We study vanishing at infinity solutions of a fourth-order nonlinear differential equation. We state sufficient and/or necessary conditions for the existence of the positive solution on the half-line which is vanishing at infinity and sufficient conditions ensuring that all eventually positive solutions are vanishing at infinity. We also discuss an oscillation problem.
Dedicated to Jean Mawhin on occasion of his seventieth birthday.
In this paper we study the fourth-order nonlinear differential equation
Jointly with (1), we consider a more general equation
By a solution of (1) we mean a function , , which satisfies (1) on . A solution is said to be nonoscillatory if for large t; otherwise, it is said to be oscillatory. Observe that if , according to [, Theorem 11.5], all nontrivial solutions of (1) satisfy for , on the contrary to the case , when nontrivial solutions satisfying for large t may exist.
Fourth-order differential equations have been investigated in detail during the last years. The periodic boundary value problem for the superlinear equation has been studied in . In , the fourth-order linear eigenvalue problem, together with the nonlinear boundary value problem , has been investigated. Oscillatory properties of solutions for self-adjoint linear differential equations can be found in . Equation (1) with can be viewed as a prototype of even-order two-term differential equations, which are the main object of monographs [1,5,6].
Equation (1′) with for is a special case of higher-order differential equations investigated in . Equation (1′) with q near to a nonzero constant as has been considered in  as a perturbation of the linear equation , and the existence of oscillatory solutions of (1′) has been proved. In , necessary and sufficient conditions for the existence of asymptotically linear solutions of (1′) have been given.
In the recent paper , the equation
where , for , and has been investigated and applications to the biharmonic PDE’s can be found there. In particular, the so called homoclinics solutions, which are defined as nontrivial solutions x such that , are studied.
The goal of this paper is to investigate asymptotic problems associated with (1) and the asymptotic boundary condition
A solution x of (1) satisfying (3) is said to be vanishing at infinity.
We establish necessary and/or sufficient conditions for the solvability of the boundary value problem (1), (3), (4). In the light of these results, as the second problem, we study when all eventually positive solutions x of (1) are vanishing at infinity assuming that and (2) is oscillatory. As a consequence, we give a bound for the set of all nonoscillatory solutions. Finally, we discuss when problem (1), (3) is not solvable and solutions to (1) are oscillatory.
A systematic analysis of solutions of (1) satisfying (3) is made according to whether (2) is nonoscillatory or oscillatory. If (2) is nonoscillatory, then the following approach will be used. Equation (1) can be rewritten as the two-term equation
where h is a positive solution of (2). According to , a solution h of (2) is said to be a principal solution if , and such a solution is determined uniquely up to a multiple constant. Since , every eventually positive solution of (2) is nondecreasing for large t. Hence there exists a principal solution h of (2) such that for and
If (2) is oscillatory, then our approach is based on the choice of a suitable transformation. The main idea is based on a transformation of (1) to the fourth-order quasilinear equation and the use of the estimates for positive solutions of such an equation on a compact interval stated in . This, together with an energy function associated with (1), enables us to state an oscillation theorem. In the final section, some extensions of our results to (1′) are given.
2 The Kneser problem
In this section we present necessary and/or sufficient conditions for solvability of boundary value problem (1), (3), (4).
To prove this theorem, we use Chanturia’s result [, Theorem 1] for the system of differential equations
Theorem A ()
Then (5) is equivalent to the system
Now we state conditions for the existence of a solution for problem (1), (3), (4).
In addition, if
then the condition
is necessary and sufficient for the solvability of problem (1), (3), (4).
For the proof, the following lemma will be needed.
Proof In view of the monotonicity of , there exist , . Since h is the principal solution, (6) holds, and integrating the first three equations in (10) from a to t, we get for . Now integrating (10) from t to ∞, we have
Proof of Theorem 1 In view of (11), (2) is disconjugate on . By Proposition 1, equation (1) has a solution x satisfying (4). Therefore, system (10) has a solution such that and for . Choose h in (10) as a principal solution of (2). The Euler equation
for large t; see, e.g., . Thus there exists such that for . Assume (12). Then (15) holds, and by Lemma 1 a solution x satisfies (3).
Assume (13). Then the principal solution h of (2) satisfies for large t (see, e.g., ). Hence, condition (15) reads as (14), and by Lemma 1 this condition is equivalent to the property (3). □
As a consequence of Lemma 1, we get the following result.
