- Research Article
- Open access
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Structure of Eigenvalues of Multi-Point Boundary Value Problems
Advances in Difference Equations volume 2010, Article number: 381932 (2010)
Abstract
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.
1. Introduction
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ1_HTML.gif)
associated with the -point boundary condition
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ2_HTML.gif)
Here and the boundary data are
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ3_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ4_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ5_HTML.gif)
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 [13] 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.
Theorem 1.1.
Given and
, then
is composed of a sequence
which satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ6_HTML.gif)
Theorem 1.2.
Given and
, then
, where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ7_HTML.gif)
For , the norm is
. For
, the
norm is denoted by
. With some restrictions on
, we are able to prove that
contains only real eigenvalues.
Theorem 1.3.
If satisfies
, then the spectrum
contains at most finitely many nonreal eigenvalues.
Theorem 1.4.
Given , there exists some constant
, depending on the norm
only, such that if
satisfies
, then one has
.
To sketch our proofs, let us denote
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ8_HTML.gif)
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 [11] 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 [11] 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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ9_HTML.gif)
where and
. We consider the eigenvalue problems (1.5)–(2.1).
Let . Complex solutions
of (1.5) satisfying
are
,
, where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ10_HTML.gif)
Notice that is an entire function of
. Define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ11_HTML.gif)
Obviously, depends on the boundary data
as well. Then
if and only if
satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ12_HTML.gif)
Example 2.1.
Let . By (2.3) and (2.4),
if and only if
satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ13_HTML.gif)
That is, either
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ14_HTML.gif)
or
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ15_HTML.gif)
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,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ16_HTML.gif)
Equation (2.7) has nonreal solutions if and only if
. In this case, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ17_HTML.gif)
For example, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ18_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ19_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ20_HTML.gif)
For more details, see [5, 6, 8].
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
In the following we consider general , based on properties of almost periodic functions [14, 15].
Definition 2.2.
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ21_HTML.gif)
Any almost periodic function admits a well-defined mean value
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ22_HTML.gif)
To study and
, let us prove some properties on almost periodic functions.
Lemma 2.3.
Let be an almost periodic function.
-
(i)
For any
, one has
(2.15)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ24_HTML.gif)
-
(ii)
Assume that
is non-zero and
. Then
is oscillatory as
, that is,
(2.17)
In particular, has a sequence of positive zeros tending to
.
Proof.
-
(i)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ27_HTML.gif)
By (2.13) again, there exists such that
. Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ28_HTML.gif)
In particular,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ29_HTML.gif)
By the choice of , one has
. Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ30_HTML.gif)
This proves (2.15).
-
(ii)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ32_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ33_HTML.gif)
Notice that is an entire function of
. Hence (2.24) has only isolated zeros in
. For
, we have the following elementary equalities:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ34_HTML.gif)
For real eigenvalues of problem , we have the following result.
Lemma 2.4.
Given , then
, where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ35_HTML.gif)
Proof.
Let us first consider possible positive eigenvalues of
, where
. By the first equality of (2.26), equation (2.24) is the same as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ36_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ37_HTML.gif)
See the first equality of (2.26). Notice that is analytic in
. As
, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ38_HTML.gif)
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
To study real eigenvalues of problem , the authors of [5, 6, 8] have imposed some restrictions on
. The typical conditions are
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ39_HTML.gif)
With some restrictions on , we will prove that
consists of only real eigenvalues.
Lemma 2.5.
Suppose that satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ40_HTML.gif)
Then contains only real eigenvalues. Moreover, one has
.
Proof.
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
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ41_HTML.gif)
It follows from the Hölder inequality that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ42_HTML.gif)
which is impossible under assumption (2.32). Thus and therefore
.
Next, by (2.2), (2.25), and the Hölder inequality, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ43_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ44_HTML.gif)
Hence (2.28) shows that .
Remark 2.6.
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ45_HTML.gif)
Notice that are entire functions of
. See [11]. To study
, let us introduce
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ46_HTML.gif)
which is an entire function of . See (2.25) for the case
. Notice that
is real for
. Then
if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ47_HTML.gif)
3.1. Basic Estimates
Lemma 3.1.
Given , one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ48_HTML.gif)
uniformly in .
Proof.
Suppose that . We have from (2.26)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ49_HTML.gif)
The uniform limits in (3.4) are evident.
For the function of (2.28), one has the following result on its amplitude.
Lemma 3.2.
Given , there exist a constant
and a sequence
of increasing positive numbers such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ50_HTML.gif)
Proof.
Recall that is quasi-periodic and has the mean value
. Denote that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ51_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ52_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ53_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ54_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ55_HTML.gif)
Remark 3.4.
For their purpose, the authors of [11] 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 [11]. 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 [16]. By the Hölder inequality, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ56_HTML.gif)
This is why the authors of [11] have used these terms in (3.8)–(3.11).
Lemma 3.5.
There holds the following estimate for :
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ57_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ58_HTML.gif)
Proof.
Define . From (3.9), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ59_HTML.gif)
By (2.25) and (3.2), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ60_HTML.gif)
This gives (3.13).
Lemma 3.6.
One has on
. Consequently, there exists
such that
.
Proof.
Otherwise, we have on
. Notice that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ61_HTML.gif)
Let in (3.13), where
is as in Lemma 3.2. We have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ62_HTML.gif)
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 [12]. By Lemma 3.6, there exists such that
. That is, the following equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ63_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ64_HTML.gif)
has nontrivial solution satisfying (1.2). In other words,
if and only if the following equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ65_HTML.gif)
has non-trivial solution , where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ66_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ67_HTML.gif)
Suppose that and
. Then
and (3.13) implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ68_HTML.gif)
We conclude that all non-zero eigenvalues satisfy
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ69_HTML.gif)
Let us derive some consequences from estimate (3.25) for .
