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Asymptotic problems for fourth-order nonlinear differential equations
Boundary Value Problemsvolume 2013, Article number: 89 (2013)
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
where , , for large t, such that for large t and .
Jointly with (1), we consider a more general equation
where satisfies for , and the associated linear second-order 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 start with the Kneser problem for (1). The Kneser problem is a problem concerning the existence of solutions of (1) subject to the boundary conditions on the half-line
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).
Proposition 1 Let , (2) be disconjugate on , and for . Then boundary value problem (1), (4) is solvable for any .
To prove this theorem, we use Chanturia’s result [, Theorem 1] for the system of differential equations
where we restrict to the case that are continuous functions, , , , and . Then this result reads as follows.
Theorem A ()
Let there exist such that
for , , (). Suppose
for , , (), where functions () are continuous and nondecreasing in the second argument such that
is a continuous function and is a continuous nondecreasing function such that
Then, for any , system (7) has a solution satisfying
Proof of Proposition 1 Assume for . Since (2) is disconjugate, it has a positive solution h on , and (1) can be written as (5) where on . Let x be a solution of (5) and denote
Then (5) is equivalent to the system
Let be from (4). We apply Theorem A choosing
where , and . By this result, system (10) has a solution such that
Now we state conditions for the existence of a solution for problem (1), (3), (4).
Theorem 1 Let , and
on . If
then problem (1), (3), (4) is solvable for any .
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.
Lemma 1 Consider system (10) on (), where for and h is a principal solution of (2). Let be a solution of (10) such that and for , and . Then for , and if
then , too. Vice versa, if and , then (15) holds.
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
Let (15) hold and assume, by contradiction, that . Then
Letting and using the change of the order of integration, we get a contradiction with the boundedness of . This proves that .
Let the integral in (15) be convergent and assume, by contradiction, that . Then we have
Since , then using the change of the order of integration, we get a contradiction for large t. This proves that . □
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
is the majorant of (2) on and has the principal solution . By the comparison theorem, for the minimal solution of the Riccati equation related to (2) and (17), we have
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.
Corollary 1 Let (2) be disconjugate on , and for . Then any solution x of (1) satisfying
is a solution of the Kneser problem, i.e., for and .
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).
A function g, defined in a neighborhood of infinity, is said to change sign if there exists a sequence such that .
Theorem 2 Let for large t. Then problem (1), (4) has no solution and the following hold:
If (2) is nonoscillatory, then every nonoscillatory solution x of (1) satisfies and is of one sign for large t.
If (2) is oscillatory, then every nonoscillatory solution x of (1) satisfies either , or changes sign. In addition, if a solution x satisfies (3), then changes sign.
Proof Claim (a). Let x be a positive solution of (1) on , or, equivalently, of (5) on , where h satisfies (6). Denote
Then (5) is equivalent to the system
We have for . Assume by contradiction that for . Let and . Since is nonincreasing, and
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
for , , . Thus, and
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
Hence, , which contradicts the nonnegativity of . Finally, the case on cannot occur, because if on , then from (1) and , we have on , which is a contradiction.
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. □
For , the analogous result to Theorem 1 is the following oscillation result.
Proposition 2 Let , for large t. Assume either (11) for large t, (12), or (13), (14). Then all the solutions of (1) are 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.
Proposition 3 Let , and for . Then the condition is necessary and sufficient for every solution of (1) to be 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.
Theorem 3 Let and (2) be oscillatory. Assume that for large t and some , the functions
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])
Assume . Let y be a positive solution of (26) defined on and
on for some constants and . Then there exists a positive constant such that
Remark 1 In [, Theorem 3.4] the constant M is explicitly calculated.
Lemma 2 Let . Assume that (27) holds on . Then any positive solution of (26) defined on satisfies
where α and M are constants from Proposition 4.
Proof Let . By Proposition 4, applied on , we have for and
Letting , we get (29). □
The next lemma describes the transformation between solutions of (1) and a certain quasi-linear equation.
