We study the spectrum structure of discrete second-order Neumann boundary value
problems (NBVPs) with sign-changing weight. We apply the properties of characteristic determinant of the
NBVPs to show that the spectrum consists of real and simple eigenvalues; the number of positive eigenvalues is
equal to the number of positive elements in the weight function, and the number of negative eigenvalues
is equal to the number of negative elements in the weight function. We also show that the eigenfunction
corresponding to the

Let

When the weight function

However, these two results do not give any information on the sign-changing of the eigenfunction of (

In 1991, Kelley and Peterson [

Furthermore, when

However, there are few results on the spectrum of discrete second-order linear eigenvalue problems when

Naturally, there are two interesting questions: (a) how to distribute of the eigenvalues of (

It is the purpose of this paper to establish the structure of eigenvalues and the oscillatory properties of the corresponding eigenfunctions of (

The main result of our paper is the following theorem.

Suppose that (A0), (A1) hold. Then one has the following.

If

If

If

It is worth remarking that the number of sign changing of eigenfunction is given in Theorem

Applying Theorem

The rest of the paper is devoted to proving Theorem

Let

Let

In fact, for any real vector

For any real vector

For

As we know, to find the eigenvalues of (

For

For

For

The roots of

Since

For the

The fact that

Two consecutive polynomials

Suppose on the contrary that there exists

Suppose that

Since

Assume that (A0), (A1) hold. Then

If

(1) From (

So, by simple computation, it follows that

(2) If

By computing and simplifying, we get that

For

The largest negative root

For

If

First, we deal with the case

Obviously,

Recall

If

If

If

Second, suppose that for

If

If

Now, we consider the case

In this case,

In this case,

In this case,

In this case,

We only deal with Case 1. The other cases could be dealt via the same method.

First, we show that (

Since

We only deal with the case that

Thus (

Next, we show that (

Obviously,

From Lemma

Using (

Now, for

Finally, for

If

if

if

if

If

if

if

if

Let

It is motivated by the proof of Strums Theorem; see [

The idea of the proof is to follow the changes in

If

It is easy to see from Lemma

Next, we show that each root of

In fact, for

For

Repeating the above argument, we may deduce that

If

Let

Let

From Lemmas

If

If

If

Now, we consider the numbers of sign changing of eigenfunction. From Lemma

Let

If

Finally, by using the above method, with obvious changes, we may prove that the number of sign changes

If one extra term

The authors are very grateful to the anonymous referees for their valuable suggestions. This study is supported by the NSFC (no. 11061030), SRFDP (no. 20126203110004) and Gansu Provincial National Science Foundation of China (no. 1208RJZA258).