- Research Article
- Open Access
Structure of Eigenvalues of Multi-Point Boundary Value Problems
Advances in Difference Equations volume 2010, Article number: 381932 (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.
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.
In this paper, we will establish some fundamental results for eigenvalue theory of multi-point boundary value problems. Precisely, for a real potential , we consider the eigenvalue problem
associated with the -point boundary condition
Here and the boundary data are and
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.
When , boundary condition (1.2) is reduced to the Dirichlet boundary condition
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.
When , (1.1) is
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.
Given and , then is composed of a sequence which satisfies
Given and , then , where
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 .
To sketch our proofs, let us denote
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.
2. Structure of Eigenvalues of the Zero Potential
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
Let . Boundary condition (1.2) is the following three-point boundary condition:
where and . We consider the eigenvalue problems (1.5)–(2.1).
Let . Complex solutions of (1.5) satisfying are , , where
Notice that is an entire function of . Define
Obviously, depends on the boundary data as well. Then if and only if satisfies
Let . By (2.3) and (2.4), if and only if satisfies
That is, either
Equation (2.6) shows that always contains positive eigenvalues , .
Equation (2.7) has real solutions if and only if . In this case, consists of non-negative eigenvalues. More precisely,
Equation (2.7) has nonreal solutions if and only if . In this case, we have
For example, one has
Notice that all eigenvalues obtained from (2.7) can be constructed explicitly as (2.10) and (2.11). For example, contains negative eigenvalues if and only if . Moreover, in this case, one has the unique negative eigenvalue given by
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
Suppose that is a bounded continuous function. One calls that is almost periodic, if for any , there exists such that for any , there exists such that
Any almost periodic function admits a well-defined mean value
To study and , let us prove some properties on almost periodic functions.
Let be an almost periodic function.
For any , one has(2.15)
Assume that is non-zero and . Then is oscillatory as , that is,(2.17)
In particular, has a sequence of positive zeros tending to .
Let us only prove (2.15) because (2.16) is similar. For any , choose such that(2.18)
By (2.13), there exists such that . For any , let us take
By (2.13) again, there exists such that . Hence
By the choice of , one has . Hence
This proves (2.15).
If and has mean value , it is easy to see that(2.23)
Now result (2.17) can be deduced simply from (2.15) and (2.16).
Like (2.3) and (2.4), all eigenvalues of problem are determined by the following equation:
Notice that is an entire function of . Hence (2.24) has only isolated zeros in . For , we have the following elementary equalities:
For real eigenvalues of problem , we have the following result.
Given , then , where
Let us first consider possible positive eigenvalues of , where . By the first equality of (2.26), equation (2.24) is the same as
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 .
Next we consider possible negative eigenvalues of , where . In this case, (2.24) is the same as
See the first equality of (2.26). Notice that is analytic in . As , one has
Thus (2.29) has at most finitely many positive solutions. Hence contains at most finitely many negative eigenvalues.
As both (2.28) and (2.29) have only isolated solutions, the above two cases show that all real eigenvalues of can be listed as in (2.27).
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.
Suppose that satisfies
Then contains only real eigenvalues. Moreover, one has .
When , problem (1.5)–(1.2) is the Dirichlet problem and . In the following, assume that .
Suppose that , where , . We assert that under assumption (2.32). Otherwise, assume that . By (2.26), equation (2.24) is the following system for :
It follows from the Hölder inequality that
which is impossible under assumption (2.32). Thus and therefore .
Next, by (2.2), (2.25), and the Hölder inequality, we have
because and . By (2.24), . Hence we have .
Finally, by the Hölder inequality, assumption (2.32) implies that . For any , the function of (2.28) satisfies
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. Structure of Eigenvalues of Non-Zero Potentials
Given and complex parameter , the fundamental solutions of (1.1) are denoted by , . That is, they are solutions of (1.1) satisfying the initial values
Notice that are entire functions of . See . To study , let us introduce
which is an entire function of . See (2.25) for the case . Notice that is real for . Then if and only if
3.1. Basic Estimates
Given , one has
uniformly in .
Suppose that . We have from (2.26)
The uniform limits in (3.4) are evident.
For the function of (2.28), one has the following result on its amplitude.
Given , there exist a constant and a sequence of increasing positive numbers such that and
Recall that is quasi-periodic and has the mean value . Denote that
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 ]).
Let and . There hold the following estimates for all
For their purpose, the authors of  have proved (3.8)–(3.11) for complex potentials . For example, in (3.8)–(3.11), the terms and are replaced by and , respectively in . Inspecting their proofs, especially the proof of [11, pages 7–9, Theorem ], one can find that estimates (3.8)–(3.11) are also true for potentials . Moreover, these estimates can be established even for linear measure differential equations with general measures . By the Hölder inequality, one has
This is why the authors of  have used these terms in (3.8)–(3.11).
There holds the following estimate for :
Define . From (3.9), we have
By (2.25) and (3.2), we have
This gives (3.13).
One has on . Consequently, there exists such that .
Otherwise, we have on . Notice that
Let in (3.13), where is as in Lemma 3.2. We have
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.
