An inverse spectral problem for integro-differential Dirac operators with general convolution kernels. (English) Zbl 1436.34012
The integro-differential Dirac-type system
\[
BY'(x)+\int_0^x M(x-t)Y(t)\,dt=\lambda Y(x),\; 0<x<\pi, \tag{1}
\]
is considered, where
\[
Y(x)=
\left[ \begin{matrix}
y_1(x) \\ y_2(x)\end{matrix}\right],\;
B=\left[\begin{matrix}
0 & 1 \\
-1 & 0
\end{matrix}\right],\;
\left[ \begin{matrix}
M_1(x) & M_2(x) \\
M_3(x) & M_4(x)
\end{matrix}\right].
\]
The authors provide the solution of the inverse problem of recovering
\(M(x)\) from the given two spectra of the boundary value problem for Eq. (1)
with the boundary conditions \(y_1(0)=y_j(\pi)=0\).
Reviewer: Vjacheslav Yurko (Saratov)
MSC:
| 34A55 | Inverse problems involving ordinary differential equations |
| 45J05 | Integro-ordinary differential equations |
| 34B09 | Boundary eigenvalue problems for ordinary differential equations |
| 34B10 | Nonlocal and multipoint boundary value problems for ordinary differential equations |
| 45G15 | Systems of nonlinear integral equations |
Keywords:
non-selfadjoint Dirac system; integro-differential operator; nonlocal operator; inverse spectral problem; nonlinear integral equationReferences:
| [1] | Borg, G., Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe, Acta Math, 78, 1-96 (1946) · Zbl 0063.00523 · doi:10.1007/BF02421600 |
| [2] | Gasymov, MG; Levitan, BM., The inverse problem for the Dirac system, Dokl Akad Nauk SSSR, 167, 5, 967-970 (1966) · Zbl 0168.12401 |
| [3] | Gasymov, MG; Dzabiev, TT., Solution of the inverse problem by two spectra for the Dirac equation on a finite interval, Akad Nauk Azerbaijan SSR Dokl, 22, 7, 3-6 (1966) |
| [4] | Marchenko, VA., Sturm-Liouville operators and their applications (1977), Kiev: Naukova Dumka, Kiev · Zbl 0399.34022 |
| [5] | Levitan, BM; Sargsyan, IS., Sturm-Liouville and Dirac operators (1988), Moscow: Nauka, Moscow · Zbl 0657.34002 |
| [6] | Malamud, MM., Uniqueness questions in inverse problems for systems of differential equations on a finite interval, Trans Moscow Math Soc, 60, 204-262 (1999) · Zbl 0939.34012 |
| [7] | Freiling, G.; Yurko, VA., Inverse Sturm-Liouville problems and their applications (2001), New York: NOVA Science Publishers, New York · Zbl 1037.34005 |
| [8] | Yang, CF; Huang, ZY., Reconstruction of the Dirac operator from nodal data, Integr Eqns Oper Theory, 66, 539-551 (2010) · Zbl 1204.34019 · doi:10.1007/s00020-010-1763-1 |
| [9] | Yang, CF., Hochstadt-Lieberman theorem for Dirac operator with eigenparameter dependent boundary conditions, Nonlinear Anal, 74, 2475-2484 (2011) · Zbl 1216.34015 · doi:10.1016/j.na.2010.12.003 |
| [10] | Mykytyuk, YV; Puyda, DV., Inverse spectral problems for Dirac operators on a finite interval, J Math Anal Appl, 386, 177-194 (2012) · Zbl 1264.34025 · doi:10.1016/j.jmaa.2011.07.061 |
| [11] | Gorbunov, OB; Yurko, VA., Inverse problem for Dirac system with singularities in interior points, Anal Math Phys, 6, 1-29 (2016) · Zbl 1357.34038 · doi:10.1007/s13324-015-0097-1 |
| [12] | Wang, YP; Yurko, VA., On the missing eigenvalue problem for Dirac operators, Appl Math Lett, 80, 41-47 (2018) · Zbl 1390.34047 · doi:10.1016/j.aml.2018.01.004 |
| [13] | Yurko, VA., Method of spectral mappings in the inverse problem theory (2002), Utrecht: VSP, Utrecht · Zbl 1098.34008 |
| [14] | Beals, R.; Deift, P.; Tomei, C., Direct and inverse scattering on the line (1988), Providence (RI): AMS, Providence (RI) · Zbl 0699.34016 |
| [15] | Yurko, VA., Inverse spectral problems for differential operators and their applications (2000), Amsterdam: Gordon and Breach Science Publishers, Amsterdam · Zbl 0952.34001 |
| [16] | Yurko, VA., An inverse spectral problem for singular non-selfadjoint differential systems, Matem Sbornik, 195, 12, 123-156 (2004) · Zbl 1091.34007 · doi:10.4213/sm869 |
| [17] | Yurko, VA., Inverse spectral problems for differential systems on a finite interval, Results Math, 48, 3-4, 371-386 (2005) · Zbl 1119.34007 · doi:10.1007/BF03323374 |
| [18] | Yurko, VA., An inverse problem for differential systems on a finite interval in the case of multiple roots of the characteristic polynomial, Diff. Uravn, 41, 6, 781-786 (2005) · Zbl 1077.34503 |
