i={n}_{0}}^{n+\tau 1}\frac{1}{{c}_{i}^{1/\gamma}}{\left(\sum _{j={n}_{0}}^{i1}\frac{1}{{b}_{j}^{1/\beta}}{\left(\sum _{k={n}_{0}}^{j1}\frac{1}{{a}_{k}^{1/\alpha}}\right)}^{1/\beta}\right)}^{1/\gamma}\right)}^{\lambda}=\mathrm{\infty}.(13)
Proof Let (x,y,z,w) be a type (a) solution of system (S), i.e., all components of the solution are positive. Since z is positive increasing, there exists k>0 such that {z}_{n}^{1/\beta}\ge k for large n, say n\ge {n}_{0}. From the first and the second equations of system (S) we get
so
Let (11) or (12) hold. By summation of the fourth equation of system (S) and using (14) we get
Passing n\to \mathrm{\infty} we get the contradiction with the boundedness of w.
Let condition (iii) hold. Taking into account that {w}_{n} is positive and decreasing, we get by summation of the third equation of system (S)
Thus
Hence
Summing this inequality from {n}_{0} to ∞ we have
By Lemma 3 the expression on the left side is finite, which is a contradiction with (13). □
The nonexistence of solutions of type (b) is ensured by the following conditions.
Lemma 5 Let (10) hold. Equation (1) has no solution of type (b) if any of the following conditions hold: (i)

(ii)
T<\mathrm{\infty} and
\sum _{n={n}_{0}}^{\mathrm{\infty}}\frac{1}{{b}_{n}^{1/\beta}}{\left(\sum _{k=n}^{\mathrm{\infty}}\frac{1}{{a}_{k}^{1/\alpha}}{\left(\sum _{i=k}^{\mathrm{\infty}}{d}_{i}\right)}^{1/\alpha}\right)}^{1/\beta}=\mathrm{\infty},(16) 
(iii)
\lambda <\alpha \beta \gamma, T<\mathrm{\infty} and
\sum _{n={n}_{0}}^{\mathrm{\infty}}\frac{1}{{b}_{n}^{1/\beta}}{\left(\sum _{k={n}_{0}}^{n+\tau 1}\frac{1}{{c}_{k}^{1/\gamma}}\right)}^{\lambda /(\alpha \beta )}{\left(\sum _{k=n}^{\mathrm{\infty}}\frac{1}{{a}_{k}^{1/\alpha}}{\left(\sum _{i=k}^{\mathrm{\infty}}{d}_{i}\right)}^{1/\alpha}\right)}^{1/\beta}=\mathrm{\infty}.(17)
Proof Let (x,y,z,w) be a solution of (S) satisfying {x}_{n}>0, {y}_{n}>0, {z}_{n}<0, {w}_{n}>0. Since the components w and −z are positive and decreasing, we have
By summation of the fourth equation of (S) we have
If (i) holds, then by summation of the third equation of (S)
which gives a contradiction with the boundedness of z.
Assume (ii). Then
Using this and the fact that y is positive decreasing, we get
which leads to a contradiction with (16).
Assume (iii). From the first equation of system (S) we have for large n
Thus using the second equation of system (S) and (18)
so
Since \alpha \beta \gamma >\lambda we get by Lemma 3
which is a contradiction. □
Remark 3 The condition (H) is not needed in Lemmas 4 and 5.
Remark 4 (i) Lemmas 4 and 5 can be viewed as a discrete counterpart of similar results for differential systems of the n thorder, see [14], Propositions 4.1, 4.5].
(ii) Oscillation criteria established in [2] are based on a different approach than that applied here, namely by comparing (1) with certain first and secondorder difference equations whose oscillatory properties are known. Comparing conditions for the nonexistence of solutions of types (a) and (b), part (iii) of Lemmas 4 and 5 extends Corollaries 2.2 and 2.1 in [2], respectively, where it is assumed that \tau \le 0 and (H). Moreover, assuming (H), part (ii) of Lemmas 4 and 5 can be obtained from Theorems 2.6 and 2.4 in [2], respectively, but our proofs are different.
Combining conditions in Lemmas 4 and 5, we get oscillation criteria in case where the operator in (1) is in the canonical form. This, together with the application of the cyclic permutation method, will form the content of the following two sections.
Oscillation criteria
In this section, we establish oscillation criteria for (1) which are based on conditions for the nonexistence of the nonoscillatory solutions given in the previous section.
First we discuss conditions (12) and (15). Assume (H), (10) and consider the double series
If P=\mathrm{\infty} and T=\mathrm{\infty}, then by Lemmas 2, 4, 5, Equation (1) with \tau \ge 1 is oscillatory.
In a special case when \alpha =\gamma =\lambda =1 and {a}_{n}={c}_{n} we have P=\mathrm{\infty} if and only if T=\mathrm{\infty}.
The interesting case occurs when \alpha =\lambda \ne 1 or \alpha \ne \lambda. The problem of comparison of conditions (12) and (15) leads to the problem of a change of summation for double series. This problem has been investigated for the case \alpha =\lambda and \alpha \ne \lambda in [17, 18], respectively.
For brevity, denote the following cases of parameters α, λ:
(C1) \alpha >\lambda or \alpha =\lambda \ge 1;
(C2) \alpha <\lambda or \alpha =\lambda \le 1.
Put
In cases (C1) and (C2) the following change of summation holds.

