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A class of fourth-order boundary value problems with transmission conditions are investigated. By constructing we prove that these class of fourth order problems consist of finite number of eigenvalues. Further, we show that the number of eigenvalues depend on the order of the equation, partition of the domain interval, and the boundary conditions (including the transmission conditions) given.

Boundary value problems with finite spectrum have been studied recently [

In recent years, boundary value problems with transmission conditions or eigenparameter dependent boundary conditions have been an important research topic for their applications in physics [

We consider the fourth-order boundary value problems (BVPs) consisting of the equation

By introducing the quasiderivatives

Note that condition (

By a trivial solution of (

We comment on the self-adjoint expressions of the fourth-order boundary value problems (

An important class of self-adjoint boundary conditions is the separated condition. It has the following canonical representation [

Let (

Suppose

From this, it follows that the top component of

In the proof of Lemma

Next we find a formula for

Let (

Firstly, for any

Hence we have

Then we can conclude that (

Consider the problem (

Note that the third and fourth row of

The fourth-order problem (

In this section we assume that (

Given (

The above mentioned notations are needed later.

Following [

Let (

Then for

Observe from (

Let (

Then for

Let (

From the transmission conditions (

This means that the fundamental matrix solution of the system (

Note that

For the fundamental matrix

Note that

For the fundamental matrix

From Lemmas

For the fundamental matrix

Now we construct regular fourth-order problems with transmission conditions which have at most

Let

Note that

In terms of (

If the BCs are separated self-adjoint boundary conditions given as (

Let

From Corollaries

In the end we remark on how the number of eigenvalues depend on the order of the equation, partition of the domain interval, and the boundary conditions (including the transmission conditions). In fact, the characteristic function

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the National Natural Science Foundation of China (Grant no. 11301259 and Grant no. 11161030), Natural Science Foundation of Inner Mongolia (Grant no. 2013MS0105), and a Grant-in-Aid for Scientific Research from Inner Mongolia University of Technology (Grant no. ZD201310).