| [19] | Yurko, VA., An inverse problem for differential systems with multiplied roots of the characteristic polynomial, J Inv Ill-Posed Probl, 13, 5, 503-512 (2005) · Zbl 1102.34006 · doi:10.1515/156939405775297425 |
| [20] | Lakshmikantham, V, Rama Mohana Rao, M.Theory of integro-differential equations, Stability and Control: Theory, Methods and Applications, v.1, Gordon and Breach Science Publishers, Singapore, 1995. · Zbl 0849.45004 |
| [21] | Malamud, MM.On some inverse problems. Kiev: Boundary Value Problems of Mathematical Physics; 1979; p. 116-124. |
| [22] | Yurko, VA., Inverse problem for integro-differential operators of the first order, Funct Anal Ul’janovsk, 22, 144-151 (1984) · Zbl 0566.47031 |
| [23] | Eremin, MS., An inverse problem for a second-order integro-differential equation with a singularity, Diff Uravn, 24, 2, 350-351 (1988) · Zbl 0662.45006 |
| [24] | Yurko, VA., An inverse problem for integro-differential operators, Mat Zametki, 50, 5, 134-146 (1991) · Zbl 0744.45004 |
| [25] | Buterin, SA., On an inverse spectral problem for a convolution integro-differential operator, Results Math, 50, 34, 173-181 (2007) · Zbl 1135.45007 · doi:10.1007/s00025-007-0244-6 |
| [26] | Kuryshova, Ju V., Inverse spectral problem for integro-differential operators, Mat Zametki, 81, 6, 855-866 (2007) · Zbl 1142.45006 · doi:10.4213/mzm3736 |
| [27] | Buterin, SA., On the reconstruction of a convolution perturbation of the Sturm-Liouville operator from the spectrum, Diff Uravn, 46, 146-149 (2010) · Zbl 1197.34017 |
| [28] | Kuryshova, Yu V.; Shieh, C-T., An inverse nodal problem for integro-differential operators, J Inverse Ill-Posed Prob, 18, 4, 357-369 (2010) · Zbl 1279.34090 |
| [29] | Wang, Y.; Wei, G., The uniqueness for Sturm-Liouville problems with aftereffect, Acta Math Sci, 32A, 6, 1171-1178 (2012) · Zbl 1289.34033 |
| [30] | Yurko, VA., An inverse spectral problems for integro-differential operators, Far East J Math Sci, 92, 2, 247-261 (2014) · Zbl 1328.47051 |
| [31] | Buterin, SA; Choque, Rivero AE., On inverse problem for a convolution integro-differential operator with Robin boundary conditions, Appl Math Lett, 48, 150-155 (2015) · Zbl 1325.45011 · doi:10.1016/j.aml.2015.04.003 |
| [32] | Buterin, SA; Sat, M., On the half inverse spectral problem for an integro-differential operator, Inverse Probl Sci Eng, 25, 10, 1508-1518 (2017) · Zbl 1390.45022 · doi:10.1080/17415977.2016.1267171 |
| [33] | Yurko, VA., Inverse problems for second order integro-differential operators, Appl Math Lett, 74, 1-6 (2017) · Zbl 1376.45018 · doi:10.1016/j.aml.2017.04.013 |
| [34] | Buterin, SA., On inverse spectral problems for first-order integro-differential operators with discontinuities, Appl Math Lett, 78, 65-71 (2018) · Zbl 1381.45026 · doi:10.1016/j.aml.2017.11.005 |
| [35] | Bondarenko, NP., An inverse problem for an integro-differential operator on a star-shaped graph, Math Meth Appl Sci, 41, 4, 1697-1702 (2018) · Zbl 1392.45014 · doi:10.1002/mma.4698 |
| [36] | Ignatyev, M., On an inverse spectral problem for the convolution integro-differential operator of fractional order, Results Math, 73, 34 (2018) · Zbl 1444.45008 · doi:10.1007/s00025-018-0800-2 |
| [37] | Bondarenko, N.; Buterin, S., On recovering the Dirac operator with an integral delay from the spectrum, Results Math, 71, 3-4, 1521-1529 (2017) · Zbl 1408.34019 · doi:10.1007/s00025-016-0568-1 |
| [38] | Bondarenko, NP.Inverse problem for the Dirac system with an integral delay of the convolution-type, in: Mathematika. Mekhanika, Vol. 19, Saratov Univ., Saratov, 2017, 9-12. |
| [39] | Bondarenko, NP.An inverse problem for the integro-differential Dirac system with partial information given on the convolution kernel. J Inverse Ill-Pose P. 2018. DOI: |
| [40] | Buterin, SA., The inverse problem of recovering the Volterra convolution operator from the incomplete spectrum of its rank-one perturbation, Inverse Probl, 22, 2223-2236 (2006) · Zbl 1114.45004 · doi:10.1088/0266-5611/22/6/019 |
| [41] | Buterin, S.; Malyugina, M., On global solvability and uniform stability of one nonlinear integral equation, Results Math (2018) · Zbl 1401.45005 · doi:10.1007/s00025-018-0879-5 |
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