Eigenvalue problems for a class of Sturm-Liouville operators on two different time scales

Year 2022, Volume: 71 Issue: 3, 720 - 730, 30.09.2022

Zeynep Durna Ahmet Sinan Özkan

https://doi.org/10.31801/cfsuasmas.1036073

Abstract

In this study, we consider a boundary value problem generated by the Sturm-Liouville equation with a frozen argument and with non-separated boundary conditions on a time scale. Firstly, we present some solutions and the characteristic function of the problem on an arbitrary bounded time scale. Secondly, we prove some properties of eigenvalues and obtain a formulation for the eigenvalues-number on a finite time scale. Finally, we give an asymptotic formula for eigenvalues of the problem on another special time scale: $\mathbb{T}=[\alpha,\delta_{1}]\bigcup[\delta_{2},\beta].$

Keywords

Dynamic equations, time scales, measure chains, eigenvalue problems, Sturm-Liouville theory

Supporting Institution

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Project Number

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References

• Adalar, ˙I., Ozkan, A. S., An interior inverse Sturm–Liouville problem on a time scale, Analysis and Mathematical Physics, 10(4) (2020), 1-10. https://doi.org/10.1007/s13324-020-00402-2
• Agarwal, R. P., Bohner, M., Wong, P. J. Y., Sturm-Liouville eigenvalue problems on time scales, Appl. Math. Comput. 99 (1999), 153–166. https://doi.org/10.1016/S0096-3003(98)00004-6
• Albeverio S., Hryniv, R. O., Nizhink, L. P., Inverse spectral problems for non-local Sturm-Liouville operators, (1975), 2007-523-535. https://doi.org/10.1088/0266-5611/23/2/005
• Albeverio, S., Nizhnik, L., Schr¨odinger operators with nonlocal point interactions, J. Math. Anal. Appl., 332(2) (2007). https://doi.org/10.1016/j.jmaa.2006.10.070
• Allahverdiev, B. P., Tuna, H., Conformable fractional Sturm–Liouville problems on time scales, Mathematical Methods in the Applied Sciences, (2021). https://doi.org/10.1002/mma.7925
• Allahverdiev, B. P., Tuna, H., Dissipative Dirac operator with general boundary conditions on time scales, Ukrainian Mathematical Journal, 72(5) (2020). https://doi.org/10.37863/umzh.v72i5.546
• Allahverdiev, B. P., Tuna, H., Investigation of the spectrum of singular Sturm–Liouville operators on unbounded time scales, S˜ao Paulo Journal of Mathematical Sciences, 14(1) (2020), 327-340. https://doi.org/10.1007/s40863-019-00137-4
• Amster, P., De Napoli, P., Pinasco, J. P., Eigenvalue distribution of second-order dynamic equations on time scales considered as fractals, J. Math. Anal. Appl., 343 (2008), 573–584. https://doi.org/10.1016/j.jmaa.2008.01.070
• Amster, P., De Napoli, P., Pinasco, J. P., Detailed asymptotic of eigenvalues on time scales, J. Differ. Equ. Appl., 15 pp. (2009), 225–231. https://doi.org/10.1080/10236190802040976
• Atkinson, F., Discrete and Continuous Boundary Problems, Academic Press, New York, 1964. https://doi.org/10.1002/zamm.19660460520
• Barilla D., Bohner, B., Heidarkhani, S., Moradi, S., Existence results for dynamic Sturm–Liouville boundary value problems via variational methods, Applied Mathematics and Computation, 409, 125614 (2021). https://doi.org/10.1016/j.amc.2020.125614
• Berezin, F. A., Faddeev, L. D., Remarks on Schr¨odinger equation, Sov. Math.—Dokl., 137 (1961), 1011–4.
• Bohner, M., Peterson, A., Dynamic Equations on Time Scales, Birkhauser, Boston, MA, 2001.
• Bohner, M., Peterson, A., Advances in Dynamic Equations on Time Scales, Birkhauser, Boston, 2003. https://doi.org/10.1007/978-1-4612-0201-1
• Bondarenko, N. P., Buterin, S. A., Vasiliev, S.V., An inverse problem for Sturm -Liouville operators with frozen argument, Journal of Mathematical Analysis and Applications, 472(1) (2019), 1028-1041. https://doi.org/10.1016/j.jmaa.2018.11.062
• Buterin, S., Kuznetsova, M., On the inverse problem for Sturm–Liouville-type operators with frozen argument, rational case, Comp. Appl. Math., 39(5) (2020). https://doi.org/10.1007/s40314-019-0972-8