(i)
Assume case (C 1). If S=\mathrm{\infty}, then T=\mathrm{\infty}.

(ii)
Assume case (C 2). If T=\mathrm{\infty}, then S=\mathrm{\infty}.
Remark 5 Observe that the opposite implications in Lemma 6 in general need not hold. For example, choosing
we have S=\mathrm{\infty} and T<\mathrm{\infty} for \lambda \ge 1 and \alpha <1; the opposite case holds for \lambda <1 and \alpha \ge 1.
Using Lemma 6 we obtain the following result.
Theorem 3 Assume (H) and (10). Equation (1) with\tau \ge 1is oscillatory if any of the following conditions holds:

(i)
Case (C 1), P=\mathrm{\infty} and
lim\hspace{0.17em}inf\frac{{c}_{n}^{1/\gamma}}{{a}_{n}^{1/\alpha}}>0;(19) 
(ii)
Case (C 2), T=\mathrm{\infty} and
lim\hspace{0.17em}sup\frac{{c}_{n}^{1/\gamma}}{{a}_{n}^{1/\alpha}}<\mathrm{\infty}.
Proof Claim (i). Clearly, condition P=\mathrm{\infty} implies the validity of (11) for any \tau \ge 1. Hence, by Lemma 4, Equation (1) with \tau \ge 1 has no type (a) solution. By comparison theorem for series and in view of (19), we have S=\mathrm{\infty}. Using Lemma 6 we get T=\mathrm{\infty}. By Lemma 5 Equation (1) has no type (b) solutions. Now, the conclusion follows from Lemma 2. Claim (ii) can be proved by a similar way. □
In the general case, when Theorem 3 cannot be applied, by Lemma 4, part (ii) and Lemma 5, parts (i), (ii) the following result holds.
Theorem 4 ([2], Theorem 2.10])
Assume (H), (10) and\tau \in \mathbb{Z}. If (12) and either (15) or (16) hold, then Equation (1) is oscillatory.
In the sublinear case, this result can be improved using part (iii) of Lemmas 4 and 5 as follows.
Theorem 5 Assume\lambda <\alpha \beta \gamma, (H), (10) and\tau \in \mathbb{Z}. If (13) and either (15) or (17) hold, then Equation (1) is oscillatory.
Remark 6 Theorems 3, 4, 5 can be compared with the results of [5] using coupled system (8). Application of Theorem 1 or Theorem 2^{′} of [5] to system (8) leads to conditions (11), (15) or (13), (15), respectively. Observe that Theorem 4^{′} of [5] ensures oscillation of (8) provided \lambda <1, (13) and certain additional assumptions on α β γ.
The following examples illustrate our results and show that conditions in Theorem 5 are optimal.
Example 2 Consider the equation
where \tau \ge 1 and (10) holds. Then
and by Theorems 3 and 5 we get that Equation (20) is oscillatory if any of the following conditions is satisfied