-
(i)
Since
, it follows from the uniform limits in (3.4) that
(3.26)
Thus there exists some such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ71_HTML.gif)
The horizontal strip of (3.27) in the
-plane is transformed by (3.23) to the following half-plane
in the
-plane:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ72_HTML.gif)
Let
. We assert that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ73_HTML.gif)
contains at most finitely many eigenvalues. Otherwise, suppose that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ74_HTML.gif)
contains infinitely many . Since (3.3) has only isolated solutions, we have necessarily
. By denoting
, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ75_HTML.gif)
In particular, . Now estimate (3.25) reads as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ76_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ77_HTML.gif)
Since , w.l.o.g., we can assume that
for all
. Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ78_HTML.gif)
By using (3.6), we conclude that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ79_HTML.gif)
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
Â
We will apply the Rouché theorem to give further results on when
is small, following the approach in [11] 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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ80_HTML.gif)
Then . Obviously,
if and only if
,
. Define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ81_HTML.gif)
where is the circle in the
-plane
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ82_HTML.gif)
Then ,
, and
for all
. Let
be the unique solution of the following equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ83_HTML.gif)
Numerically, . The following facts can be verified by elementary arguments.
Lemma 4.2.
One has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ84_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ85_HTML.gif)
Let . Equation (3.3) is the same as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ86_HTML.gif)
which is considered as a perturbation of the following equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ87_HTML.gif)
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 [11].
Let us derive another consequence from estimate (3.25) with some restriction on . Suppose that
satisfies
. Define the positive function
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ88_HTML.gif)
where is as in (3.14). Then
is decreasing in
.
Lemma 4.3.
Suppose that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ89_HTML.gif)
Then for any , where
, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ90_HTML.gif)
Proof.
We keep the notations in (3.23) Let . If
, it follows from (2.26) and (3.25) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ91_HTML.gif)
Using the function in (4.9), we obtain
. This proves (4.11).
Consider the following circles of the -plane:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ92_HTML.gif)
Lemma 4.4.
Let be as in (4.13). one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ93_HTML.gif)
Proof.
Let , where
. Then
. By (2.26) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ94_HTML.gif)
See (4.1) and (4.2). Notice that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ95_HTML.gif)
By (2.25) and (4.15), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ96_HTML.gif)
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ97_HTML.gif)
Compared with (4.1) and (4.2), it follows from (3.13) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ98_HTML.gif)
Thus one has (4.14).
Proof of Theorem 1.3.
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ99_HTML.gif)
where the constant is as in Lemma 4.2. One has
and
. Denote by
the disc enclosed by the circle
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ100_HTML.gif)
Since ,
intersects
. See Figure 3.
In the following, we always assume that satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ101_HTML.gif)
Suggested by (3.14), (4.9), and (4.11), we denote
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ102_HTML.gif)
Then, for all as in (4.22), by (4.9) one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ103_HTML.gif)
Suppose that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ104_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ105_HTML.gif)
For any , one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ106_HTML.gif)
It follows from (4.5) that . By (4.14), we have the estimate
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ107_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ108_HTML.gif)
See (4.28). We have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ109_HTML.gif)
Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ110_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ111_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ112_HTML.gif)
Denote that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ113_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ114_HTML.gif)
Notice that equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ115_HTML.gif)
has (simple) solutions ,
. By the Rouché theorem, we conclude that, if
, the following problem:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ116_HTML.gif)
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 [11]. Hence solutions of problem (4.37) can be denoted by
,
, where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ117_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ118_HTML.gif)
In particular,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ119_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ120_HTML.gif)
Since is a Dirichlet eigenvalue of problem (1.1), we have
. Moreover, the Liouville theorem for (1.1) implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ121_HTML.gif)
In particular, . Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ122_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ123_HTML.gif)
Due to (4.38)–(4.44) and the continuity of , one can assume that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ124_HTML.gif)
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 [11], 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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ125_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F381932/MediaObjects/13662_2010_Article_1283_Equ126_HTML.gif)
or to more general Stieltjes boundary conditions [17]. In this sense, eigenvalue theory can be established for these nonsymmetric problems.
References
Anderson DR, Ma R:Second-order
-point eigenvalue problems on time scales. Advances in Difference Equations 2006, 2006:-17.
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.2004.05.002
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)90137-6
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.
Ma R:Nodal solutions for a second-order
-point boundary value problem. Czechoslovak Mathematical Journal 2006,56(131)(4):1243-1263. 10.1007/s10587-006-0092-7
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.2005.07.007
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.2008.11.026
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.2006.10.014
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.2007.03.023
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.
Pöschel J, Trubowitz E: The Inverse Spectrum Theory. Academic Press, New York, NY, USA; 1987.
Zettl A: Sturm-Liouville Theory, Mathematical Surveys and Monographs. Volume 121. American Mathematical Society, Providence, RI, USA; 2005:xii+328.
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.2010.01.054
Fink AM: Almost Periodic Differential Equations, Lecture Notes in Mathematics, Vol. 377. Springer, Berlin, Germany; 1974:viii+336.
Hale JK: Ordinary Differential Equations. 2nd edition. John Wiley & Sons, New York, NY, USA; 1969:xvi+332.
Meng G: Continuity of solutions and eigenvalues in measures with weak* topology, Ph.D. dissertation. Tsinghua University, Beijing, China; 2009.
GarcÃa-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.
Acknowledgments
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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Gao, J., Sun, D. & Zhang, M. Structure of Eigenvalues of Multi-Point Boundary Value Problems. Adv Differ Equ 2010, 381932 (2010). https://doi.org/10.1155/2010/381932
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DOI: https://doi.org/10.1155/2010/381932