Lemma 3 Let on be such that
and consider the transformation
Then is a solution of equation (1) on if and only if is a solution of the equation
where is the inverse function to .
Proof We have
Substituting into (1), we get the conclusion. □
Proof of Theorem 3 Let x be a positive solution of (1) on (). Suppose, for simplicity, that for . Let
on for some positive constants , , and
Define the function such that
for . Then, according to , we have
Let be nondecreasing on and put . Choose arbitrarily fixed. Since , we can consider the transformation from Lemma 3 with , i.e.,
and Q is defined by (31) and (32). Choose arbitrarily. We apply Lemma 2 to equation (30) with
Hence estimate (29) with reads as
where . Letting , we have by (35) that and the conclusion follows from (25) and (37). □
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.
Corollary 2 Let , , (24) and (25) hold. Then, for any , there exists a positive constant and such that every nonoscillatory solution x of (1) satisfies
Proof Let be fixed and let be such that
where α is given by (33). Let be fixed. Using estimate (37) with , we have
where is given by (34), i.e.,
Therefore and estimate (38) follows from (25) and (39). □
Example 1 Consider the equation
Then and by Theorem 3 all eventually positive solutions are vanishing at infinity. One can check that is such a solution of (40).
Open problem It is an open problem to find conditions for the solvability of boundary value problem (1), (3), (4) in case and (2) is oscillatory.
In view of Theorem 2, Corollary 2 and Proposition 1, it is a question whether (1) can have vanishing at infinity solutions in case and (2) is oscillatory.
In the next section, we show that under certain additional assumptions the answer is negative.
Here we consider (1) in case for large t. When (2) is nonoscillatory, we have established the oscillation criterion in Proposition 2. When (2) is oscillatory, the following oscillation theorem holds.
Theorem 4 Let , and assumptions (24), (25) hold. Assume
for some and . Then problem (1), (3) is not solvable and all the solutions of (1) are oscillatory.
Proof Suppose that (25) holds on . First, observe that the assumption (42) implies that
Indeed, putting , we have
and thus, in view of (42), we get (43). Consider a solution x of (1) such that for . According to Corollary 2, there exists such that
and in view of (25) we get . Consider the function
and in view of (41) the function F is increasing for large t. Hence, there exists such that either
According to Theorem 2(b), oscillates. Define by an increasing sequence of zeros of tending to ∞ with .
In view of (44) and (43) the function Z is well defined and
on . Moreover, we have from (42), (44) and (47)
If (45) holds, then and because
we get for . This is a contradiction with (49), so (45) is impossible.
If (46) holds, then and . This is again a contradiction with (49), so also this case is impossible. □
Example 2 Consider the equation
where . If and , then by Theorem 1 this equation has a solution satisfying (3) and (4). If and , then by Theorem 3 any nonoscillatory solution (if any) satisfies (3).
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
where q, r are as for (1) and for such that
for some and . The prototype of such an extension is the function for .
Theorems 1-4 read for (1′) as follows.
Theorem 1′ Let , and (11) hold for . Assume that either (i) (12), or (ii) (13) and (14) hold. Then problem (1′), (3), (4) has a solution for any .
Proof of Theorem 1′ It is analogous to the proofs of Proposition 1 and Theorem 1 replacing the nonlinearity in system (10) by . Lemma 1 remains to hold as a sufficient condition for (3). □
Theorem 2′ Theorem 2 remains to hold for (1′) without assuming (50).
Proof of Theorem 2′ In the proof of claim (a) of Theorem 2, we consider system (20) where the nonlinearity is replaced by . The proof of claim (b) of Theorem 2 is the same for the nonlinearity f. □
Theorem 3′ Theorem 3 remains to hold for (1′).
Proof of Theorem 3′ Let x be a positive solution of (1′) on . Then is a solution of the equation
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. □
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Supported by the grant GAP 201/11/0768 of the Czech Grant Agency.
The authors declare that they have no competing interests.
Both authors contributed equally to the manuscript and read and approved the final draft.