(i)
\lambda <\alpha or \alpha =\lambda \ge 1, P=\mathrm{\infty};

(ii)
\lambda >\alpha or \alpha =\lambda \le 1, T=\mathrm{\infty};

(iii)
\lambda <\alpha, {\sum}_{n=1}^{\mathrm{\infty}}{n}^{3\lambda}{d}_{n}=\mathrm{\infty}, T<\mathrm{\infty} and
\sum _{n=1}^{\mathrm{\infty}}{n}^{\lambda /\alpha}\sum _{j=n}^{\mathrm{\infty}}{\left(\sum _{k=j}^{\mathrm{\infty}}{d}_{k}\right)}^{1/\alpha}=\mathrm{\infty}.
The claim (iii) of Example 2 is not true for \alpha =\lambda =1 as the next example shows.
Example 3 Consider the Eulertype difference equation
One can check that {x}_{n}=n(n+1)(n+2)(n+3) is a positive solution of (21) and {\sum}_{n=1}^{\mathrm{\infty}}{n}^{3}{d}_{n}=\mathrm{\infty}.
Another oscillation criteria can be obtained using the cyclic permutation described in Lemma 1 and Theorem 2. For instance, in the case when
we can apply Theorems 35 to Equation (R2). We show the application of Theorem 5 to a special case of (1) in the next section.
Applications
Consider equation
where \tau \in \mathbb{Z} and
Then the cyclic permutated Equation (R2) to (22) is
whose difference operator is in the canonical form. In Equation (24), we have \alpha =1, \beta =1/\lambda, \gamma =1, \lambda =1/\beta, thus the condition \lambda <\alpha \beta \gamma reads \lambda <\beta and the series P and T as
Since {lim}_{n\to \mathrm{\infty}}\frac{n+\tau}{n}=1, we have \overline{P}=\mathrm{\infty} if and only if
Similarly, since {lim}_{n\to \mathrm{\infty}}\frac{n+\tau}{n{n}_{0}+1}=1, we get \overline{T}=\mathrm{\infty} if and only if
Observe that if \beta \ge 1 and \tilde{P}=\mathrm{\infty}, then \tilde{T}=\mathrm{\infty}, while if \beta \le 1 and \tilde{T}=\mathrm{\infty}, then \tilde{P}=\mathrm{\infty}. Hence, under these assumptions both series \tilde{P}, \tilde{T} are divergent and Equation (22) with \tau \ge 1 is oscillatory by Theorem 2.
Applying Theorem 5 to Equation (24) and using Theorem 2, we get the following result.
Corollary 1 Assume (23) and\lambda <\beta, \tau \in \mathbb{R}. If
and either \tilde{T}=\mathrm{\infty} or
then Equation (22) is oscillatory.
Remark 7 Corollary 1 completes the oscillation criteria for Equation (22) with \tau =3 given in [10, 12] where instead of the condition \sum {d}_{n}=\mathrm{\infty} it is assumed that both series \tilde{P}\tilde{T} are divergent or convergent, respectively.
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Acknowledgements
Zuzana Došlá was supported by the Grant P201/10/1032 of the Czech Grant Agency, and Jana Krejčová was supported by the Grant MUNI/A/0964/2009. The authors thank the referees for their useful comments.
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Došlá, Z., Krejčová, J. Oscillation of a class of the fourthorder nonlinear difference equations. Adv Differ Equ 2012, 99 (2012). https://doi.org/10.1186/16871847201299
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DOI: https://doi.org/10.1186/16871847201299