Research article

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

  • 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.

    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

    Related Papers:

  • 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.



    加载中


    [1] E. C. Titchmarsh, Eigenfunctions expansion associated with second order differential equations, London: Oxford University Press, 1962.
    [2] M. A. Naimark, Linear differential operators, part 2, London: Harrap, 1968.
    [3] Z. J. Cao, Ordinary differential operator (in Chinese), Beijing: Science Press, 1986.
    [4] A. Zettl, Sturm-Liouville theory, Providence, Rhode Island: American Mathematics Society, Mathematical Surveys and Monographs, 2005. http://dx.doi.org/10.1090/surv/121
    [5] A. P. Wang, Research on weimann conjecture and differential operators with transmission conditions (in Chinese), Ph. D. thesis, Inner Mongolia University, 2006, 66–74.
    [6] D. Buschmann, G. Stolz, J. Weidmann, One-dimensional schrodinger operators with local point interactions, J. Reine Angew. Math., 467 (1995), 169–186. https://doi.org/10.1515/crll.1995.467.169 doi: 10.1515/crll.1995.467.169
    [7] O. S. Mukhtarov, M. Kadakal, Some spectral properties of one Sturm-Liouville type problem with discontinuous weight, Sib. Math. J., 46 (2005), 681–694. https://doi.org/10.1007/s11202-005-0069-z doi: 10.1007/s11202-005-0069-z
    [8] O. S. Mukhtarov, E. Tunc, Eigenvalue problems for Sturm-Liouville equations with transmission conditions, Israel J. Math., 144 (2004), 367–380. https://doi.org/10.1007/BF02916718 doi: 10.1007/BF02916718
    [9] O. S. Mukhtarov, M. Kadakal, On a Sturm-Liouville type problem with discontinuous in two-points, Far East J. Appl. Math., 19 (2005), 337–352. Available from: https://www.researchgate.net/publication/267089496
    [10] A. V. Likov, Y. A. Mikhailov, The theory of heat and mass transfer (Russian), Qosenerqoizdat, 1963.
    [11] Q. Yang, W. Wang, Asymptotic behavior of a differential operator with discontinuities at two points, Math. Methods Appl. Sci., 34 (2011), 373–383. http://dx.doi.org/10.1002/mma.1361 doi: 10.1002/mma.1361
    [12] E. Tunc, O. S. Mukhtarov, Fundamental solutions and eigenvalues of one boundary-value problem with transmission conditions, Appl. Math. Comput., 157 (2004), 347–355. http://dx.doi.org/10.1016/j.amc.2003.08.039 doi: 10.1016/j.amc.2003.08.039
    [13] M. Demirci, Z. Akdoǧan, O. S. Mukhtarov, Asymptotic behavior of eigenvalues and eigenfunctions of one discontinuous boundary-value problem, Int. J. Comput. Cognition, 2 (2004), 101–113.
    [14] M. Kadakal, O. S. Mukhtarov, Sturm-Liouville problems with discontinuities at two points, Comput. Math. Appl., 54 (2007), 1367–1379. http://dx.doi.org/10.1016/j.camwa.2006.05.032 doi: 10.1016/j.camwa.2006.05.032
    [15] N. Altinisik, M. Kadakal, O. S. Mukhtarov, Eigenvalues and eigenfunctions of discontinuous Sturm-Liouville problems with eigenparameter in the boundary conditions, Acta Math. Hungar., 102 (2004), 159–193. http://dx.doi.org/10.1023/B:AMHU.0000023214.99631.52 doi: 10.1023/B:AMHU.0000023214.99631.52
    [16] C. T. Fulton, Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A, 77 (1977), 293–308. http://dx.doi.org/10.1017/s030821050002521x doi: 10.1017/s030821050002521x
    [17] D. B. Hinton, J. K. Shaw, Differential operators with spectral parameter incompletely in the boundary conditions, Funkcial. Ekvac., 33 (1990), 363–385. Available from: https://www.researchgate.net/publication/266135408.
