Oscillation for higher order differential equations with a middle term
© Bartušek et al.; licensee Springer. 2014
Received: 25 November 2013
Accepted: 12 February 2014
Published: 26 February 2014
We study the existence of bounded oscillatory solutions for a higher order differential equation, considered as a perturbation of an associated linear equation. Jointly with this, we study the nonexistence of solutions vanishing at infinity and, as an application, we obtain in the linear case an asymptotic equivalency criterion.
Hence, q is bounded from above and . Note that the function r may change sign.
By a solution of (1) we mean a function x differentiable up to order n which satisfies (1) on , . A solution of (1) is said to be proper if for any . As usual, a solution x of (1) is said to be oscillatory if x changes sign for large t.
where c, φ are suitable constants and ε is a continuous function for which vanishes at infinity. According to , in this case (4) is said to have property A′.
The existence of solutions vanishing at infinity, and the closely related problem on existence of bounded oscillatory solutions, has attracted the attention in many papers, see, e.g. the monograph  and references therein. In particular, nonlinear equations with middle term have been investigated in many directions, as a perturbation of an associated linear equation. For third and fourth order equations, we refer especially to [3–5], in which the property A, or its generalizations, has been studied and to [6–10] for oscillation problems. In particular, in  a good and detailed discussion of known oscillation criteria is given as well.
has been considered. In  sufficient conditions are obtained for the existence of solutions, which are equivalent to a polynomial. In  a criterion is given for existence of nonoscillatory solutions with non-zero limit at infinity. Moreover, for n even, an oscillation result is obtained too. In , (1) is studied as a perturbation of (2), under the assumption (H). Finally, the case of higher order equations with forcing term has been considered in the recent papers [14–16], see also the last section.
Observe that if q is a positive constant, then (2) has oscillatory and bounded solutions not vanishing at infinity. If q is not constant and (H) is satisfied, then these properties remain to hold for the second order equation (3), see, e.g. [, Chapter 6], or [, Theorem 2]. Thus, it is natural to ask under which assumptions these properties are valid also for (2) (when q is not constant) and for the more general case (1).
In this paper, we answer both these questions. In particular, our main result yields the existence of oscillatory solutions of (1), which are bounded and not vanishing at infinity. Our results complete recent ones in [, Corollary 1, Corollary 3], extend similar ones in [, Theorem 1.3, Theorem 1.4], which are proved for (4), and generalize ones in [3, 10], which are stated for the particular cases .
In Section 3 we study the problem of the nonexistence of solutions vanishing at infinity of (1). This result will be employed in Section 4 and Section 5, namely to prove the existence of oscillatory solutions of (2) (Section 4) and the uniqueness of solutions of (1), which have the same asymptotics as solutions of (2) (Section 5).
More precisely, we will give conditions under which (2) and (5) are asymptotically equivalent. Some suggestions for further researches complete the paper.
We start with some basic properties of solutions of (3), which will be useful in the sequel. Obviously, assumption implies that (3) is oscillatory. Moreover, the following holds.
Proof Since q is of bounded variation for , all solutions of (3) are bounded together with their derivatives, see e.g. [, Theorem 2]. Moreover, the function q can be represented as , where a, b are positive nonincreasing and differentiable functions such that and , for details see [, Lemma 5.4.1]. Hence, taking such that and applying [, Lemma 2], we get the assertion. □
Then w and ζ are bounded in .
and u, v are two independent solutions of (3). The question whenever the sets (7) and (8) are, roughly speaking, close as is considered in Section 5.
The following holds.
Theorem A ([, Theorem 1])
where are functions of bounded variation for large t and , .
for some , then every nontrivial solution of (1) is proper. This follows from [, Theorem 11.5] with and for .
3 Solutions vanishing at infinity
Our main result in this section is concerned with the nonexistence of solutions vanishing at infinity.
Proof If for large t, then the assertion holds. Thus, assume that for large t.
where . If , then (26) holds by (25).
Hence, z is well defined. A direct computation shows that z is a solution of (15).
which yields (17). □
Remark 2 Under the stronger assumption , Lemma 2 follows with a standard calculation.
which gives the assertion, because . □
The next auxiliary result is a Gronwall type lemma, which proof is elementary and so it is omitted.
In particular, if , then for each .
Proof of Theorem 1 Assume that there exists a nontrivial solution x of (1) defined on and satisfying (14).
where , are suitable constants.
We claim that . Indeed, if , then is nontrivial solution of (3) and applying Lemma 1 we get a contradiction.
By Lemma 4, we find that x is identically zero for large t. Since , (34) and (35) imply (12), by Remark 1 every nontrivial solution x of (1) is proper and so x is identically zero for . The proof is complete. □
Remark 3 If , , then the assumption in Theorem 1 is satisfied. If , then and Theorem 1 is not valid. The following example illustrates that assumptions of Theorem 1 are optimal.
where and .
