Asymptotic distribution of eigenvalues and eigenfunctions of a nonlocal boundary value problem
Abstract
In this work, we obtain asymptotic formulas for eigenvalues and eigenfunctions of the second order boundary-value problem with a Bitsadze–Samarskii type nonlocal boundary condition.
Keyword : differential equation, nonlocal conditions, asymptotics of eigenvalues and eigenfunctions
How to Cite
Şen, E., & Štikonas, A. (2021). Asymptotic distribution of eigenvalues and eigenfunctions of a nonlocal boundary value problem. Mathematical Modelling and Analysis, 26(2), 253-266. https://doi.org/10.3846/mma.2021.13056
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
K. Aydemir and O.Sh. Mukhtarov. Asymptotic distribution of eigenvalues and eigenfunctions for a multi-point discontinuous Sturm–Liouville problem. Electron. J. Differential Equations, 2016(131):1–14, 2016. Available from Internet: https://ejde.math.txstate.edu/Volumes/2016/131/aydemir.pdf
K. Aydemir and O.Sh. Mukhtarov. Class of Sturm–Liouville problems with eigenparameter dependent transmission conditions. Numer. Funct. Anal. Optim., 38(10):1260–1275, 2017. https://doi.org/10.1080/01630563.2017.1316995
S.A. Beilin. Existence of solutions for one-dimensional wave equations with nonlocal conditions. Electron. J. Diff. Eqns., 2001(76):1–8, 2001. Available from Internet: https://ejde.math.txstate.edu/Volumes/2001/76/abstr.html
K. Bingelė, A. Bankauskienė and A. Štikonas. Spectrum curves for a discrete Sturm–Liouville problem with one integral boundary condition. Nonlinear Anal. Model. Control, 24(5):755–774, 2019. https://doi.org/10.15388/NA.2019.5.5
K. Bingelė, A. Bankauskienė and A. Štikonas. Investigation of spectrum curves for a Sturm–Liouville problem with two-point nonlocal boundary conditions. Math. Model. Anal., 25(1):53–70, 2020. https://doi.org/10.3846/mma.2020.10787
R. Čiegis, A. Štikonas, O. Štikonienė and O. Suboč. Stationary Problems with Nonlocal Boundary Conditions. Math. Model. Anal., 6(2):178–191, 2001. https://doi.org/10.3846/13926292.2001.9637157
R. Courant and D. Hilbert. Methods Of Mathematical Physics, vol. 1. Interscience, New York, 1953.
L. Crocco and S. Chang. Theory of combustion instability in liquid propellant rocket motors. Butterworths, London, 1956.
C.T. Fulton. Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions. Proc. Edinb. Math. Soc. A, 77(3–4):293– 308, 1977. https://doi.org/10.1017/S030821050002521X
F. Gesztesy and W. Kirsch. One-dimensional schrdinger operators with interactions singular on a discrete set. J. Reine Angew. Math., 362:28–50, 1985. https://doi.org/10.1515/crll.1985.362.28
G. Infante. Eigenvalues of some nonlocal boundary-value problems. Proc. Edinb. Math. Soc. (Series 2), 46:75–86, 2003. https://doi.org/10.1017/S0013091501001079
N.I. Ionkin and E.A. Valikova. On eigenvalues and eigenfunctions of a nonclassical boundary value problem. Matem. Mod., 8(1):53–63, 1996. (in Russian)
A.G. Kostyuchenko and I.S. Sargsjan. Distribution of eigenvalues. Selfadjoint ordinary differential operators. Nauka, Moscow, 1979. (in Russian)
B.M. Levitan and I.S. Sargsjan. Sturm–Liouville and Dirac operators. Kluwer, Dordrecht, 1991.
V. Mityushev and P.M. Adler. Darcy flow around a two-dimensional permeable lens. J. Phys. A: Math. Gen., 39(14):3545–3560, 2006. https://doi.org/10.1088/0305-4470/39/14/004
O.Sh. Mukhtarov, H. Olğar and K. Aydemir. Resolvent operator and spectrum of new type boundary value problems. Filomat, 29(7):1671–1680, 2015. https://doi.org/10.2298/FIL1507671M
S.B. Norkin. Differential equations of the second order with retarded argument, Translations of Mathematical Monographs, volume 31. AMS, Providence, RI, 1972.