• Davidson, F. A., Rynne, B. P., Global bifurcation on time scales, J. Math. Anal. Appl., 267 (2002), 345–360. https://doi.org/10.1006/jmaa.2001.7780
• Davidson, F. A., Rynne, B. P., Self-adjoint boundary value problems on time scales, Electron. J. Differ. Equ., 175 (2007), 1–10.
• Davidson, F. A., Rynne, B. P., Eigenfunction expansions in $L^2$ spaces for boundary value problems on time-scales, J. Math. Anal. Appl., 335 (2007), 1038–1051. https://doi.org/10.1016/j.jmaa.2007.01.100
• Erbe, L., Hilger, S., Sturmian theory on measure chains, Differ. Equ. Dyn. Syst., 1 (1993), 223–244.
• Erbe, L., Peterson, A., Eigenvalue conditions and positive solutions, J. Differ. Equ. Appl., 6 (2000), 165–191. https://doi.org/10.1080/10236190008808220
• Guseinov, G. S., Eigenfunction expansions for a Sturm-Liouville problem on time scales, Int. J. Differ. Equ., 2 (2007), 93–104. https://doi.org/10.37622/000000
• Guseinov, G. S., An expansion theorem for a Sturm-Liouville operator on semi-unbounded time scales, Adv. Dyn. Syst. Appl., 3 (2008), 147–160. https://doi.org/10.37622/000000
• Heidarkhani, S., Bohner, B., Caristi, G., Ayazi F., A critical point approach for a second-order dynamic Sturm–Liouville boundary value problem with p -Laplacian, Applied Mathematics and Computation, 409 (2021), 125521. https://doi.org/10.1016/j.amc.2020.125521
• Heidarkhani, S., Moradi, S., Caristi G., Existence results for a dynamic Sturm–Liouville boundary value problem on time scales, Optimization Letters, 15 (2021), 2497–2514. https://doi.org/10.1007/s11590-020-01646-4
• Hu, Y. T., Bondarenko, N. P., Yang, C. F., Traces and inverse nodal problem for Sturm–Liouville operators with frozen argument, Applied Mathematics Letters, 102 (2020), 106096. https://doi.org/10.1016/j.aml.2019.106096
• Hilscher, R. S., Zemanek, P., Weyl-Titchmarsh theory for time scale symplectic systems on half line, Abstr. Appl. Anal., 738520, (2011), 41 pp. https://doi.org/10.1155/2011/738520
• Huseynov, A., Limit point and limit circle cases for dynamic equations on time scales, Hacet. J. Math. Stat., 39 (2010), 379–392.
• Huseynov, A., Bairamov, E., On expansions in eigenfunctions for second order dynamic equations on time scales, Nonlinear Dyn. Syst. Theo., 9 (2009), 7–88.
• Kong, Q., Sturm-Liouville problems on time scales with separated boundary conditions, Results Math., 52 (2008), 111–121. https://doi.org/10.1007/s00025-007-0277-x
• Krall, A. M., The development of general differential and general differential-boundary systems, Rocky Mount. J. Math., 5 (1975), 493–542.
• Lakshmikantham, V., Sivasundaram, S., Kaymakcalan, B., Dynamic Systems on Measure Chains, Kluwer Academic Publishers, Dordrecht, 1996. https://doi.org/10.1007/978-1-4757- 2449-3
• Nizhink, L. P., Inverse eigenvalue problems for non-local Sturm Liouville problems, Methods Funct. Anal. Topology, 15(1) (2009), 41-47.
• Ozkan, A. S., Sturm-Liouville operator with parameter-dependent boundary conditions on time scales, Electron. J. Differential Equations, 212 (2017), 1-10.
• Ozkan, A. S., Adalar, I., Half-inverse Sturm-Liouville problem on a time scale, Inverse Probl. 36 (2020), 025015. https://doi.org/10.1088/1361-6420/ab2a21
• Rynne, B. P., L2 spaces and boundary value problems on time-scales, J. Math. Anal. Appl., 328 (2007), 1217–1236. https://doi.org/10.1016/j.jmaa.2006.06.008
• Sun, S., Bohner, M., Chen, S., Weyl-Titchmarsh theory for Hamiltonian dynamic systems, Abstr. Appl. Anal. Art., 514760 (2010), 18 pp . https://doi.org/10.1155/2010/514760
• Wentzell A. D., On boundary conditions for multidimensional diffusion processes, Theory Probab., 4 (1959), 164–77. (Engl. Transl.) https://doi.org/10.1137/1104014
• Yurko, V. A., Inverse problems for Sturm-Liouville differential operators on closed sets, Tamkang Journal of Mathematics, 50(3) (2019), 199-206. https://doi.org/10.5556/j.tkjm.50.2019.3343