- Research Article
- Open Access
Structure of Eigenvalues of Multi-Point Boundary Value Problems
© Jie Gao et al. 2010
- Received: 7 January 2010
- Accepted: 29 March 2010
- Published: 9 May 2010
The structure of eigenvalues of , , and , will be studied, where , , and . Due to the nonsymmetry of the problem, this equation may admit complex eigenvalues. In this paper, a complete structure of all complex eigenvalues of this equation will be obtained. In particular, it is proved that this equation has always a sequence of real eigenvalues tending to . Moreover, there exists some constant depending on , such that when satisfies , all eigenvalues of this equation are necessarily real.
- Entire Function
- Dirichlet Problem
- Implicit Function Theorem
- Negative Eigenvalue
- Real Eigenvalue
In the recent years, multi-point boundary value problems of ordinary differential equations have received much attention.Some remarkable results have been obtained,especially for the existence and multiplicity of (positive) solutions for nonlinear second-order ordinary differential equations [1–10]. However, as noted in [5, 6], although it is important in many nonlinear problems, the corresponding eigenvalue theory for linear problems is incomplete. The main reason is that the linear operators are no longer symmetric with respect to multi-point boundary conditions.
As usual, is called an eigenvalue of (1.1) and (1.2) if (1.1) has a nonzero complex solution satisfying conditions of (1.2). The set of all eigenvalues of problem (1.1) and (1.2) is denoted by called the spectrum.
Problem (1.1)–(1.4) is symmetric and has only real eigenvalues [11, 12]. However, in case , problem (1.1) and (1.2) is not symmetric, thus may contain nonreal eigenvalues. A simple example is given by Example 2.1.
Eigenvalues of problem (1.5)–(1.2) can be analyzed using elementary method, because all solutions of (1.5) can be found explicitly. However, as far as the authors know, even for this simple eigenvalue problem, the spectrum theory is incomplete in the literature. In [5, 6], Ma and O'Regan have constructed allreal eigenvalues of problem (1.5)–(1.2) when all are rational, and satisfies certain nondegeneracy condition. In [8, 9], Rynne has obtained all real eigenvalues for general . See  for further extension.
The main topic of this paper is the structure of . Much attention will be paid to the real eigenvalues due to important applications in nonlinear problems.
For , the norm is . For , the norm is denoted by . With some restrictions on , we are able to prove that contains only real eigenvalues.
If satisfies , then the spectrum contains at most finitely many nonreal eigenvalues.
Given , there exists some constant , depending on the norm only, such that if satisfies , then one has .
Basically, eigenvalues of are zeros of some entire functions. See (2.24) and (3.3). In order to study the distributions of eigenvalues, we will consider as a perturbation of or of . To obtain the existence of infinitely many real eigenvalues as in Theorem 1.2, some properties of almost periodic functions [14, 15] will be used. See Lemmas 2.3 and 3.2. In order to pass the results of to general potentials , many techniques like implicit function theorem and the Rouché theorem will be exploited. Moreover, some basic estimates in  for fundamental solutions of (1.1) play an important role, especially in the proofs of Theorems 1.3 and 1.4. Due to the non-symmetry of problem , the proofs are complicated than that in  where the Dirichlet problem is considered.
The paper is organized as follows. In Section 2, we will give some detailed analysis on problem . In Section 3, after developing some basic estimates, we will prove Theorems 1.1 and 1.2. In Section 4, we will develop some techniques to exclude nonreal eigenvalues and complete the proofs of Theorems 1.3 and 1.4. Some open problem on the spectrum of will be mentioned.
In order to motivate our consideration for with non-zero potentials , in this section we consider the spectrum with the zero potential.
2.1. An Example of Nonreal Eigenvalues
where and . We consider the eigenvalue problems (1.5)–(2.1).
Equation (2.6) shows that always contains positive eigenvalues , .
Results (2.10) and (2.11) show that to guarantee that contains only real eigenvalues, some restrictions on parameters are necessary.
2.2. Real Eigenvalues with General Parameters
To study and , let us prove some properties on almost periodic functions.
In particular, has a sequence of positive zeros tending to .
Now result (2.17) can be deduced simply from (2.15) and (2.16).
For real eigenvalues of problem , we have the following result.
The function is a non-zero, almost periodic function and has mean value . In fact, is quasiperiodic. By Lemma 2.3(ii), has infinitely many positive zeros tending to . See Figure 1. Hence contains a sequence of positive eigenvalues tending to .
Thus (2.29) has at most finitely many positive solutions. Hence contains at most finitely many negative eigenvalues.
The quasi-periodic function is as in Figure 1.