S. Roman and A. Štikonas. Green’s functions for stationary problems with nonlocal boundary conditions. Lith. Math. J., 49(2):190–202, 2009. https://doi.org/10.1007/s10986-009-9041-0
E. Şen and A. Bayramov. Calculation of eigenvalues and eigenfunctions of a discontinuous boundary value problem with retarded argument which contains a spectral parameter in the boundary condition. Math. Comput. Modelling, 54(11– 12):3090–3097, 2011. https://doi.org/10.1016/j.mcm.2011.07.039
E. Şen and A. Štikonas. Computation of eigenvalues and eigenfunctions of a nonlocal boundary value problem with retarded argument. Complex Var. Elliptic Equ., pp. 1–16, 2021. https://doi.org/10.1080/17476933.2021.1890054 (Online First)
A. Štikonas. The Sturm–Liouville problem with a nonlocal boundary condition. Lith. Math. J.,47(3):336–351,2007. https://doi.org/10.1007/s10986-007-0023-9
A. Štikonas. Investigation of characteristic curve for Sturm–Liouville problem with nonlocal boundary conditions on torus. Math. Model. Anal., 16(1):1–22, 2011. https://doi.org/10.3846/13926292.2011.552260
A. Štikonas. A survey on stationary problems, Green’s functions and spectrum of Sturm–Liouville problem with nonlocal boundary conditions. Nonlinear Anal. Model. Control, 19(3):301–334, 2014. https://doi.org/10.15388/NA.2014.3.1
A. Štikonas and O. Štikonienė. Characteristic functions for Sturm–Liouville problems with nonlocal boundary conditions. Math. Model. Anal., 14(2):229– 246, 2009. https://doi.org/10.3846/1392-6292.2009.14.229-246
A.I. Sherstyuk. Problems of Theoretical Physics. Leningrad. Gos. Univ., Leningrad, 1988.
E.C. Titchmarsh. Eigenfunction expansions associated with second-order differential equations. Clarendon Press, Oxford, 1946.
Z. Zheng, J. Cai, K. Li and M. Zhang. A discontinuous Sturm–Liouville problem with boundary conditions rationally dependent on the eigenparameter. Bound. Value Probl., 2018:103.1–15, 2018. https://doi.org/10.1186/s13661-018-1023-x
K. Aydemir and O.Sh. Mukhtarov. Class of Sturm–Liouville problems with eigenparameter dependent transmission conditions. Numer. Funct. Anal. Optim., 38(10):1260–1275, 2017. https://doi.org/10.1080/01630563.2017.1316995
S.A. Beilin. Existence of solutions for one-dimensional wave equations with nonlocal conditions. Electron. J. Diff. Eqns., 2001(76):1–8, 2001. Available from Internet: https://ejde.math.txstate.edu/Volumes/2001/76/abstr.html
K. Bingelė, A. Bankauskienė and A. Štikonas. Spectrum curves for a discrete Sturm–Liouville problem with one integral boundary condition. Nonlinear Anal. Model. Control, 24(5):755–774, 2019. https://doi.org/10.15388/NA.2019.5.5
K. Bingelė, A. Bankauskienė and A. Štikonas. Investigation of spectrum curves for a Sturm–Liouville problem with two-point nonlocal boundary conditions. Math. Model. Anal., 25(1):53–70, 2020. https://doi.org/10.3846/mma.2020.10787
R. Čiegis, A. Štikonas, O. Štikonienė and O. Suboč. Stationary Problems with Nonlocal Boundary Conditions. Math. Model. Anal., 6(2):178–191, 2001. https://doi.org/10.3846/13926292.2001.9637157