We argue as in general spectrum theory . By Lemma 3.6, there exists such that . That is, the following equation:
has only the trivial solution satisfying boundary condition (1.2). Let be the Green function associated with problem (3.19)-(1.2). Then if and only if and
has nontrivial solution satisfying (1.2). In other words, if and only if the following equation:
has non-trivial solution , where
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 .
For , denote that
Suppose that and . Then and (3.13) implies that
We conclude that all non-zero eigenvalues satisfy
Let us derive some consequences from estimate (3.25) for .
Since , it follows from the uniform limits in (3.4) that(3.26)
Thus there exists some such that
The horizontal strip of (3.27) in the -plane is transformed by (3.23) to the following half-plane in the -plane:
Let . We assert that
contains at most finitely many eigenvalues. Otherwise, suppose that
contains infinitely many . Since (3.3) has only isolated solutions, we have necessarily . By denoting , one has
In particular, . Now estimate (3.25) reads as
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.
We need to only consider positive eigenvalues of . Let in (3.13), where is as in Lemma 3.2. By using (3.17), we have
Since , w.l.o.g., we can assume that for all . Thus
By using (3.6), we conclude that
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).
4. Nonexistence of Nonreal Eigenvalues for Small
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.
For later use, let us introduce the following elementary function:
Then . Obviously, if and only if , . Define
where is the circle in the -plane
Then , , and for all . Let be the unique solution of the following equation:
Numerically, . The following facts can be verified by elementary arguments.
For the graph of , see Figure 2.
4.1. Large Eigenvalues
In the following we apply the Rouché theorem to study the spectrum , that is, the zeros of the function in the -plane. To this end, we consider problem as a perturbation of the Dirichlet problem , whose eigenvalues are zeros of the function
Let . Equation (3.3) is the same as
which is considered as a perturbation of the following equation:
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 .
Let us derive another consequence from estimate (3.25) with some restriction on . Suppose that satisfies . Define the positive function
where is as in (3.14). Then is decreasing in .
Then for any , where , one has
We keep the notations in (3.23) Let . If , it follows from (2.26) and (3.25) that
Using the function in (4.9), we obtain . This proves (4.11).
Consider the following circles of the -plane:
Let be as in (4.13). one has
Let , where . Then . By (2.26) we have
See (4.1) and (4.2). Notice that
By (2.25) and (4.15), we have
Compared with (4.1) and (4.2), it follows from (3.13) that
Thus one has (4.14).
Proof of Theorem 1.3.
where the constant is as in Lemma 4.2. One has and . Denote by the disc enclosed by the circle , that is,
Since , intersects . See Figure 3.
In the following, we always assume that satisfies
Suggested by (3.14), (4.9), and (4.11), we denote
Then, for all as in (4.22), by (4.9) one has
Let us show that must be positive. In fact, (4.25) implies that . By result (4.11), we have . Hence is a zero of inside some disc . See Figure 3. W.l.o.g., let us assume that . Then satisfies
For any , one has
It follows from (4.5) that . By (4.14), we have the estimate
Since (4.8) has the unique, simple zero in , by the Rouché theorem, we conclude from estimate (4.28) that (4.7) has the unique, simple zero in . Furthermore, denote that
See (4.28). We have
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.
Finally, it follows from Theorem 1.1 that contains at most finitely many which do not satisfy (4.25). Thus the proof of Theorem 1.3 is completed.
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.
By (4.4), we can fix some such that
In the following we assume that satisfies (4.22), that is, . From the proof of Theorem 1.3, consists of positive eigenvalues. See conditions (4.25) and (4.32). Moreover, for , that is, , we obtain from estimate (4.14) and condition (4.33) that
Notice that equation
has (simple) solutions , . By the Rouché theorem, we conclude that, if , the following problem:
has precisely solutions, counted multiplicity. Here has been written as to emphasize the dependence on .
Suppose that . Equation (4.37) corresponds to the Dirichlet eigenvalue problem (1.1)–(1.4), which has only real eigenvalues. Moreover, all solutions of (4.37) are simple in this case . Hence solutions of problem (4.37) can be denoted by , , where
They are the first eigenvalues of problem (1.1)–(1.4).
In the following, we apply the implicit function theorem to prove that solutions of (4.37) inside are actually real when is small. Notice that is a smooth real-valued function of . By [11, page 21, Theorem ], the derivative of w.r.t. is
Since is a Dirichlet eigenvalue of problem (1.1), we have . Moreover, the Liouville theorem for (1.1) implies that
In particular, . Hence
Now the implicit function theorem is applicable to (4.37). In conclusion, there exist some constant and a continuously differentiable real-valued functions of such that
Due to (4.38)–(4.44) and the continuity of , one can assume that
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 .
We end the paper with an open problem. Given , for any , due to Theorem 1.2, problem has always a sequence of real eigenvalues which tends to . In applications of eigenvalues to nonlinear problems, the smallest (real) eigenvalues are of great importance. The main reason is that solutions of problem (1.1)–(1.2) are oscillatory only when . As for the smallest eigenvalue of the Dirichlet problem (1.1)–(1.4), denoted by , one has the following variational characterization:
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 .
Finally, let us remark that the approaches in this paper also can be applied to other multi-point boundary conditions like
or to more general Stieltjes boundary conditions . In this sense, eigenvalue theory can be established for these nonsymmetric problems.
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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.