Principal Eigenvalues of a Second-Order Difference Operator with Sign-Changing Weight and Its Applications

Let T > 2 be an integer and T = {1, 2, . . . , T}. We show the existence of the principal eigenvalues of linear periodic eigenvalue problem −Δ2u(j − 1) + q(j)u(j) = λg(j)u(j), j ∈ T , u(0) = u(T), u(1) = u(T+1), and we determine the sign of the corresponding eigenfunctions, where λ is a parameter, q(j) ≥ 0 and q(j) ̸ ≡ 0 in T , and theweight functiong changes its sign in T . As an application of our spectrum results, we use the global bifurcation theory to study the existence of positive solutions for the corresponding nonlinear problem.


Introduction
In 1997, Constantin [1] studied the following linear periodic eigenvalue problem: (0) = ( ) , where , ∈ [0, ], ( ) ≥ 0, and ( ) ̸ ≡ 0. He obtained that if changes its sign, then (1) and (2) have infinite real eigenvalues, ± , such that Equation (1) with ≡ 1/4 plays a crucial role in the study of the water shallow equation; see [2][3][4][5]. Let > 2 be an integer and T = {1, 2, . . . , }. In 2005, Wang and Shi [6] discussed the eigenvalues of a discrete periodic boundary value problem (0) = ( ) , where ( ), ( ), and ( ) are real functions with ( ) > 0 for ∈ {0, 1, . . . , }, ( ) > 0 for ∈ T, and (0) = ( ) = 1, and is the spectral parameter. They showed the existence of eigenvalues of (4) and (5) and calculated the numbers of eigenvalues. Ji and Yang [7] considered a class of boundary value problems of the second-order difference equation (4) with the more general boundary conditions The class of problems considered include those with antiperiodic, Dirichlet, and periodic boundary conditions. They focused on the structure of eigenvalues and comparisons of all eigenvalues of (4) and (6), as the coefficients ( ), ( ), and ( ) change their signs. They got a very interesting result: the numbers of positive eigenvalues are equal to the numbers of positive elements in the weight function, and the numbers of negative eigenvalues are equal to the numbers of negative elements in the weight function. Gao and Ma [8] studied the eigenvalues of periodic and antiperiodic eigenvalue problems of discrete linear secondorder difference equation (4) with sign-changing weight. They find that these two problems have real eigenvalues (including the multiplicity), respectively. Furthermore, the 2 Discrete Dynamics in Nature and Society numbers of positive eigenvalues are equal to the numbers of positive elements in the weight function, and the numbers of negative eigenvalues are equal to the numbers of negative elements in the weight function. Furthermore, these eigenvalues, including the eigenvalues of Neumann problem, satisfy the certain order relation.
However, all of above papers provide no information about the sign of the eigenfunctions of (4) and (5). In particular, they give no information about the sign of the eigenfunctions corresponding to two simple eigenvalues + 1 and − 1 . It is the purpose of this paper to show the existence of the principal eigenvalues and determine the sign of the corresponding eigenfunctions for linear periodic eigenvalue problem where ( ) ≥ 0 and ( ) ̸ ≡ 0 in T, and the weight function changes its sign in T. Our approach is motivated by Brown and Lin [9] and Smoller [10], where the infimum of Rayleigh quotient is used to characterize the principal eigenvalues for diverse linear eigenvalue problems of elliptic equations with infinite weight.
For the other recent results on the spectrum structure of discrete linear eigenvalue problems with one-sign weight, see Sun and Shi [11], Shi and Chen [12], Jirari [13], Bohner [14], and Agarwal et al. [15] and the references therein.
The rest of the paper is organized as follows. In Section 2, we show the existence of the principal eigenvalues of (7) and determine the sign of the corresponding eigenfunctions. In Section 3, we apply our spectrum theory and the wellknown Rabinowitz bifurcation theorem to show the existence of positive solutions for nonlinear discrete periodic boundary value problem −Δ 2 ( − 1) + ( ) ( ) = ( , ( )) , ∈ T, (8)

