Incomplete and Interior Inverse Problem for a Discontinuous Differential Pencil with the Spectral Boundary Condition on the Half-Line

Document Type : Original Scientific Paper

Authors

1 Department of Basic Sciences, Sari Agricultural Sciences and Natural Resources University, 578 Sari, Iran

2 Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran

Abstract

Differential pencils on the half-line with a spectral boundary condition having a discontinuity in an interior point are investigated. We prove two uniqueness theorems: (i) knowing β1 ,β0 and potentials p,q on (0,a), only eigenvalues suffice to determine the boundary value problem B. (ii) some in formation on eigenfunctions at x = a and eigenvalues establish the boundary value problem B.

Keywords


[1] R. S. Anderssen, The effect of discontinuities in density and shear velocity on the asymptotic overtone structure of tortional eigenfrequencies of the Earth, Geophys J. R. Astr. Soc. 50 (1997) 303 − 309.
[2] E. Bairamov and C. Coskun, Jost solutions and the spectrum of the system of difference equations, Appl. Math. Lett. 17 (2004) 1039 − 1045.
[3] S. A. Buterin and C. T. Shieh, Incomplete inverse spectral and nodal problems for differential pencils, Results Math. 62 (2012) 167 − 179.
[4] S. A. Buterin and V. A. Yurko, Inverse problems for second-order differential pencils with Dirichlet boundary conditions, J. Inverse Ill-Posed Probl. 20 (5-6) (2012) 855 − 881.
[5] J. B. Conway, Functions of One Complex Variable, 2nd ed., Vol. I, Springer-Verlag, New York, 1995.
[6] G. Freiling and V. Yurko, Inverse spectral problems for singular non-selfadjoint differential operators with discontinuities in an interior point, Inverse Problems 18 (3) (2002) 757.
[7] G. Freiling and V. A. Yurko, Inverse Sturm-Liouville Problems and their Applications, NOVA Science Publ., Inc., Huntington, New York, 2001.
[8] N. J. Guliyev, Inverse eigenvalue problems for Sturm-Liouville equations with spectral parameter linearly contained in one of the boundary conditions, Inverse problem 21 (2005) 1315 − 1330.
[9] H. Hochstadt and B. Lieberman, An inverse Sturm-Liouville problem with mixed given data, SIAM J. Appl. Math. 34 (4) (1978) 676 − 680.
[10] M. Jaulent and C. Jean, The inverses-wave scattering problem for a class of potentials depending on energy, Commun. Math. Phys. 28 (1972) 177 − 220.
[11] Y. Khalili and D. Baleanu, A uniqueness result for differential pencils with discontinuities from interior spectral data, Analysis 38 (4) (2018) 195 − 202.
[12] Y. Khalili, M. Yadollahzadeh and M. K. Moghadam, Half inverse problems for the impulsive operator with eigenvalue-dependent boundary conditions, Electronic. J. Diff. Equs. 2017 (190) (2017) 1 − 5.
[13] R. J. Krueger, Inverse problems for nonabsorbing media with discontinuous material properties, J. Math. Phys. 23 (3) (1982) 396 − 404.
[14] Y. Liu, G. Shi and J. Yan, An inverse problem for non-selfadjoint Sturm-Liouville operator with discontinuity conditions inside a finite interval, Inver. Probl. Sci. Eng. 27 (3) (2019) 407 − 421.
[15] V. Marchenko (in Russian), Sturm-Liouville Operators and their Applications, Izd Nauk Dumka, Kiev, 8, 1977.
[16] K. Mochizuki and I. Trooshin, Inverse problem for interior spectral data of Sturm-Liouville operator, J. Inverse Ill-Posed Probl. 9 (4) (2001) 425 − 433.
[17] M. A. Naimark, Linear Differential Operators, Parts I, II, Ungar, New York, 1968.
[18] A. Neamaty and Y. Khalili, Determination of a differential operator with discontinuity from interior spectral data, Inver. Probl. Sci. Eng. 22 (6) (2013) 1002 − 1008.
[19] A. Neamaty and Y. Khalili, The inverse problem for pencils of differential operators on the half-line with discontinuity, Malays. J. Math. Sci. 9 (2) (2015) 175 − 186.
[20] A. S. Ozkan, Inverse Sturm-Liouville problems with eigenvalue-dependent boundary and discontinuity conditions, Inverse Probl. Sci. Eng. 20 (6) (2012) 857 − 868.
[21] A. S. Ozkan, B. Keskin and Y. Cakmak, Uniqueness of the solution of half inverse problem for the impulsive Sturm-Liouville operator, Bull. Korean Math. Soc. 50 (2) (2013) 499 − 506.
[22] C. -T. Shieh and V. A. Yurko, Inverse nodal and inverse spectral problems for discontinuous boundary value problems, J. Math. Anal. Appl. 347 (1) (2008) 266 − 272.
[23] Y. P. Wang, Inverse problems for Sturm-Liouville operators with interior discontinuities and boundary conditions dependent on the spectral parameter, Math. Methods Appl. Sci. 36 (7) (2013) 857 − 868.
[24] Y. P. Wang, The inverse problem for differential pencils with eigenparameter dependent boundary conditions from interior spectral data, Appl. Math. Lett. 25 (2012) 1061 − 1067.
[25] C. F. Yang, Inverse problems for the Sturm-Liouville operator with discontinuity, Inverse Probl. Sci. Eng. 347 (2014) 266 − 272.
[26] C. F. Yang and Y. X. Guo, Determination of a differential pencil from interior spectral data, J. Math. Anal. Appl. 375 (2011) 284 − 293.
[27] C. F. Yang and X. P. Yang, An interior inverse problem for the Sturm- Liouville operator with discontinuous conditions, Appl. Math. Lett. 22 (2009) 1315 − 1319.
[28] C. F. Yang and A. Zettl, Half inverse problems for quadratic pencils of Sturm-Liouville operators, Taiwan J. Math. 16 (5) (2012) 1829 − 1846.
[29] V. Yurko, Inverse spectral problems for differential pencils on the half-line with turning points, J. Math. Anal. Appl. 320 (1) (2006) 439 − 463.