A dissipative third-order boundary value problem with distributional potentials and eigenparameter-dependent boundary conditions

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Abstract

This paper investigates a class of dissipative boundary value problems arising from a third-order differential equation with distributional potentials and eigenparameter-dependent boundary conditions. Initially, we transform the boundary value problem into the corresponding operator problem. We then demonstrate that the operator is dissipative and examine certain eigenvalue properties of the operator. Furthermore, by applying Krein’s theorem, we establish the completeness theorems for both the boundary value problem and the corresponding operator.

Keywords

third-order boundary value problem / dissipative differential operator / distributional potentials / completeness theorems

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References

[1] S. Albeverio, F. Gesztesy, R. Hoegh-Krohn, H. Holden, Solvable Models in Quantum Mechanics , Second Edition, AMS Chelsea Publ., Providence, RI, 2005.

[2] P. Kurasov, On the Coulomb potential in one dimension, J. Phys. A Math. Gen., 29 (1996), 1767. https://doi.org/10.1088/0305-4470/29/8/023

[3] N. J. Guliyev, Schrödinger operators with distributional potentials and boundary conditions dependent on the eigenvalue parameter, J. Math. Phys., 60 (2019), 063501. https://doi.org/10.1063/1.5048692

[4] J. Yan, G. L. Shi, Inequalities among eigenvalues of Sturm-Liouville problems with distribution potentials, J. Math. Anal. Appl., 409 (2014), 509-520. https://doi.org/10.1016/j.jmaa.2013.07.024

[5] H. Y. Zhang, J. J. Ao, F. Z. Bo, Eigenvalues of fourth-order boundary value problems with distributional potentials, AIMS Math., 7 (2022), 7294-7317. https://doi.org/10.3934/math.2022407

[6] J. Eckhardt, F. Gesztesy, R. Nichols, G. Teschl, Inverse spectral theory for Sturm-Liouville operators with distributional potentials, J. London Math. Soc. Second Ser., 88 (2013), 801-828. https://doi.org/10.1112/jlms/jdt041

[7] E. L. Korotyaev, Resonances of third order differential operators, J. Math. Anal. Appl., 478 (2019), 82-107. https://doi.org/10.1016/j.jmaa.2019.05.007

[8] A. Badanin, E. L. Korotyaev, Third-order operators with three-point conditions associated with Boussinesqs equation, Appl. Anal., 100 (2021), 527-560. https://doi.org/10.1080/00036811.2019.1610941

[9] H. Y. Zhang, J. J. Ao, Some eigenvalue properties of third-order boundary value problems with distributional potentials, Acta Math. Appl. Sin. Engl. Ser., 41 (2025), 179-199. https://doi.org/10.1007/s10255-023-1064-5

[10] N. P. Bondarenko, Inverse spectral problem for the third-order differential equation, Results Math., 78 (2023), 179. https://doi.org/10.1007/s00025-023-01955-x

[11] M. Gregus, Third Order Linear Differential Equations , Mathematics and Its Applications, Reidel, Dordrecht, 1987.

[12] F. Bernis, L. A. Peletier, Two problems from draining flows involving third-order ordinary differential equations, SIAM J. Math. Anal., 27 (1996), 515-527. https://doi.org/10.1137/S0036141093260847

[13] E. O. Tuck, L. W. Schwartz, A numerical and asymptotic study of some third-order ordinary differential equations relevant to draining and coating flows, SIAM Rev., 32 (1990), 453-469. https://doi.org/10.1137/1032079

[14] C. Tretter, Boundary eigenvalue problems with differential equations Nη = λPη with λ-polynomial boundary conditions , J. Differ. Equations, 170 (2001), 408-471. https://doi.org/10.1006/jdeq.2000.3829

[15] C. Fulton, S. Pruess, Numerical methods for a singular eigenvalue problem with eigenparameter in the boundary conditions, J. Math. Anal. Appl., 71 (1979), 431-462. https://doi.org/10.1016/0022-247X(79)90203-8