Existence of Principal Eigenvalues
In this section, we consider the linear eigenvalue problem Define a linear operator 0 : D → D by Then, it is easy to see that 0 : D → D is isomorphism. Moreover, 0 : D → D is a self-adjoint operator whose spectrum consists only of the real eigenvalues. Let us denote the norm and the inner product of D by respectively. For V ∈ D, we have from [16, Lemma 2.1] that Thus, we may define a functional To study the principal eigenvalues of (7), we need the following preliminary lemmas.
Proof. Suppose that is a nonnegative eigenfunction corresponding to the eigenvalue . Then, is a nonnegative eigenfunction corresponding to the eigenvalue = of (0) = ( ) , Discrete Dynamics in Nature and Society (0) = ( ) , Since : D → D is a self-adjoint operator, its spectrum contains only real eigenvalues: Moreover, by the well-known Krein-Rutman theorem [17, Theorem 19.2], 1 > 0 is simple, and its corresponding eigenfunction 1 is of one sign. So, the eigenvalue 1 (= 1 − ) is simple and the corresponding eigenfunction 1 does not change sign in T. Notice and are eigenvalue and the corresponding eigenfunction of (16) and (17), respectively, and is not orthogonal to 1 . This together with the fact that eigenfunctions corresponding to distinct eigenvalues are orthogonal implies that must be an eigenfunction of corresponding to 1 (= 1 + ). Hence, that is, (V) ≥ 0 for all V ∈ D.
Proof. By the spectral theorem, ⟨ 0 V, V⟩ ≥ 1 ⟨V, V⟩ for all V ∈ D, where 1 is the first eigenvalue of 0 . Note that (H0) implies that Lemma 3. If > + 1 , then is not an eigenvalue of (7) possessing a nonnegative eigenfunction.
and so (V) < 0. The required result is now an immediate consequence of Lemma 1. Proof. Let = (1 − ) + 1 , where 0 < < 1. We claim that In fact, for V ∈ D, we have from the fact that

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Proof. If V ∈ D with ∑ =1 ( )V 2 ( ) < 0, then it follows from the fact that Hence, Lemma 6. If < − 1 , then is not an eigenvalue of (7) possessing a nonnegative eigenfunction.
and so (V) < 0. The required result is now an immediate consequence of Lemma 1.

Theorem 8. If (H0) and (H1) hold, then
(1) − 1 < 0 < + 1 ; (2) the algebraic multiplicity of − 1 and + 1 is 1; It is easy to see that + 1 is an eigenvalue for (7) with corresponding eigenfunction if and only if 0 is an eigenvalue of 0 , and, accordingly, 0 is an eigenvalue of (38) with corresponding eigenfunction . The least eigenvalue of 0 is given by Since Therefore, and so 1 ≤ 0. Hence, 1 = 0 is the least eigenvalue of (38) and so 1 is simple and the corresponding eigenfunction can be chosen to be positive on T.
Using the same method, with obvious changes, we may prove the algebraic multiplicity of − 1 is 1 and the eigenfunction − 1 corresponding to the eigenvalue − 1 is of one sign.

Existence of Positive Solutions
As an application, we consider the existence of positive solutions of the discrete nonlinear problem (8) and (9). In this section, we assume that and let be an eigenfunction corresponding to 1 ( ).
We extend the function to a continuous functiond efined on T × R by setting, for ∈ T, Obviously, within the context of positive solutions, problem (8) and (9) is equivalent to the same problem with replaced bỹ. Furthermore,̃( , ) is an odd function for ∈ T. In the sequel of the proof, we shall replace with̃. However, for the sake of simplicity, the modified functioñwill still be denoted by .
Recall that Let us consider as a bifurcation problem from the trivial solution ≡ 0. A solution of (48) is a pair ( , ) ∈ R × D which satisfies (48). It is easy to see that any solution of (48) of the form (1, ) yields a solution of (8) and (9). Equation (48) can be converted into the equivalent equation with respect to varying in bounded intervals. Notice that if is the eigenvalue of 0 , then it also is the characteristic value of L.
We say that a solution ( , ) ∈ R × D of (51) is nontrivial if there exists 0 ∈ T such that ( 0 ) ̸ = 0. Denote by S the closure in R × D of the set of all nontrivial solutions ( , ) of (51) with > 0. Theorem 1.3 in [19] yields the existence of a maximal closed connected set C in S such that ( + 1 , 0) ∈ C and at least one of the following conditions holds: (i) C is unbounded in R × D.
Step 2. In what follows, we prove several properties which will eventually lead to the fact that condition (ii) above does not hold.
Hence, we conclude by (53) that Therefore, we have with ‖̃‖ = 1 and in particular ̸ = 0. Accordingly,̃is a characteristic value of L.
and conclude that, possibly passing to a subsequence, This follows from the fact that , and hence H, is odd with respect to the second variable.
In the sequel, we denote by the positive cone in D; that is, and we denote by int its interior and by its boundary.
It is an immediate consequence of the fact that + 1 ( ) > 0 for all ∈ T and the well-known Crandall-Rabinowitz local bifurcation theorem; see Crandall and Rabinowitz [20] and Kielhöfer [21].