Asymptotic analysis of Sturm-Liouville problem with Dirichlet and nonlocal two-point boundary conditions
Abstract
In this study, we obtain asymptotic expansions for eigenvalues and eigenfunctions of the one–dimensional Sturm–Liouville equation with one classical Dirichlet type boundary condition and two-point nonlocal boundary condition. We analyze the characteristic equation of the boundary value problem for eigenvalues and derive asymptotic expansions of arbitrary order. We apply the obtained results to the problem with two-point nonlocal boundary condition.
Keyword : Sturm–Liouville problem, Dirichlet condition, two-point nonlocal conditions, asymptotics of eigenvalues and eigenfunctions
How to Cite
Štikonas, A., & Şen, E. (2023). Asymptotic analysis of Sturm-Liouville problem with Dirichlet and nonlocal two-point boundary conditions. Mathematical Modelling and Analysis, 28(2), 308–329. https://doi.org/10.3846/mma.2023.17617
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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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
A. Boumenir. Eigenvalues of periodic Sturm–Liouville problems by the Shannon–Whittaker sampling theorem. Math. Comput., 68(227):1057–1066, 1999. https://doi.org/10.1090/S0025-5718-99-01053-4
J. Cai, K. Li and Z. Zheng. A singular Sturm–Liouville problem with limit circle endpoint and boundary conditions rationally dependent on the eigenparameter. Mediterr. J. Math., 19(184):1–15, 2022. https://doi.org/10.1007/s00009-022-02109-z
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
H. Coşkun and E. Başkaya. Asymptotics of eigenvalues of regular Sturm– Liouville problems with eigenvalue parameter in the boundary condition for integrable potential. Math. Scand., 107(2):209–223, 2010.
M. Eastham. The Spectral Theory of Periodic Differential Equations. Scottish Academic Press, Edinburgh, 1973.
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
N.J. Guliyev. Essentially isospectral transformations and their applications. Ann. Mat. Pura Appl., 199(4):1621–1648, 2020. https://doi.org/10.1007/s10231-019-00934-w
N.J. Guliyev. On two-spectra inverse problems. Proc. Amer. Math. Soc., 148(10):4491–4502, 2020. https://doi.org/10.1090/proc/15155
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
G. Infante. Positive solutions of systems of perturbed Hammerstein integral equations with arbitrary order dependence. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci, 379(2191):20190376(1–10), 2021. https://doi.org/10.1098/rsta.2019.0376
G. Infante, P. Pietramala and F.A.F. Tojo. Non-trivial solutions of local and non-local Neumann boundary-value problems. Proc. Roy. Soc. Edinburgh Sect. A, 146(2):337–369, 2016. https://doi.org/10.1017/S0308210515000499
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. Pečiulytė, O. Štikonienė and A. Štikonas. Investigation of negative critical points of the characteristic function for problems with nonlocal boundary conditions. Nonlinear Anal. Model. Control, 13(4):467–490, 2008. https://doi.org/10.15388/NA.2008.13.4.14552
S. Pečiulytė and A. Štikonas. On positive eigenfunctions of Sturm–Liouville problem with nonlocal two-point boundary condition. Math. Model. Anal., 12(2):215– 226, 2007. https://doi.org/10.3846/1392-6292.2007.12.215-226
M. Pikula, E. Čatrnja and I. Kalčo. Spectral problems for operators with deviating arguments. Hacettepe J. Math. Stat., 47(5):1172–1183, 2018. https://doi.org/10.15672/HJMS.2017.494
S. Roman and A. Štikonas. Third-order linear differential equation with three additional conditions and formula for Green’s function. Lith. Math. J., 50(4):426– 446, 2010. https://doi.org/10.1007/s10986-010-9097-x
S. Roman and A. Štikonas. Green’s function for discrete second-order problems with nonlocal boundary conditions. Bound. Value Probl., 2011(Article ID 767024):1–23, 2011. https://doi.org/10.1155/2011/767024
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. Asymptotic distribution of eigenvalues and eigenfunctions of a nonlocal boundary value problem. Math. Model. Anal., 26(2):253–266, 2021. https://doi.org/10.3846/mma.2021.13056
E. Şen and A. Štikonas. Computation of eigenvalues and eigenfunctions of a non-local boundary value problem with retarded argument. Complex Var. Elliptic Equ., 67(7):1662–1676, 2022. https://doi.org/10.1080/17476933.2021.1890054
A. Skučaitė and A. Štikonas. Spectrum curves for Sturm–Liouville Problem with Integral Boundary Condition. Math. Model. Anal., 20(6):802–818, 2015. https://doi.org/10.3846/13926292.2015.1116470
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 E. Şen. Asymptotic analysis of Sturm–Liouville problem with nonlocal integral-type boundary condition. Nonlinear Anal. Model. Control, 26(5):969–991, 2021. https://doi.org/10.15388/namc.2021.26.24299
A. Štikonas and E. Şen. Asymptotic analysis of Sturm–Liouville problem with Neumann and nonlocal two-point boundary conditions. Lith. Math. J, 62(4):519– 541, 2022. https://doi.org/10.1007/s10986-022-09577-6
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
F. Sun, K. Li and J. Cai. Bounds on the non-real eigenvalues of nonlocal indefinite Sturm–Liouville problems with coupled boundary conditions. Complex Anal. Oper. Theory, 16(30):1–12, 2022. https://doi.org/10.1007/s11785-022-01202-1
E.C. Titchmarsh. Eigenfunction expansions associated with second-order differential equations. Clarendon Press, Oxford, 1946.
