Asymptotic problems for fourth-order nonlinear differential equations
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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.
KeywordsQuasilinear Equation Homoclinics Solution Nonoscillatory Solution Oscillation Problem Asymptotic Boundary Condition
Dedicated to Jean Mawhin on occasion of his seventieth birthday.
where , , for large t, such that for large t and .
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.
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.
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.
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 .
where we restrict to the case that are continuous functions, , , , and . Then this result reads as follows.
Theorem A ()
Now we state conditions for the existence of a solution for problem (1), (3), (4).
then problem (1), (3), (4) is solvable for any .
is necessary and sufficient for the solvability of problem (1), (3), (4).
For the proof, the following lemma will be needed.
then , too. Vice versa, if and , then (15) holds.
Letting and using the change of the order of integration, we get a contradiction with the boundedness of . This proves that .
Since , then using the change of the order of integration, we get a contradiction for large t. This proves that . □
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.
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 .
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.
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].
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.
Then any eventually positive solution of (1) is vanishing at infinity.
The proof of Theorem 3 is based on the following auxiliary results.
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.
where α and M are constants from Proposition 4.
Letting , we get (29). □
The next lemma describes the transformation between solutions of (1) and a certain quasi-linear equation.
where is the inverse function to .
Substituting into (1), we get the conclusion. □
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.
Therefore and estimate (38) follows from (25) and (39). □
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.
for some and . Then problem (1), (3) is not solvable and all the solutions of (1) are oscillatory.
According to Theorem 2(b), oscillates. Define by an increasing sequence of zeros of tending to ∞ with .
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. □
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).
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′).
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. □
Supported by the grant GAP 201/11/0768 of the Czech Grant Agency.
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