2.3. Nonexistence of Nonreal Eigenvalues
With some restrictions on , we will prove that consists of only real eigenvalues.
Then contains only real eigenvalues. Moreover, one has .
When , problem (1.5)–(1.2) is the Dirichlet problem and . In the following, assume that .
which is impossible under assumption (2.32). Thus and therefore .
because and . By (2.24), . Hence we have .
Hence (2.28) shows that .
Condition (2.32) is sharp. For example, let and . Example 2.1 shows that contains nonreal eigenvalues if . Similarly, by letting and , one can verify that contains nonreal eigenvalues when .
3.1. Basic Estimates
uniformly in .
The uniform limits in (3.4) are evident.
For the function of (2.28), one has the following result on its amplitude.
Then . The construction for is as follows. By (2.15), one has some such that . By letting in (2.16), we have some such that . Then, by letting in (2.15), we have some such that . Inductively, we can use (2.15) and (2.16) to find a sequence such that , and (3.6) is satisfied for all .
Lemma 3.3 (basic estimates, [11, page 13, Theorem ]).
This is why the authors of  have used these terms in (3.8)–(3.11).
This gives (3.13).
One has on . Consequently, there exists such that .
Hence , a contradiction with (3.6).
3.2. Eigenvalues with General Parameters
The most general results on spectrum of are stated as in Theorem 1.1.
Proof of Theorem 1.1.
Since is a compact linear operator, one sees that this happens when and only when , where is the spectrum of . Hence consists of a sequence of eigenvalues which can accumulate only at infinity of .
This is impossible because we have the uniform limits (3.4).
Combining (i) and (ii), we know that can be listed as in (1.6).
Though problem is not symmetric, always contains infinitely many real eigenvalues, as stated in Theorem 1.2.
Proof of Theorem 1.2.
Hence (3.3) has at least one positive solution in each interval , . Combining with Theorem 1.1, consists of a sequence of real eigenvalues tending to . Hence can be listed as in (1.7).
We will apply the Rouché theorem to give further results on when is small, following the approach in  for the Dirichlet problem (1.1)-(1.4), which corresponds to with . Let us recall the Rouché theorem.
Lemma 4.1 (Rouché theorem).
Suppose that are entire functions of . If on a Jordan curve , then and have the same number of zeros inside , counted multiplicities.
Numerically, . The following facts can be verified by elementary arguments.
4.1. Large Eigenvalues
Due to the form of (4.7) and (4.8), one needs to only consider solutions in the right half-plane of . Notice that all solutions of (4.8) are , , which are simple zeros of . For any , we do not know whether all zeros of (4.7) are real. In order to overcome this, the proof is complicated than that in .
where is as in (3.14). Then is decreasing in .
Using the function in (4.9), we obtain . This proves (4.11).
Thus one has (4.14).
Proof of Theorem 1.3.
Since , intersects . See Figure 3.
Hence (4.7) has at least one real solution in the interval . Due to the uniqueness, all eigenvalues as in (4.25) must be positive.
4.2. Small Eigenvalues
In order to prove Theorem 1.4, we need to show that all "small eigenvalues" are also real provided that is small. The proof below is a modification of the proof of Theorem 1.3.
Proof of Theorem 1.4.
has precisely solutions, counted multiplicity. Here has been written as to emphasize the dependence on .
They are the first eigenvalues of problem (1.1)–(1.4).
Thus are different eigenvalues of located in the interval . Since (4.37) has precisely solutions inside , we conclude that all solutions of (4.37) inside are necessarily real. Now we have proved that for all .
Notice that the constant in (4.44) is constructed from the implicit function theorem. Generally speaking, depends on and all information of the potential . However, during the application of the implicit function theorem to (4.37), the derivatives of can be well controlled using estimates in , like (3.8)–(3.11). It is possible to choose some such that it depends on the norm only. We will not give the detailed construction. Note that this has been already observed for large eigenvalues. For example, and depend only on the norm of .
An open problem is what is the characterization like (4.46) for the smallest eigenvalue of . Once this is clear, some results on nonlinear problems in [5, 6, 8] can be extended by using eigenvalues of .
or to more general Stieltjes boundary conditions . In this sense, eigenvalue theory can be established for these nonsymmetric problems.
The third author is supported by the Major State Basic Research Development Program (973 Program) of China (no. 2006CB805903), the Doctoral Fund of Ministry of Education of China (no. 20090002110079), the Program of Introducing Talents of Discipline to Universities (111 Program) of Ministry of Education and State Administration of Foreign Experts Affairs of China (2007), and the National Natural Science Foundation of China (no. 10531010). The authors would like to express their thanks to Ping Yan for her help during the preparation of the paper.
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