If , then , i.e. condition (13) is satisfied, however, we have . If , condition (13) is not satisfied.
One can check that in both cases is a solution vanishing at infinity of (36). This shows the strictness of both assumptions of Theorem 1, that is, and (13).
4 Oscillation in the linear case
In this section we prove the existence of oscillatory solutions of (2), which are bounded and not vanishing at infinity.
To prove this theorem, we give asymptotic expressions of the integrals in (8).
where ε is a continuous function on and .
When , the assertion follows from (38) with . Similarly, when , the assertion follows from (39) with .
in virtue of Lemma 1 and (37), we get .
where are suitable constants and is a continuous function on and .
Choosing in (38), from (41) the assertion follows for n odd, .
where are suitable constants and is a continuous function on and .
Choosing in (39), from (42) the assertion follows. □
Proof of Theorem 2 By Lemma 5, (2) has a solution if n is odd, and if n is even, where . According to Lemma 1 this solution is oscillatory, bounded, and not vanishing at infinity. □
5 Oscillation in the nonlinear case
Our main results are given by the following.
Theorem 3 Let and u be a nontrivial solution of (3).
where ε is a continuous function on and .
Proof By Theorem 2, (2) has an oscillatory solution ϕ, which is bounded and not vanishing at infinity. Applying Theorem A with , (1) has a solution with the same asymptotic properties as that one of (2). □
where the function is defined in the Preliminaries.
where ε is a continuous function on and .
then the solution x given by (45) is unique.
Proof Existence. As noticed above, the set (8) is a basis for the space of solutions of (2). Applying Theorem A with and Theorem 3, (1) has a solution x satisfying (45).
where , for and a suitable constant .
where . Hence we can apply Theorem 1 to (48), which gives a contradiction with (47). □
Thus, according to Theorem 4, for a fixed vector there exists a unique solution of (49) which has the asymptotic representation (45).
Now, consider (49), where n is even, for and . By [, Corollary 1.6], if , then every solution of (49) is oscillatory. Obviously, condition (37) is satisfied. Therefore, the condition (43) in Theorem 3 is optimal.
From this and Theorem 3 we get the following.
Corollary 1 Let , n even, and for . Then (49) has oscillatory solutions.
6 Asymptotic equivalence of linear equations
In this section we present another consequence of our results.
and denote by and the solution space of (5) and (2), respectively.
Applying Theorem A and Theorem 1 we get the following.
Then (2) and (5) are asymptotically equivalent.
Proof As we noticed above, functions , and are linearly independent solutions of (2).
are solutions of (5). Then is also a solution of (5) and tends to zero as . This is a contradiction with Theorem 1. For we proceed by the same way.
and , we have . Proceeding in the same way we get for .
Since are uniquely determined, there exists a 1-1 map between the sets and and so the assertion follows. □
Remark 5 The assumption (37) is not needed in Theorem 5.
7 Open problems
Does Corollary 1 hold for n even and ?
Let q be bounded, but not of bounded variation on . Then (3) can have unbounded oscillatory solutions, see, e.g., [, Chapter VI-5, Theorem 3]. Similarly, if (3) is oscillatory and , then (3) can have again unbounded oscillatory solutions, as the Euler equation illustrates. In these cases, it should be interesting to find conditions under which (1) has unbounded oscillatory solutions too.
Let q be unbounded. Thus, as the Armellini-Tonelli-Sansone theorem shows, (3) can have (oscillatory) solutions which tend to zero as , see  for a detailed survey on this topic. In particular, in [20, 21] the boundedness and the existence of vanishing at infinity solutions are investigated for second order linear equations with advanced arguments. As before, also in this situation, it should be an interesting problem studying asymptotic properties of possible oscillatory solutions of (1). Note that for (1) with , conditions which ensure that all solutions are oscillatory, can be found in [6, 7] or in [, Theorem 4].
- (4)Let g be a continuous function on ℝ such thatUnder assumptions (H), any solution of the nonlinear equationis bounded together with its first derivative, see, e.g., [, Theorem 4]. Thus, motivated by the results here obtained, we can ask under which conditions the nonlinear equation
has oscillatory bounded solutions.
- (5)(5)Recently, oscillation of equations with a forcing term have been studied. For example, the boundedness of any solutions is studied in  and the periodic case in . Moreover, in  a two-term equation with forcing term e has been considered and the oscillation is studied under additional conditions on the function r. Thus, it seems interesting to extend our study to the existence of oscillatory solutions and solutions vanishing at infinity for the equation with the forcing term
The first and second authors were supported by the grant GAP201/11/0768 of the Czech Grant Agency. The authors thank both referees for their valuable comments to the paper.
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