Proof Let h be a positive solution on satisfying (6), and let x be a solution of (1) satisfying (18). Then , where are defined by (9), is a solution of system (10). Since for and (6) holds, we have by the Kiguradze lemma (see, e.g., ) that either or for and large t, say for . Since x is positive and tends to zero, we have for , so also () for . By Lemma 1, we get () for . Since for , we have for and is positive and decreasing on . Hence, proceeding by the same argument, () is positive and decreasing on . Now the conclusion follows from (9). □
First we show that the sign condition posed on r is necessary for the solvability of problem (1), (4).
Then (5) is equivalent to the system
Letting , we get a contradiction with the positiveness of . The remaining case can be eliminated in a similar way using (6). Observe that system (20) is a special case of the Emden-Fowler system investigated in , and the proof follows also from [, Lemma 2.1].
Claim (b). Without loss of generality, suppose that for and there exists a solution x of (1) such that and on , . Borůvka  proved that if (2) is oscillatory, then there exists a function , called a phase function, such that and
Using this result, we can consider the change of variables
Substituting into (1), we obtain the second-order equation
From here and (21), we obtain
Since , (22) yields and so , that is, is decreasing. If there exists such that , X becomes eventually negative, which is a contradiction. Then and is nondecreasing. Let be such that on . Thus, using (23) we obtain
Finally, let x be a positive solution of (1) satisfying (3). Then is either oscillatory or for large t. Assume on some , then is decreasing and either or for large t. If for large t, then we get a contradiction with (3). If , then for and x becomes negative for large t. Hence must be oscillatory. □
Proof Let x be a solution of (1) and h be the principal solution of (2). Then , where are given by (19), is a solution of system (20). Proceeding by the similar way as in the proof of Theorem 1, we have that (15) holds. Using the change of the order of integration in (15), we can check that conditions of Theorem 4.3 in  applied to system (20) are verified. Hence by this result all the solutions of (20) are oscillatory, which gives the conclusion. □
The following result follows from [, Theorem 1.5] and completes Proposition 2 in the case when (2) is oscillatory.
In the light of these results, in the sequel, we study asymptotic and oscillation problems to (1) when (2) is oscillatory.
3 Vanishing at infinity solutions
In this section we study when all nonoscillatory solutions of (1) are vanishing at infinity.
Then any eventually positive solution of (1) is vanishing at infinity.
The proof of Theorem 3 is based on the following auxiliary results.
Consider the fourth-order quasi-linear differential equation
where and R are continuous functions on . In [, Theorem 2.4], the following uniform estimate for positive solutions of (26) with a common domain was proved.
Proposition 4 ([, Theorem 3.4, Corollary 3.6])
Remark 1 In [, Theorem 3.4] the constant M is explicitly calculated.
whereαandMare constants from Proposition 4.
The next lemma describes the transformation between solutions of (1) and a certain quasi-linear equation.
and consider the transformation
Proof We have
Substituting into (1), we get the conclusion. □
Then equation (1) is transformed into equation (30) which is a quasilinear equation of the form (26), where
From the proof of Theorem 3, we get the estimate for the set of all nonoscillatory solutions of (1) which will be used in the next section.
Example 1 Consider the equation
In the next section, we show that under certain additional assumptions the answer is negative.
In view of (44) and (43) the function Z is well defined and
Example 2 Consider the equation
As it was mentioned in , a certain nonlinear PDE leads to the fourth-order equation with the exponential nonlinearity. In the sequel, we show that the results of this paper can be extended to the nonlinear equation
Theorems 1-4 read for (1′) as follows.
Theorem 2′Theorem 2 remains to hold for (1′) without assuming (50).
Theorem 3′Theorem 3 remains to hold for (1′).
Now we apply Theorem 3 to (51). □
Theorem 4′Let the assumptions of Theorem 4 hold. Then (1′) has no eventually positive solutions.
Proof of Theorem 4′ It is similar to the one of Theorem 4. In view of (52), the estimate (38) holds and the energy function F is the same. □
The authors declare that they have no competing interests.
Both authors contributed equally to the manuscript and read and approved the final draft.
Supported by the grant GAP 201/11/0768 of the Czech Grant Agency.
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