[16] J. Walter, Regular eigenvalue problems with eigenvalue parameter in the boundary conditions, Math. Z., 133 (1973), 301-312. https://doi.org/10.1007/BF01177870

[17] Z. S. Aliyev, A. A. Dunyamalieva, Defect basis property of a system of root functions of a Sturm-Liouville problem with spectral parameter in the boundary conditions, Differ. Equations, 51 (2015), 1249-1266. https://doi.org/10.1134/S0012266115100018

[18] R. S. Phillips, Dissipative operators and hyperbolic systems of partial differential equations, Trans. Am. Math. Soc., 90 (1959), 193-254. https://doi.org/10.2307/1993202

[19] P. D. Lax, R. S. Phillips, Scattering theory for dissipative hyperbolic systems, J. Funct. Anal., 14 (1973), 172-235. https://doi.org/10.1016/0022-1236(73)90049-9

[20] E. Bairamov, A. M. Krall, Dissipative operators generated by the Sturm-Liouville differential expression in the Weyl limit circle case, J. Math. Anal. Appl., 254 (2001), 178-190. https://doi.org/10.1006/jmaa.2000.7233

[21] Z. Wang, H. Wu, Dissipative non-self-adjoint Sturm-Liouville operators and completeness of their eigenfunctions, J. Math. Anal. Appl., 394 (2012), 1-12. https://doi.org/10.1016/j.jmaa.2012.04.071

[22] E. Bairamov, E. Uğurlu, The determinants of dissipative Sturm-Liouville operators with transmission conditions, Math. Comput. Modell., 53 (2011), 805-813. https://doi.org/10.1016/j.mcm.2010.10.017

[23] E. Bairamov, E. Uğurlu, On the characteristic values of the real component of a dissipative boundary value transmission problem, Appl. Math. Comput., 218 (2012), 9657-9663. https://doi.org/10.1016/j.amc.2012.02.079

[24] E. Bairamov, E. Uğurlu, Krein's theorems for a dissipative boundary value transmission problem, Complex Anal. Oper. Theory, 7 (2013), 831-842. https://doi.org/10.1007/s11785-011-0180-z

[25] B. P. Allahverdiev, H. Tuna, Y. Yalcinkaya, A completeness theorem for dissipative conformable fractional Sturm-Liouville operator in singular case, Filomat, 36 (2022), 2461-2474. https://doi.org/10.2298/FIL2207461A

[26] T. Wang, J. J. Ao, A. Zettl, A class of dissipative differential operators of order three, AIMS Math., 6 (2021), 7034-7043. https://doi.org/10.3934/math.2021412

[27] X. Y. Zhang, J. Sun, The determinants of fourth order dissipative operators with transmission conditions, J. Math. Anal. Appl., 410 (2014), 55-69. https://doi.org/10.1016/j.jmaa.2013.08.004

[28] E. Uğurlu, Extensions of a minimal third-order formally symmetric operator, Bull. Malays. Math. Sci. Soc., 43 (2020), 453-470. https://doi.org/10.1007/s40840-018-0696-8

[29] B. P. Allahverdiev, A nonself-adjoint singular Sturm-Liouville problem with a spectral parameter in the boundary condition, Math. Nachr., 278 (2005), 743-755. https://doi.org/10.1002/mana.200310269

[30] H. Tuna, On spectral properties of dissipative fourth order boundary-value problem with a spectral parameter in the boundary condition, Appl. Math. Comput., 219 (2013), 9377-9387. https://doi.org/10.1016/j.amc.2013.03.010

[31] I. Gohberg, M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators , American Mathematical Soc., Providence R.I., 1969.

[32] M. G. Krein, On the indeterminate case of the Sturm-Liouville boundary problem in the interval (0,∞), Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 16 (1952), 293-324.