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Oscillation for higher order differential equations with a middle term
Boundary Value Problems volume 2014, Article number: 48 (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.
Consider the higher order nonlinear differential equation with a middle term
where and . Throughout the paper, we assume that such that for , , , and
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.
The aim of this paper is to study (1) as a perturbation of the linear differential equation
An important role in our approach is played by the relationship between solutions of (2) and those of the second order linear equation
This approach is mainly motivated by previous results, obtained by Kiguradze in  for the special case , i.e., for the equation
It was shown in  that, if r is positive and sufficient large in some sense, then for n even every proper solution of (4) is oscillatory. Moreover, for n odd every proper solution of (4) is oscillatory, or is vanishing at infinity together with its derivatives, or admits the asymptotic representation
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).
We close the paper with an application that concerns the influence of the perturbing term r on the change of the oscillatory character, passing from (2) to the linear equation
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.
Lemma 1 Let (H) hold and let u be a nontrivial solution of (3). Then u is oscillatory and
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. □
Let u, v be two linearly independent solutions of (3) with Wronskian equal to 1. Set
Then w and ζ are bounded in .
In the general case, that is, when q is not constant, it is easy to see that a basis of the space of solutions of (2) is given by
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.
We close this section by recalling the main result from , which plays a crucial role in our further consideration. Let
The symbol as means, as usual, that there exists a constant M such that in a neighborhood of infinity
The following holds.
Theorem A ([, Theorem 1])
Assume . Let for any positive constant μ and for some
Then, for any solution y of (2) such that as , there exists a solution x of (1) such that for large t
where are functions of bounded variation for large t and , .
Remark 1 If f satisfies in a neighborhood I of zero
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.
Theorem 1 Let , and
Then (1) does not have nontrivial solutions x (oscillatory or nonoscillatory) satisfying
To prove Theorem 1, some lemmas will be needed. The first result concerns the linear nonhomogeneous equation
Lemma 2 Assume . Let satisfy
where w is defined by (6), is a solution of (15). Moreover, there exists such that for
Proof If for large t, then the assertion holds. Thus, assume that for large t.
First, we prove that z is well defined. Choose large so that
Let m be a constant such that , for . From (16) we have
Hence, for fixed the function
is well defined for . In particular, we have
Consider the function
Integrating this equality on , we get
From this, (20), and (21), we obtain
or, from (18),
In view of (16), we have or according to or , respectively. Moreover,
Hence, from (21) we obtain
For we have and
From (22), in case using (23) and letting , and in case using (24) we get
where . Thus
When , since for , using (25) we get
From this, taking into account , we obtain
where . If , then (26) holds by (25).
Now, for the sake of clarity, denote by the function α given in (19). Fix . In virtue of we have
where , . Thus for large so that
and from (26) we obtain
for any , . Using the Cauchy criterion, in virtue of (26), there exists the finite limit
Hence, z is well defined. A direct computation shows that z is a solution of (15).
It remains to prove that z satisfies the estimation (17). Using (26) we have
which yields (17). □
Remark 2 Under the stronger assumption , Lemma 2 follows with a standard calculation.
Lemma 3 Let and , , be such that
Proof Let . Since y and are bounded, we can apply [, Lemma 5.2] with and we obtain
where and , are suitable positive constants. Passing b to ∞, there exists a positive constant such that
which gives the assertion, because . □
The next auxiliary result is a Gronwall type lemma, which proof is elementary and so it is omitted.
Lemma 4 Let g and Ψ be nonnegative continuous functions for such that Ψ and Ψg belong to . If
for some nonnegative constant A, then
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).
Let u, v be linearly independent solutions of (3) with Wronskian 1. By Lemma 1,
Hence, , defined by (6), is bounded. Put . Then h is a solution of the second order equation
where . Since , from (14) and (28), we obtain
Taking into account (28), (29) and using the variation constant formula, we have
where , are suitable constants.
From this and (28), we see that is bounded on . By Lemma 3 we obtain
Hence, from (30)
We claim that . Indeed, if , then is nontrivial solution of (3) and applying Lemma 1 we get a contradiction.
Therefore, (30) implies
Integrating times and using (32), we get
for . Since , in view of the mean value theorem, there exists a function such that
and . We have for
thus, because , the function is continuous and there exists such that
Thus, from (33), we obtain
Hence, according to Lemma 2 with we get for large t
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.
Example 1 Consider the third order equation
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.
Theorem 2 Let , u be a nontrivial solution of (3) and
Then (2) has an oscillatory solution ϕ such that
where ε is a continuous function on and . In particular,
To prove this theorem, we give asymptotic expressions of the integrals in (8).
Lemma 5 Let and (37) hold. If u is a nontrivial (oscillatory) solution of (3), then there exist constants , , , and a function ε such that
where ε is a continuous function on and .
Proof In view of (H), we have , . Let μ be an integer, . Using (3) we get
Similarly, let ν be an integer, . Using (3) we have
When , the assertion follows from (38) with . Similarly, when , the assertion follows from (39) with .
Now, let . Let j be an integer and . Fixed and integrating by parts, we obtain
Applying Lemma 1 and (37), we have
where are constants and
in virtue of Lemma 1 and (37), we get .
Let n be odd, . By using recursively (40) with
we obtain the following estimation for the function given by (9):
where are suitable constants and is a continuous function on and .
Choosing in (38), from (41) the assertion follows for n odd, .
Finally, let n be even, . By using a similar argument to the one above given and applying recursively (40) with
we obtain the following estimation for the function :
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).
Assume (37) and
Then for any real numbers , , (1) has a solution x, defined on , , such that
where ε is a continuous function on and .
Consequently, (1) has oscillatory solutions x such that
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). □
Theorem 4 Let and u, v be two linearly independent solutions of (3). Assume (37) and for any positive constant μ
where the function is defined in the Preliminaries.
Then for any vector there exists a solution x of (1), defined on , , such that
where ε is a continuous function on and .
If, in addition, and there exists such that
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).
Uniqueness. For the sake of simplicity, let n be even. The case n odd follows in a similar way. Suppose, by contradiction, that for there exist two different solutions x and of (1) satisfying (45). Then for we have
In view of the mean value theorem, there exists a function such that
Then we have
Therefore is a solution of
where , for and a suitable constant .
Let (46) hold for . Then, in view of (44), the condition (13) for (48) is satisfied, because
where . Hence we can apply Theorem 1 to (48), which gives a contradiction with (47). □
An application of Theorems 3 and 4 is the following. Consider the Emden-Fowler type equation
where . Then (46) is satisfied for any and (44) reads
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.
Consider the linear equation
and denote by and the solution space of (5) and (2), respectively.
We say that (2) and (5) are asymptotically equivalent, if there exists a map such that for every there exists a unique such that and
Applying Theorem A and Theorem 1 we get the following.
Theorem 5 Assume and
Then (2) and (5) are asymptotically equivalent.
Proof As we noticed above, functions , and are linearly independent solutions of (2).
By Theorem A, there exist functions , , which tend to zero as , such that
are solutions of (5). Hence, applying again Theorem A, we get
We show that are uniquely determined. Without loss of generality, assume by contradiction that there exist and such that and
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.
A standard argument shows that solutions , , are linearly independent. Set
and let us prove that implies that is the zero vector in . From (53) we obtain
From this, (52) with and Lemma 1, we get . Therefore,
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].
Let g be a continuous function on ℝ such that
Under assumptions (H), any solution of the nonlinear equation
is 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)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
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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.
The authors declare that they have no competing interests.
All authors contributed equally to the manuscript and read and approved the final draft.