E. Başkaya. Periodic and semi-periodic eigenvalues of Hill’s equation with symmetric double well potential. TWMS J. App. and Eng. Math., 10(2):346–352, 2020.
A. Bayramov, S. Uslu and S. Ҫalıṣkan. Computation of eigenvalues and eigenfunctions of a discontinuous boundary value problem with retarded argument. Appl. Math. Comput., 191(2):592–600, 2007. https://doi.org/10.1016/j.amc.2007.02.118
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
A. Boumenir. Eigenvalues of periodic Sturm–Liouville problems by the Shannon–Whittaker sampling theorem. Math. Comput., 68(227):1057–1066, 1999. https://doi.org/10.1090/S0025-5718-99-01053-4
J. Cai, K. Li and Z. Zheng. A singular Sturm–Liouville problem with limit circle endpoint and boundary conditions rationally dependent on the eigenparameter. Mediterr. J. Math., 19(184):1–15, 2022. https://doi.org/10.1007/s00009-022-02109-z
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
H. Coşkun and E. Başkaya. Asymptotics of eigenvalues of regular Sturm– Liouville problems with eigenvalue parameter in the boundary condition for integrable potential. Math. Scand., 107(2):209–223, 2010.
M. Eastham. The Spectral Theory of Periodic Differential Equations. Scottish Academic Press, Edinburgh, 1973.
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
N.J. Guliyev. Essentially isospectral transformations and their applications. Ann. Mat. Pura Appl., 199(4):1621–1648, 2020. https://doi.org/10.1007/s10231-019-00934-w
N.J. Guliyev. On two-spectra inverse problems. Proc. Amer. Math. Soc., 148(10):4491–4502, 2020. https://doi.org/10.1090/proc/15155
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
G. Infante. Positive solutions of systems of perturbed Hammerstein integral equations with arbitrary order dependence. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci, 379(2191):20190376(1–10), 2021. https://doi.org/10.1098/rsta.2019.0376
G. Infante, P. Pietramala and F.A.F. Tojo. Non-trivial solutions of local and non-local Neumann boundary-value problems. Proc. Roy. Soc. Edinburgh Sect. A, 146(2):337–369, 2016. https://doi.org/10.1017/S0308210515000499
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. Pečiulytė, O. Štikonienė and A. Štikonas. Investigation of negative critical points of the characteristic function for problems with nonlocal boundary conditions. Nonlinear Anal. Model. Control, 13(4):467–490, 2008. https://doi.org/10.15388/NA.2008.13.4.14552
S. Pečiulytė and A. Štikonas. On positive eigenfunctions of Sturm–Liouville problem with nonlocal two-point boundary condition. Math. Model. Anal., 12(2):215– 226, 2007. https://doi.org/10.3846/1392-6292.2007.12.215-226
M. Pikula, E. Čatrnja and I. Kalčo. Spectral problems for operators with deviating arguments. Hacettepe J. Math. Stat., 47(5):1172–1183, 2018. https://doi.org/10.15672/HJMS.2017.494
S. Roman and A. Štikonas. Third-order linear differential equation with three additional conditions and formula for Green’s function. Lith. Math. J., 50(4):426– 446, 2010. https://doi.org/10.1007/s10986-010-9097-x
S. Roman and A. Štikonas. Green’s function for discrete second-order problems with nonlocal boundary conditions. Bound. Value Probl., 2011(Article ID 767024):1–23, 2011. https://doi.org/10.1155/2011/767024
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. Asymptotic distribution of eigenvalues and eigenfunctions of a nonlocal boundary value problem. Math. Model. Anal., 26(2):253–266, 2021. https://doi.org/10.3846/mma.2021.13056
E. Şen and A. Štikonas. Computation of eigenvalues and eigenfunctions of a non-local boundary value problem with retarded argument. Complex Var. Elliptic Equ., 67(7):1662–1676, 2022. https://doi.org/10.1080/17476933.2021.1890054
A. Skučaitė and A. Štikonas. Spectrum curves for Sturm–Liouville Problem with Integral Boundary Condition. Math. Model. Anal., 20(6):802–818, 2015. https://doi.org/10.3846/13926292.2015.1116470
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 E. Şen. Asymptotic analysis of Sturm–Liouville problem with nonlocal integral-type boundary condition. Nonlinear Anal. Model. Control, 26(5):969–991, 2021. https://doi.org/10.15388/namc.2021.26.24299
A. Štikonas and E. Şen. Asymptotic analysis of Sturm–Liouville problem with Neumann and nonlocal two-point boundary conditions. Lith. Math. J, 62(4):519– 541, 2022. https://doi.org/10.1007/s10986-022-09577-6
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
F. Sun, K. Li and J. Cai. Bounds on the non-real eigenvalues of nonlocal indefinite Sturm–Liouville problems with coupled boundary conditions. Complex Anal. Oper. Theory, 16(30):1–12, 2022. https://doi.org/10.1007/s11785-022-01202-1
E.C. Titchmarsh. Eigenfunction expansions associated with second-order differential equations. Clarendon Press, Oxford, 1946.