    [18] E. Şen, O. S. Mukhtarov, Spectral properties of discontinuous Sturm-Liouville problems with a finite number of transmission conditions, Mediterr. J. Math., 13 (2016), 153–170. http://dx.doi.org/10.1007/s00009-014-0487-x doi: 10.1007/s00009-014-0487-x
    [19] Q. X. Yang, W. Y. Wang, A class of fourth order differential operators with transmission conditions, IJST Trans. A, 35 (2011), 323–332. http://dx.doi.org/10.22099/IJSTS.2011.2158 doi: 10.22099/IJSTS.2011.2158
    [20] E. Şen, On spectral properties of a fourth order boundary value problem, Ain Shams Eng. J., 4 (2013), 531–537. http://dx.doi.org/10.1016/j.asej.2012.11.006 doi: 10.1016/j.asej.2012.11.006
    [21] A. Zettl, Adjoint and self-adjoint boundary value problems with interface conditions, SIAM J. Appl. Math., 16 (1968), 851–859. Available from: https://www.jstor.org/stable/2099133.
    [22] P. A. Binding, P. J. Browne, B. A. Watson, Sturm-Liouville problems with boundary conditions raditionally dependent on the eigenparameter, J. Comput. Appl. Math., 148 (2002), 147–168. http://dx.doi.org/10.1016/S0377-0427(02)00579-4 doi: 10.1016/S0377-0427(02)00579-4
    [23] I. Dehghani, A. J. Akbarfam, Resolvent operator and self-adjointness of Sturm-Liouville operators with a finite number of transmission conditions, Mediterr. J. Math., 11 (2014), 447–462. http://dx.doi.org/10.1007/s00009-013-0338-1 doi: 10.1007/s00009-013-0338-1
    [24] K. Aydemir, O. S. Mukhtarov, Second-order differential operators with interior singularity, Adv. Difference Equ., 2015 (2015), 1–10. http://dx.doi.org/10.1186/s13662-015-0360-7 doi: 10.1186/s13662-015-0360-7
    [25] K. Aydemir, O. S. Mukhtarov, Spectrum and Green's function of a many-interval Sturm-Liouville problem, Z. Naturforsch., 70 (2015), 301–308. http://dx.doi.org/10.1515/zna-2015-0071 doi: 10.1515/zna-2015-0071
    [26] J. Cai, Z. Zheng, Inverse spectral problems for discontinuous Sturm-Liouville problems of Atkinson type, Appl. Math. Comput., 327 (2018), 22–34. http://dx.doi.org/10.1016/j.amc.2018.01.010 doi: 10.1016/j.amc.2018.01.010
    [27] J. Cai, Z. Zheng, A singular Sturm-Liouville problem with limit circle endpoints and eigenparameter dependent boundary conditions, Discrete Dyn. Nat. Soc., 2017 (2017), 1–12. http://dx.doi.org/10.1155/2017/9673846 doi: 10.1155/2017/9673846
    [28] J. Cai, Z. Zheng, Matrix representations of Sturm-Liouville problems with coupled eigenparameter-dependent boundary conditions and transmission conditions, Math. Methods Appl. Sci., 41 (2018), 3495–3508. https://doi.org/10.1002/mma.4842 doi: 10.1002/mma.4842
    [29] Z. Zheng, J. Cai, K. Li, M. Zhang, A discontinuous Sturm-Liouville problem with boundary conditions rationally dependent on the eigenparameter, Bound. Value Probl., 2018 (2018) 1–15. http://dx.doi.org/10.1186/s13661-018-1023-x
    [30] M. Kandemir, O. Sh. Mukhtarov, Nonclassical Sturm-Liouville problems with integral terms in the boundary conditions, Electron. J. Differ. Equ., 2017 (2017), 1–12. Available from: http://ejde.math.txstate.edu.
    [31] E. Şen, M. Acikgoz, S. Araci, Computation of eigenvalues and fundamental solutions of a fourth-order boundary value problem, Proc. Jangjeon Math. Soc., 15 (2012), 455–464. Available from: https://www.researchgate.net/publication/265570498.
    [32] E. Sȩn, K. Orucoglu, On spectral properties of a discontinuous fourth-order boundary value problem with transmission conditions, Ann. Univ. Oradea Math. Fasc., 21 (2014), 95–107.
  • Reader Comments
  • © 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(1472) PDF downloads(103) Cited by(2)

Article outline

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog