An inverse problem for the second-order integro-differential pencil
Main Article Content
Abstract
We consider the second-order (Sturm-Liouville) integro-differential pencil with polynomial dependence on the spectral parameter in a boundary condition. The inverse problem is solved, which consists in reconstruction of the convolution kernel and one of the polynomials in the boundary condition by using the eigenvalues and the two other polynomials. We prove uniqueness of solution, develop a constructive algorithm for solving the inverse problem, and obtain necessary and sufficient conditions for its solvability.
Article Details
How to Cite
Bondarenko, N. . P. (2019). An inverse problem for the second-order integro-differential pencil. Tamkang Journal of Mathematics, 50(3), 223–231. https://doi.org/10.5556/j.tkjm.50.2019.3348
Issue
Section
Papers
References
[1] Marchenko, V. A. Sturm-Liouville Operators and their Applications, Naukova Dumka, Kiev (1977) (Russian); English transl., Birkhauser (1986).
[2] Levitan, B. M. Inverse Sturm-Liouville Problems, Nauka, Moscow (1984) (Russian); En- glish transl., VNU Sci. Press, Utrecht (1987).
[3] P ̈oschel, J; Trubowitz, E. Inverse Spectral Theory, New York, Academic Press (1987).
[4] Freiling, G.; Yurko, V. Inverse Sturm-Liouville Problems and Their Applications. Hunt- ington, NY: Nova Science Publishers, 2001.
[5] 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.
[6] Malamud, M. M. On some inverse problems, Boundary Value Problems of Mathematical Physics, Kiev (1979), 116-124.
7
[7] Eremin, M. S. An inverse problem for a second-order integro-differential equation with a singularity, Differ. Uravn. 24:2 (1988), 350–351.
[8] Yurko, V. A. Inverse problem for integrodifferential operators, Mathematical Notes 50:5 (1991), 1188–1197.
[9] Buterin, S. A. The invese problem of recovering the Volterra convolution operator from the incomplete spectrum of its rank-one perturbation, Inverse Problems 22 (2006), 2223–2236.
[10] Buterin, S. A. On an inverse spectral problem for a convolution integro-differential opera- tor, Results Math. 50 (2007), no. 3-4, 73–181.
[11] Buterin, S. A.; Choque Rivero, A. E. On inverse problem for a convolution integrodiffer- ential operator with Robin boundary conditions, Appl. Math. Lett. 48 (2015) 150–155.
[12] Buterin, S.; Sat, M. On the half inverse spectral problem for an integro-differential opera- tor, Inverse Problems in Science and Engineering (2017), Vol. 25, Issue 10, 1508–1518.
[13] Buterin, S. A. On an inverse spectral problem for first-order integro-differential operators with discontinuities, Applied Math. Letters 78 (2018), 65–71.
[14] Bondarenko, N.; Buterin S. On recovering the Dirac operator with an integral delay from the spectrum, Results Math. (2017), Vol. 71, issue 3-4, 1521–1529.
[15] Bondarenko, N.P.; Buterin, S.A. An inverse spectral problem for integro-differential Dirac operators with general convolution kernels, Applicable Analysis (2018), published online, DOI: https://doi.org/10.1080/00036811.2018.1508653.
[16] Bondarenko, N.P. An inverse problem for the integro-differential Dirac system with partial information given on the convolution kernel, Journal of Inverse and Ill-Posed Problems (2018), published online: 2018-06-19, DOI: https://doi.org/10.1515/jiip-2017-0058.
[17] Ignatiev, M. On an inverse spectral problem for one integro-differential opera- tor of fractional order, J. Inverse Ill-Posed Probl. (2018), published online, DOI: https://doi.org/10.1515/jiip-2017-0121.
[18] Ignatyev, M. On an inverse spectral problem for the convolution integro-differential opera- tor of fractional order, Results Math. (2018), 73:34, DOI: https://doi.org/10.1007/s00025- 018-0800-2.
[19] Buterin, S.; Malyugina, M. On global solvability and uniform stability of one nonlinear integral equation, Results Math. (2018), 73:117.
[20] Kuryshova, Yu.V. The inverse spectral problem for integro-differential Operators, Math. Notes 81:6 (2007), 767–777.
[21] Kuryshova, Yu. V.; Shieh, C.-T. An inverse nodal problem for integro-differential operators, J. Inverse Ill-Posed Prob. 18:4 (2010), 357–369.
[22] Wang, Y., Wei, G. The uniqueness for SturmLiouville problems with aftereffect, Acta Math. Sci. 32A:6 (2012), 1171–1178.
[23] Keskin, B.; Ozkan, A. S. Inverse nodal problems for Dirac-type integro-differential opera- tors, Journal of Differential Equations (2017), Vol. 63, Issue 12, 8838–8847.
8
[24] Zolotarev, V. A. Inverse spectral problem for the operators with non-local potential, Math- ematische Nachrichten (2018), 1–21, DOI: https://doi.org/10.1002/mana.201700029.
[25] Yurko, V.A. Inverse problems for second order integro-differential operators, Appl. Math. Lett. 74 (2017), 1–6.
[26] Yurko, V. Inverse problems for arbitrary order integral and integro-differential operators, Results Math. (2018), 73:72.
[27] Freiling G., Yurko V.A. Inverse problems for Sturm-Liouville equations with boundary conditions polynomially dependent on the spectral parameter, Inverse Problems 26 (2010) 055003 (17pp).
[28] Freiling G., Yurko V. Determination of singular differential pencils from the Weyl function, Advances in Dynamical Systems and Applications, Vol. 7, No. 2, pp. 171–193 (2012).
[2] Levitan, B. M. Inverse Sturm-Liouville Problems, Nauka, Moscow (1984) (Russian); En- glish transl., VNU Sci. Press, Utrecht (1987).
[3] P ̈oschel, J; Trubowitz, E. Inverse Spectral Theory, New York, Academic Press (1987).
[4] Freiling, G.; Yurko, V. Inverse Sturm-Liouville Problems and Their Applications. Hunt- ington, NY: Nova Science Publishers, 2001.
[5] 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.
[6] Malamud, M. M. On some inverse problems, Boundary Value Problems of Mathematical Physics, Kiev (1979), 116-124.
7
[7] Eremin, M. S. An inverse problem for a second-order integro-differential equation with a singularity, Differ. Uravn. 24:2 (1988), 350–351.
[8] Yurko, V. A. Inverse problem for integrodifferential operators, Mathematical Notes 50:5 (1991), 1188–1197.
[9] Buterin, S. A. The invese problem of recovering the Volterra convolution operator from the incomplete spectrum of its rank-one perturbation, Inverse Problems 22 (2006), 2223–2236.
[10] Buterin, S. A. On an inverse spectral problem for a convolution integro-differential opera- tor, Results Math. 50 (2007), no. 3-4, 73–181.
[11] Buterin, S. A.; Choque Rivero, A. E. On inverse problem for a convolution integrodiffer- ential operator with Robin boundary conditions, Appl. Math. Lett. 48 (2015) 150–155.
[12] Buterin, S.; Sat, M. On the half inverse spectral problem for an integro-differential opera- tor, Inverse Problems in Science and Engineering (2017), Vol. 25, Issue 10, 1508–1518.
[13] Buterin, S. A. On an inverse spectral problem for first-order integro-differential operators with discontinuities, Applied Math. Letters 78 (2018), 65–71.
[14] Bondarenko, N.; Buterin S. On recovering the Dirac operator with an integral delay from the spectrum, Results Math. (2017), Vol. 71, issue 3-4, 1521–1529.
[15] Bondarenko, N.P.; Buterin, S.A. An inverse spectral problem for integro-differential Dirac operators with general convolution kernels, Applicable Analysis (2018), published online, DOI: https://doi.org/10.1080/00036811.2018.1508653.
[16] Bondarenko, N.P. An inverse problem for the integro-differential Dirac system with partial information given on the convolution kernel, Journal of Inverse and Ill-Posed Problems (2018), published online: 2018-06-19, DOI: https://doi.org/10.1515/jiip-2017-0058.
[17] Ignatiev, M. On an inverse spectral problem for one integro-differential opera- tor of fractional order, J. Inverse Ill-Posed Probl. (2018), published online, DOI: https://doi.org/10.1515/jiip-2017-0121.
[18] Ignatyev, M. On an inverse spectral problem for the convolution integro-differential opera- tor of fractional order, Results Math. (2018), 73:34, DOI: https://doi.org/10.1007/s00025- 018-0800-2.
[19] Buterin, S.; Malyugina, M. On global solvability and uniform stability of one nonlinear integral equation, Results Math. (2018), 73:117.
[20] Kuryshova, Yu.V. The inverse spectral problem for integro-differential Operators, Math. Notes 81:6 (2007), 767–777.
[21] Kuryshova, Yu. V.; Shieh, C.-T. An inverse nodal problem for integro-differential operators, J. Inverse Ill-Posed Prob. 18:4 (2010), 357–369.
[22] Wang, Y., Wei, G. The uniqueness for SturmLiouville problems with aftereffect, Acta Math. Sci. 32A:6 (2012), 1171–1178.
[23] Keskin, B.; Ozkan, A. S. Inverse nodal problems for Dirac-type integro-differential opera- tors, Journal of Differential Equations (2017), Vol. 63, Issue 12, 8838–8847.
8
[24] Zolotarev, V. A. Inverse spectral problem for the operators with non-local potential, Math- ematische Nachrichten (2018), 1–21, DOI: https://doi.org/10.1002/mana.201700029.
[25] Yurko, V.A. Inverse problems for second order integro-differential operators, Appl. Math. Lett. 74 (2017), 1–6.
[26] Yurko, V. Inverse problems for arbitrary order integral and integro-differential operators, Results Math. (2018), 73:72.
[27] Freiling G., Yurko V.A. Inverse problems for Sturm-Liouville equations with boundary conditions polynomially dependent on the spectral parameter, Inverse Problems 26 (2010) 055003 (17pp).
[28] Freiling G., Yurko V. Determination of singular differential pencils from the Weyl function, Advances in Dynamical Systems and Applications, Vol. 7, No. 2, pp. 171–193 (2012).