R. Courant and D. Hilbert. Methods Of Mathematical Physics, vol. 1. Interscience, New York, 1953.
L. Crocco and S. Chang. Theory of combustion instability in liquid propellant rocket motors. Butterworths, London, 1956.
C.T. Fulton. Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions. Proc. Edinb. Math. Soc. A, 77(3–4):293– 308, 1977. https://doi.org/10.1017/S030821050002521X
F. Gesztesy and W. Kirsch. One-dimensional schrdinger operators with interactions singular on a discrete set. J. Reine Angew. Math., 362:28–50, 1985. https://doi.org/10.1515/crll.1985.362.28
G. Infante. Eigenvalues of some nonlocal boundary-value problems. Proc. Edinb. Math. Soc. (Series 2), 46:75–86, 2003. https://doi.org/10.1017/S0013091501001079
N.I. Ionkin and E.A. Valikova. On eigenvalues and eigenfunctions of a nonclassical boundary value problem. Matem. Mod., 8(1):53–63, 1996. (in Russian)
A.G. Kostyuchenko and I.S. Sargsjan. Distribution of eigenvalues. Selfadjoint ordinary differential operators. Nauka, Moscow, 1979. (in Russian)
B.M. Levitan and I.S. Sargsjan. Sturm–Liouville and Dirac operators. Kluwer, Dordrecht, 1991.
V. Mityushev and P.M. Adler. Darcy flow around a two-dimensional permeable lens. J. Phys. A: Math. Gen., 39(14):3545–3560, 2006. https://doi.org/10.1088/0305-4470/39/14/004
O.Sh. Mukhtarov, H. Olğar and K. Aydemir. Resolvent operator and spectrum of new type boundary value problems. Filomat, 29(7):1671–1680, 2015. https://doi.org/10.2298/FIL1507671M
S.B. Norkin. Differential equations of the second order with retarded argument, Translations of Mathematical Monographs, volume 31. AMS, Providence, RI, 1972.
S. Roman and A. Štikonas. Green’s functions for stationary problems with nonlocal boundary conditions. Lith. Math. J., 49(2):190–202, 2009. https://doi.org/10.1007/s10986-009-9041-0
E. Şen and A. Bayramov. Calculation of eigenvalues and eigenfunctions of a discontinuous boundary value problem with retarded argument which contains a spectral parameter in the boundary condition. Math. Comput. Modelling, 54(11– 12):3090–3097, 2011. https://doi.org/10.1016/j.mcm.2011.07.039
E. Şen and A. Štikonas. Computation of eigenvalues and eigenfunctions of a nonlocal boundary value problem with retarded argument. Complex Var. Elliptic Equ., pp. 1–16, 2021. https://doi.org/10.1080/17476933.2021.1890054 (Online First)
A. Štikonas. The Sturm–Liouville problem with a nonlocal boundary condition. Lith. Math. J.,47(3):336–351,2007. https://doi.org/10.1007/s10986-007-0023-9
A. Štikonas. Investigation of characteristic curve for Sturm–Liouville problem with nonlocal boundary conditions on torus. Math. Model. Anal., 16(1):1–22, 2011. https://doi.org/10.3846/13926292.2011.552260
A. Štikonas. A survey on stationary problems, Green’s functions and spectrum of Sturm–Liouville problem with nonlocal boundary conditions. Nonlinear Anal. Model. Control, 19(3):301–334, 2014. https://doi.org/10.15388/NA.2014.3.1
A. Štikonas and O. Štikonienė. Characteristic functions for Sturm–Liouville problems with nonlocal boundary conditions. Math. Model. Anal., 14(2):229– 246, 2009. https://doi.org/10.3846/1392-6292.2009.14.229-246
A.I. Sherstyuk. Problems of Theoretical Physics. Leningrad. Gos. Univ., Leningrad, 1988.
E.C. Titchmarsh. Eigenfunction expansions associated with second-order differential equations. Clarendon Press, Oxford, 1946.
Z. Zheng, J. Cai, K. Li and M. Zhang. A discontinuous Sturm–Liouville problem with boundary conditions rationally dependent on the eigenparameter. Bound. Value Probl., 2018:103.1–15, 2018. https://doi.org/10.1186/s13661-018-1023-x