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.
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.
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
2.1. An Example of Nonreal Eigenvalues
2.2. Real Eigenvalues with General Parameters
Now result (2.17) can be deduced simply from (2.15) and (2.16).
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 .
The quasi-periodic function is as in Figure 1.
2.3. Nonexistence of Nonreal Eigenvalues
3. Structure of Eigenvalues of Non-Zero Potentials
3.1. Basic Estimates
The uniform limits in (3.4) are evident.
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).
3.2. Eigenvalues with General Parameters
Proof of Theorem 1.1.
This is impossible because we have the uniform limits (3.4).
Proof of Theorem 1.2.
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).
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 .
Thus one has (4.14).
Proof of Theorem 1.3.
Since , intersects . See Figure 3.
4.2. Small Eigenvalues
Proof of Theorem 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.
- Anderson DR, Ma R:Second-order -point eigenvalue problems on time scales. Advances in Difference Equations 2006, 2006:-17.Google Scholar
- Agarwal RP, Kiguradze I: On multi-point boundary value problems for linear ordinary differential equations with singularities. Journal of Mathematical Analysis and Applications 2004,297(1):131-151. 10.1016/j.jmaa.2004.05.002MathSciNetView ArticleMATHGoogle Scholar
- Gupta CP, Ntouyas SK, Tsamatos PCh:On an -point boundary-value problem for second-order ordinary differential equations. Nonlinear Analysis: Theory, Methods & Applications 1994,23(11):1427-1436. 10.1016/0362-546X(94)90137-6MathSciNetView ArticleMATHGoogle Scholar
- Liu B, Yu J: Solvability of multi-point boundary value problems at resonance. I. Indian Journal of Pure and Applied Mathematics 2002,33(4):475-494.MathSciNetMATHGoogle Scholar
- Ma R:Nodal solutions for a second-order -point boundary value problem. Czechoslovak Mathematical Journal 2006,56(131)(4):1243-1263. 10.1007/s10587-006-0092-7View ArticleGoogle Scholar
- Ma R, O'Regan D:Nodal solutions for second-order -point boundary value problems with nonlinearities across several eigenvalues. Nonlinear Analysis: Theory, Methods & Applications 2006,64(7):1562-1577. 10.1016/j.na.2005.07.007MathSciNetView ArticleMATHGoogle Scholar
- Meng F, Du Z: Solvability of a second-order multi-point boundary value problem at resonance. Applied Mathematics and Computation 2009,208(1):23-30. 10.1016/j.amc.2008.11.026MathSciNetView ArticleMATHGoogle Scholar
- Rynne BP:Spectral properties and nodal solutions for second-order, -point, boundary value problems. Nonlinear Analysis: Theory, Methods & Applications 2007,67(12):3318-3327. 10.1016/j.na.2006.10.014MathSciNetView ArticleMATHGoogle Scholar
- Rynne BP: Second-order, three-point boundary value problems with jumping non-linearities. Nonlinear Analysis: Theory, Methods & Applications 2008,68(11):3294-3306. 10.1016/j.na.2007.03.023MathSciNetView ArticleMATHGoogle Scholar
- Zhang M, Han Y: On the applications of Leray-Schauder continuation theorem to boundary value problems of semilinear differential equations. Annals of Differential Equations 1997,13(2):189-207.MathSciNetGoogle Scholar
- Pöschel J, Trubowitz E: The Inverse Spectrum Theory. Academic Press, New York, NY, USA; 1987.Google Scholar
- Zettl A: Sturm-Liouville Theory, Mathematical Surveys and Monographs. Volume 121. American Mathematical Society, Providence, RI, USA; 2005:xii+328.Google Scholar
- Rynne BP:Spectral properties of second-order, multi-point, -Laplacian boundary value problems. Nonlinear Analysis: Theory, Methods & Applications 2010,72(11):4244-4253. 10.1016/j.na.2010.01.054MathSciNetView ArticleMATHGoogle Scholar
- Fink AM: Almost Periodic Differential Equations, Lecture Notes in Mathematics, Vol. 377. Springer, Berlin, Germany; 1974:viii+336.Google Scholar
- Hale JK: Ordinary Differential Equations. 2nd edition. John Wiley & Sons, New York, NY, USA; 1969:xvi+332.MATHGoogle Scholar
- Meng G: Continuity of solutions and eigenvalues in measures with weak* topology, Ph.D. dissertation. Tsinghua University, Beijing, China; 2009.Google Scholar
- García-Huidobro M, Manásevich R, Yan P, Zhang M:A -Laplacian problem with a multi-point boundary condition. Nonlinear Analysis: Theory, Methods & Applications 2004,59(3):319-333.MathSciNetView ArticleMATHGoogle Scholar
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