Properties for fourth order discontinuous differential operators with eigenparameter dependent boundary conditions

Kun Li; Peng Wang; Kun Li; Peng Wang

doi:10.3934/math.2022640

AIMS Mathematics

2022, Volume 7, Issue 6: 11487-11508. doi: 10.3934/math.2022640

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Research article

Properties for fourth order discontinuous differential operators with eigenparameter dependent boundary conditions

Kun Li ^,,
Peng Wang

School of Mathematical Sciences, Qufu Normal University, Qufu, 273165, Shandong, China

Received: 09 December 2021 Revised: 31 March 2022 Accepted: 07 April 2022 Published: 13 April 2022
MSC : 34B05, 34L30, 47E05

In this paper, a class of fourth order differential operators with eigenparameter-dependent boundary conditions and transmission conditions is considered. A new operator associated with the problem is established, and the self-adjointness of this operator in an appropriate Hilbert space $ H $ is proved. The fundamental solutions are constructed. Sufficient and necessary conditions of the eigenvalues are investigated. Then asymptotic formulas for the fundamental solutions and the characteristic functions are given. Finally, the completeness of eigenfunctions in $ H $ is given and the Green function is also involved.
- transmission conditions,
- eigenparameter-dependent boundary conditions,
- completeness,
- Green function
Citation: Kun Li, Peng Wang. Properties for fourth order discontinuous differential operators with eigenparameter dependent boundary conditions[J]. AIMS Mathematics, 2022, 7(6): 11487-11508. doi: 10.3934/math.2022640

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Abstract

In this paper, a class of fourth order differential operators with eigenparameter-dependent boundary conditions and transmission conditions is considered. A new operator associated with the problem is established, and the self-adjointness of this operator in an appropriate Hilbert space $ H $ is proved. The fundamental solutions are constructed. Sufficient and necessary conditions of the eigenvalues are investigated. Then asymptotic formulas for the fundamental solutions and the characteristic functions are given. Finally, the completeness of eigenfunctions in $ H $ is given and the Green function is also involved.

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