Spectrum of Discrete Second-Order Difference Operator with Sign-Changing Weight and Its Applications

This result has been extended to one-dimensional p-Laplacian operator by Anane et al. [2] and to the highdimensional case by Hess and Kato [3], Bongsoo and Brown [4], and Afrouzi and Brown [5]. Meanwhile, these spectrum results have been used to deal with several nonlinear problems; see, for example, [4–7] and the references therein. For the discrete case, Atkinson [8] studied the discrete linear eigenvalue problems

This result has been extended to one-dimensional -Laplacian operator by Anane et al. [2] and to the highdimensional case by Hess and Kato [3], Bongsoo and Brown [4], and Afrouzi and Brown [5].Meanwhile, these spectrum results have been used to deal with several nonlinear problems; see, for example, [4][5][6][7] and the references therein.

Theorem B. Problem
and the eigenfunction corresponding to   has exactly  − 1 simple generalized zeros.
Furthermore, when () ≡ 1, Agarwal et al. [10] generalized the results of Theorem B to the dynamic equations on time scales with Sturm-Liouville boundary condition.Moreover, under the assumption that the weight functions are not changing sign, several important results on linear Hamiltonian difference systems have also been established by Shi and Chen [11], Bohner [12], and the references therein.
However, there are few results on the spectrum of discrete second-order linear eigenvalue problems when () changes its sign on T. In 2008, Shi and Yan [13] discussed the spectral theory of left definite difference operators when () may change its sign.However, they provided no information about the sign of the eigenvalues and no information about the corresponding eigenfunctions.Recently, Ma et al. [14] obtained that (1) and ( 2) have two principal eigenvalues  1,− < 0 <  1,+ and they studied some corresponding discrete nonlinear problems.
It is the purpose of this paper to establish the discrete analogue of Theorem A for the discrete problems (1) and (2).More precisely, we will prove the following.
The rest of the paper is arranged as follows.In Section 2, some preliminaries will be given including Lagrange-type identities.In Section 3, we develop a new method to count the number of negative and positive eigenvalues of (1) and (2), which enable us to prove Theorem 1. Finally in Section 4, we apply our spectrum theory and the Rabinowitz's bifurcation theorem to consider the existence of sign-changing solutions of discrete nonlinear problems Δ 2  ( − 1) +  ()  ( ()) = 0,  ∈ T, where  ̸ = 0 is a real parameter,  : T → R changes its sign, () ̸ = 0 on T, and  : R → R is continuous.
The nodes and simple zeros of  are called the simple generalized zeros of .
Proof.Dividing ( 17) by  −  and making  →  for fixed , then we get the desired result.
Proof.This proof is divided into two steps.
Step 1 (each zero of (, ) is real).Suppose on the contrary that (, ) has a complex zero  0 ; then (,  0 ) = (,  0 ) = 0. Furthermore, On the other hand, if  0 is a zero of (, ), then  0 is an eigenvalue of the linear eigenvalue problem Now, by Lemma 7 and Corollary 6, we get that the above determinant does not equal zero, which is a contradiction.Hence, the zeros of (, ) are all real for  ∈ T.
Step 2 (all of the zeros of (, ) are simple).Suppose on the contrary that (, ) has a multiple zeros  * , necessarily real.Then (,  * ) = 0 and   (,  * ) = 0.Moreover, However, by Lemma 7 and Corollary 5, we get that the above determinant does not equal zero, a contradiction.
From Lemma 8, it follows that the spectra of ( 1) and (2) consist of  real eigenvalues.Furthermore, by Lemma 9, there exists  ∈ {1, . . .,  − 1} such that such  real eigenvalues can be ordered as follows: Define  :  →  by Then we get the following.
where  denotes the identity operator.
By Lemmas 8-10, all of the eigenvalues of ( 1) and ( 2 Proof.Suppose that   is a zero of (, ).Now, the proof can be divided into two cases.
This completes the proof.
where   () is a polynomial of degree precisely  − 1 of .
Next, using the result of first step and Lemma 14, it follows that  = .
Proof of Theorem 1. From Lemma 7-Lemma 15, the results of Theorem 1 hold.

Application
As an application, we consider the existence of sign-changing solutions of the discrete nonlinear boundary value problems (12), (13).
Since  ̸ ≡ 0 on T, we may assume that On the other hand, it follows from (52) and (0 which implies that However, by (59) and the fact ( 0 ) = 0, we get which contradicts (58).Now, the results of Rabinowitz [23] for (52) can be stated as follows.
In the following we will investigate other sign-changing solutions of problems ( 1) and (2).
as a bifurcation problem from infinity.We note that (80) is equivalent to (12) and (13).Now, the results of Rabinowitz [25] for (80) can be stated as follows.Proof.It is similar to the proof of Lemma 20, so we omit it.