Abstract
In this paper, the signchanging solution of a thirdorder twopoint boundaryvalue problem is considered. By calculating the eigenvalues and the algebraic multiplicity of the linear problem and using a new fixed point theorem in an ordered Banach space with lattice structure, we give some conditions to guarantee the existence for a signchanging solution.
Keywords:
thirdorder boundaryvalue problem; signchanging solution; Green’s function; fixed point; lattice1 Introduction
In this paper, we consider the following nonlinear thirdorder twopoint boundaryvalue problem
where
The study on the existence of the signchanging solutions for the boundaryvalue problem is very useful and interesting both in theory and in application. Recently, there has been much attention focused on the problem, especially to the twopoint or multipoint boundaryvalue problem. For the secondorder twopoint or multipoint boundaryvalue problem, many beautiful results have been given on the existence and multiplicity of the signchanging solutions (see [15] and the references therein). For example, Xu and Sun [1] obtained an existence result of the signchanging solutions for the secondorder threepoint boundaryvalue problem
where
where
where
For the thirdorder boundaryvalue problem, the existence and multiplicity of solutions have also been discussed in many papers (see [611] and the references therein). However, the research on the signchanging solutions has been proceeded slowly. For the problem (1.1), Yao and Feng [10,11] established several existence results for the solutions including the positive solutions using the lower and upper solutions and a maximum principle, respectively. To our knowledge, however, there are fewer papers considered the signchanging solutions of the problem (1.1). Motivated by the work mentioned above, using the eigenvalues of linear operator, we give an existence result for the signchanging solutions of the problem (1.1).
The main contribution of this paper are as follows: (a) for the signchanging solutions of the problem (1.1), to our knowledge, there is no result using the eigenvalues of the linear operator until now; (b) we obtain the eigenvalues and the algebraic multiplicity of the linear problem corresponding the problem (1.1), which is one of the key points that we can use to prove our main result; (c) some conditions are given to guarantee the existence for a signchanging solution of the problem (1.1).
2 Notations and preliminaries
The following results will be used throughout the paper.
Let
Let the operators K, F, A be defined by
and
Remark 1 (1)
Definition 2.1[12]
We call E a lattice under the partial ordering ≤, if
Remark 2
Definition 2.2[12]
Let E be a Banach space with a cone
L is said to be the derived operator of A along
Definition 2.3[12]
Let
where
Remark 3 It is easy to see that the operators F and
Let us list some conditions and preliminary lemmas to be used in this paper.
(H_{1})
(H_{2})
where
(H_{3})
Lemma 2.1For any
if and only if
where
Proof On the one hand, integrating the equation
over
Then
Combining them with boundary condition
Therefore,
On the other hand, since
therefore,
and
Moreover, we get
Remark 4 Considering Lemma 2.1, we find that u is a solution of the problem (1.1) if and only if u is a fixed point of the operator
From the following lemma, we can obtain the eigenvalues and the algebraic multiplicity of the linear operator K.
Lemma 2.2The eigenvalues of the linear operatorKare
and the algebraic multiplicity of each positive eigenvalue
Proof Let
The auxiliary equation of the differential equation (2.4) has roots −μ,
Then
Applying the condition
Applying the second condition
Considering (2.3), we see that μ is one of
are eigenvalues of the linear operator K and the eigenfunction corresponding to the eigenvalue
where C is a nonzero constant.
Next we prove that the algebraic multiplicity of the eigenvalue
Now we show that
Obviously, we only need to show that
In fact, for any
By direct computation, we have
It is easy to see that the solution for the corresponding homogeneous equation of (2.7) is of the form
Then, by an ordinary differential equation method, we see that the general solution of (2.7) is of the form
where
is the special solution of the equation
and
is the special solution of the equation
Then
Applying the condition
From (2.5), we have
which implies that
That is
which is a contradiction of
Therefore, the algebraic multiplicity of the eigenvalue
Lemma 2.3Suppose that (H_{1}) holds and
Proof The proof is obvious. □
Lemma 2.4Suppose that (H_{1})(H_{3}) hold. Then the operatorAis Fréchet differentiable atθand ∞, and
Proof Since (H_{3}):
From (H_{1}), it is easy to see that
Then
Thus,
which means
Since (H_{2}):
Let
Thus,
Then
Therefore,
Remark 5 Suppose (H_{2}) holds. Similar to Lemma 2.4, we have
Lemma 2.5[13]
Suppose thatEis an ordered Banach space with a lattice structure, Pis a normal solid cone inE, and the nonlinear operatorAis quasiadditive on the lattice. Assume that
(i) Ais strongly increasing onPand −P;
(ii) both
(iii)
(iv) the Fréchet derivative
ThenAhas at least three nontrivial fixed points containing one signchanging fixed point.
3 Main result
We state the main result of this paper.
Theorem 3.1Suppose that (H_{1})(H_{3}) hold. Then the problem (1.1) has at least three solutions including a signchanging solution.
We need only to prove that
Proof Noticing
(i) A is strongly increasing on P and −P. In fact, from (H_{1}) and
(ii) From
(iii) From
(iv) Since
Therefore, from Lemma 2.5, we see that A has at least three nontrivial fixed points including one signchanging fixed point. Then, the problem (1.1) has at least three solutions, including one signchanging solution. □
Example 3.1 Consider the following thirdorder boundaryvalue problem
where
By simple calculations, we have
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.
Acknowledgements
The authors are highly grateful for the referees’ careful reading and comments on this paper. The research is supported by Program for Scientific research innovation team in Colleges and universities of Shandong Province, the Doctoral Program Foundation of Education Ministry of China (20133705110003), the Natural Science Foundation of Shandong Province of China (ZR2011AM008, ZR2012AM010, ZR2012AQ024).
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