Eigenvalues of discrete linear second-order periodic and antiperiodic eigenvalue problems with sign-changing weight

Gao, Chenghua; Ma, Ruyun

doi:10.1016/j.laa.2014.11.002

Cited by 13 publications

(8 citation statements)

References 15 publications

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“…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.…”

Section: Theorem 9 Assume That (H0)-(h3) Hold Thenmentioning

confidence: 99%

“…Therefore, C joins ( + 1 , 0) to ( 1 ( ), ∞). Therefore, C crosses the hyperplane {1}×D in R×D, and, accordingly, (8) and (9) …”

Section: Claimmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Yan

2018

Discrete Dynamics in Nature and Society

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Let > 2 be an integer and T = {1, 2, . . . , }. We show the existence of the principal eigenvalues of linear periodic eigenvalue problem −Δ 2 ( − 1) + ( ) ( ) = ( ) ( ), ∈ T, (0) = ( ), (1) = ( + 1), and we determine the sign of the corresponding eigenfunctions, where is a parameter, ( ) ≥ 0 and ( ) ̸ ≡ 0 in T, and the weight function 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.

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Section: Theorem 9 Assume That (H0)-(h3) Hold Thenmentioning

confidence: 99%

“…Therefore, C joins ( + 1 , 0) to ( 1 ( ), ∞). Therefore, C crosses the hyperplane {1}×D in R×D, and, accordingly, (8) and (9) …”

Section: Claimmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Yan

2018

Discrete Dynamics in Nature and Society

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“…It is worth pointing out that only partial information is known to the spectrum of the linear eigenvalue problem (1.3); see Gao and Ma [10] and Ji and Yang [11]. More precisely, from the results in [10] and [11] it follows that (1.3) has T real eigenvalues, including i positive eigenvalues λ + 1 , . .…”

Section: Introductionmentioning

confidence: 99%

Positive solutions of the periodic problems for quasilinear difference equation with sign-changing weight

2018

Adv Differ Equ

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“…In 2015, C. Gao, R. Ma [8] studied the problem (1) (4), they find that the problems have T real eigenvalues (including the multiplicity). Furthermore, 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.…”

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confidence: 99%

Spectrum of a Class of Difference Operators with Indefinite Weights

Yang¹,

Yun-lan²,

Sun³

2020

JAMP

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In this study, we use analytical methods and Sylvester inertia theorem to research a class of second order difference operators with indefinite weights and coupled boundary conditions. The eigenvalue problem with sign-changing weight has lasted a long time. The number of eigenvalues and the number of sign changes of the corresponding eigenfunctions of discrete equations under different boundary conditions are mainly studied. For the discrete Sturm-Liouville problems, similar conclusions about the properties of eigenvalues and the number of sign changes of the corresponding eigenfunctions are obtained under different boundary conditions, such as periodic boundary conditions, antiperiodic boundary conditions and separated boundary conditions etc. The purpose of this paper is to extend the similar conclusion to the coupled boundary conditions, which is of great significance to the perfection of the theory of the discrete Sturm-Liouville problems. We came to the following conclusions: first, the eigenvalues of the problem are real and single, the number of the 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. Second, under some conditions, we obtain the sign change of the eigenfunction corresponding to the j-th positive/negative eigenvalue.

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Eigenvalues of discrete linear second-order periodic and antiperiodic eigenvalue problems with sign-changing weight

Cited by 13 publications

References 15 publications

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

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

Positive solutions of the periodic problems for quasilinear difference equation with sign-changing weight

Spectrum of a Class of Difference Operators with Indefinite Weights

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