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Oscillation of a class of the fourth-order nonlinear difference equations
Advances in Difference Equations volume 2012, Article number: 99 (2012)
In this article, a class of fourth-order difference equations with quasi-differences 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 four-dimensional difference system and on the cyclic permutation of the coefficients in the difference equations.
In this article, we consider a class of fourth-order nonlinear difference equations of the form
where α, β, γ, λ are the ratios of odd positive integers, is a deviating argument and , , , are positive real sequences defined for , is a positive integer, and Δ is the forward difference operator defined by .
By a solution of Equation (1) we mean a real sequence satisfying Equation (1) for . A non-trivial solution of (1) is said to be non-oscillatory 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.
If and , then (1) takes the form
Equation (1) is a special case of nonlinear fourth-order equation with deviating argument investigated in the recent articles [1, 2]. In , necessary and sufficient conditions for the oscillation of all bounded solutions of (1) (the so-called B-oscillation) have been given. In , oscillation criteria for (1) have been established using the analysis of non-oscillatory solutions and by comparison with certain first- and second-order difference equations.
Equation (1) with can be seen as a coupled system of two second-order 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 .
Our approach here is to consider (1) as a four-dimensional system. If
then Equation (1) can be written as the nonlinear system
Obviously, if 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 four-dimensional differential system investigated by Kusano et al., and by Chanturia . In that article, the oscillation of the n-dimensional 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  we have used this approach to study Property B for system (S) assuming .
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 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 non-existence of non-oscillatory solutions and on the change of summation for double series. Due to our approach considering (1) as a four-dimensional system, we extend for any some results of  stated for a delay . 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 be a quickly oscillatory solution of (1). Then
From the first equation of system (S) we have
where . From the second equation of (S) we obtain
where . Repeating argument, we get from the third equation of (S)
where . 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 . Indeed, , , , and the value of follows from the relation .
In this section, we describe the left-ordered cyclic permutation of coefficients in (1).
Lemma 1 The following statements are equivalent:
x is a solution of (1).
, where , is a solution of(R1)
, where , is a solution of(R2)
, where is a solution of(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 for.
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 .
Remark 2 The cyclic permutation for the coupled system (3) means that equations in (3) are considered in the opposite order. Hence, is a solution of (3) if and only if is a solution of the system
which is again system of the form (3). Oscillation results of  for (3) assume
Throughout this and the next sections, we use the convention
The aim of this section is to study non-oscillatory solutions of (1). If (S) has a solution , then is a solution of (S), too. Hence, when studying the non-existence conditions for non-oscillatory solutions, we can consider solutions such that for large n.
We start with the classification of non-oscillatory solutions of (S). This has been presented in  without the proof, so we formulate this statement including the proof.
Lemma 2 Assume (H). Then any solutionof system (S) such thatfor large n is one of the following types:
Type (a) , , , for all large n,
Type (b) , , , for all large n.
Proof Let be a non-oscillatory solution of (S). Assume that there exists a solution such that , , for large n. Since , there exists such that for large n. Using the summation of the second equation of system (S) we get
Passing , we get , which is a contradiction.
Let there exist a solution so that , , for large n. Since z is positive increasing there exists so that for large n. Summation of the second equation of system (S) leads to , which is a contradiction with the fact .
Let there exist a solution so that , for large n. Since y is negative decreasing there exists so that for large n. By summation of the first equation of system (S) and passing , we get a contradiction.
The case when and 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 , and . Hence exists and such that for . By summation of the fourth equation of system (S) we find that (9) leads to a contradiction with the positiveness of . □
Hence, under assumptions (H), if (1) has a non-oscillatory solution, then
We say that a solution x of (1) is of type (a) (type (b)) if the corresponding solution of system (S) is of type (a) (type (b)).
In the next, we give sufficient conditions for the non-existence of both types of non-oscillatory solutions of (1). To this goal, the following lemma will be used.
Lemma 3 Letandbe a sequence such thatand. Then
Proof We have
Summing this from N to ∞
The non-existence 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)
Proof Let be a type (a) solution of system (S), i.e., all components of the solution are positive. Since z is positive increasing, there exists such that for large n, say . From the first and the second equations of system (S) we get
Let (11) or (12) hold. By summation of the fourth equation of system (S) and using (14) we get
Passing we get the contradiction with the boundedness of w.
Let condition (iii) hold. Taking into account that is positive and decreasing, we get by summation of the third equation of system (S)
Summing this inequality from to ∞ we have
By Lemma 3 the expression on the left side is finite, which is a contradiction with (13). □
The non-existence 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)
Proof Let be a solution of (S) satisfying , , , . 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)
Since 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 th-order, see , Propositions 4.1, 4.5].
(ii) Oscillation criteria established in  are based on a different approach than that applied here, namely by comparing (1) with certain first- and second-order 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 , respectively, where it is assumed that and (H). Moreover, assuming (H), part (ii) of Lemmas 4 and 5 can be obtained from Theorems 2.6 and 2.4 in , 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.
In this section, we establish oscillation criteria for (1) which are based on conditions for the non-existence of the non-oscillatory solutions given in the previous section.
First we discuss conditions (12) and (15). Assume (H), (10) and consider the double series
If and , then by Lemmas 2, 4, 5, Equation (1) with is oscillatory.
In a special case when and we have if and only if .
The interesting case occurs when or . 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 and in [17, 18], respectively.
For brevity, denote the following cases of parameters α, λ:
(C1) or ;
(C2) or .
In cases (C1) and (C2) the following change of summation holds.
Assume case (C 1). If , then .
Assume case (C 2). If , then .
Remark 5 Observe that the opposite implications in Lemma 6 in general need not hold. For example, choosing
we have and for and ; the opposite case holds for and .
Using Lemma 6 we obtain the following result.
Theorem 3 Assume (H) and (10). Equation (1) withis oscillatory if any of the following conditions holds:
Case (C 1), and(19)
Case (C 2), and
Proof Claim (i). Clearly, condition implies the validity of (11) for any . Hence, by Lemma 4, Equation (1) with has no type (a) solution. By comparison theorem for series and in view of (19), we have . Using Lemma 6 we get . 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 (, Theorem 2.10])
Assume (H), (10) and. 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, (H), (10) and. 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  using coupled system (8). Application of Theorem 1 or Theorem 2′ of  to system (8) leads to conditions (11), (15) or (13), (15), respectively. Observe that Theorem 4′ of  ensures oscillation of (8) provided , (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 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
or , ;
or , ;
, , and
The claim (iii) of Example 2 is not true for as the next example shows.
Example 3 Consider the Euler-type difference equation
One can check that is a positive solution of (21) and .
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 3-5 to Equation (R2). We show the application of Theorem 5 to a special case of (1) in the next section.
Then the cyclic permutated Equation (R2) to (22) is
whose difference operator is in the canonical form. In Equation (24), we have , , , , thus the condition reads and the series P and T as
Since , we have if and only if
Similarly, since , we get if and only if
Observe that if and , then , while if and , then . Hence, under these assumptions both series , are divergent and Equation (22) with 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, . If
and either or
then Equation (22) is oscillatory.
Remark 7 Corollary 1 completes the oscillation criteria for Equation (22) with given in [10, 12] where instead of the condition it is assumed that both series are divergent or convergent, respectively.
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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.
The authors declare that they have no competing interests.
Both authors contributed equally to the manuscript and read and approved the final draft.