Sturm-Liouville differential operators with deviating argument

Article Sidebar

PDF
Published: Mar 30, 2017
DOI: https://doi.org/10.5556/j.tkjm.48.2017.2264
Keywords:
differential operators deviating argument spectral properties inverse spectral problems

Main Article Content

Vjacheslav Anatoljevich Yurko
Sergey Alexandrovich Buterin
Milenko Pikula

Abstract

Non-selfadjoint second-order differential operators with a constant delay are studied. We establish properties of the spectral characteristics and investigate the inverse problem of recovering operators from their spectra. For this inverse problem the uniqueness theorem is proved.

Article Details

How to Cite
Yurko, V. A., Buterin, S. A., & Pikula, M. (2017). Sturm-Liouville differential operators with deviating argument. Tamkang Journal of Mathematics, 48(1), 49–59. https://doi.org/10.5556/j.tkjm.48.2017.2264
Issue
Vol. 48 No. 1 (2017)
Section
Papers
Author Biographies

Vjacheslav Anatoljevich Yurko

Department ofMathematics, Saratov University, Astrakhanskaya 83, Saratov 410012, Russia.

Sergey Alexandrovich Buterin

Department of Mathematics, Saratov University, Astrakhanskaya 83, Saratov 410012, Russia.

Milenko Pikula

Department of Mathematics, Informatics and Physics, University East Sarajevo, Alekse Šantica 1, East Sarajevo, Bosnia and Herzegovina

References

V. A. Marchenko, Sturm-Liouville Operators and Their Applications. Naukova Dumka, Kiev, 1977; English transl., Birkhauser, 1986.

B. M. Levitan, Inverse Sturm-Liouville Problems, Nauka, Moscow, 1984 (Russian); English transl., VNU Sci.Press, Utrecht, 1987.

G. Freiling and V. A. Yurko, Inverse Sturm-Liouville Problems and Their Applications. NOVA Science Publishers, New York, 2001.

V. A. Yurko, Introduction to the Theory of Inverse Spectral Problems, Moscow, Fizmatlit, 2007.

V. A. Yurko, Method of Spectral Mappings in the Inverse Problem Theory, Inverse and Ill-posed Problems Series. VSP, Utrecht, 2002.

V. A. Yurko, Inverse Spectral Problems for Differential Operators and Their Applications, Gordon and Breach Science Publishers, Amsterdam, 2000.

R. Beals, P. Deift and C. Tomei, Direct and Inverse Scattering on the Line, Mathematica Surveys and Monographs, 28. AMS, Providence, RI, 1988.

V. A. Yurko, An inverse problem for integral operators, Mat. Zametki, 37(1985), no.5, 690--701; English transl. in Math. Notes 37 (1985) no 5-6, 378--385.

M. S. Eremin, An inverse problem for a second-order integro-differential equation with a singularity, Diff. Uravn., 24(1988), no.2, 350--351.

M. Pikula, Determination of a Sturm-Liouville-type differential operator with delay argument from two spectra, Mat. Vesnik, 43(1991), no.3-4, 159--171 (Russian).

V. A. Yurko, An inverse problem for integro-differential operators, Matem. Zametki, 50(1991),no.5, 134--146 (Russian); English transl. in Math. Notes, 50(1991), no.5-6, 1188--1197.

K. V. Kravchenko, On differential operators with nonlocal boundary conditions, Diff. Uravn., 36(2000), no. 4, 464--469; English transl. in Diff. Eqns., 36(2000), no.4, 517--523.

S. A. Buterin, The inverse problem of recovering the Volterra convolution operator from the incomplete spectrum of its rank-one perturbation, Inverse Problems, 22(2006), 2223--2236.

S. A. Buterin, Inverse spectral reconstruction problem for the convolution operator perturbed by a one-dimensional operator, Matem. Zametki, 80(2006), no.5, 668--682 (Russian); English transl. in Math. Notes, 80(2006), no.5, 631--644.

S. A. Buterin, On an inverse spectral problem for a convolution integro-differential operator, Results in Math., 50(2007), no.3-4, 173--181.

Yu. V. Kuryshova, The inverse spectral problem for integro-differential operators, Matem. Zametki, 81(2007), no. 6, 855--866 (Russian); English translation in Math. Notes, 81(2007), no. 5-6,767--777.

S. A. Buterin, On the reconstruction of a convolution perturbation of the Sturm-Liouville operator from the spectrum, Diff. Uravn., 46(2010), 146--149 (Russian); English transl. in Diff. Eqns., 46(2010), 150--154.

G. Freiling and V. A. Yurko, Inverse problems for Sturm-Liouville differential operators with a constant delay, Appl. Math. Lett., 25(2012), 1999--2004.

V. A. Yurko, An inverse spectral problems for integro-differential operators, Far East J. Math. Sci., 92(2014), no.2, 247--261.

S. A. Buterin and A. E. Choque Rivero, On inverse problem for a convolution integro-differential operator with Robin boundary conditions, Appl. Math. Lett., 48(2015), 150--155.

N. P. Bondarenko and S. A. Buterin, On recovering the Dirac operator with an integral delay from the spectrum, Res. Math. (2016), DOI 10.1007/s00025-016-0568-1.

A. D. Myshkis, Linear Differential Equations with a Delay Argument, Moscow, Nauka, 1951.

R. Bellman and K. L. Cooke, Differential-Difference Equations, The RAND Corp. R-374-PR, 1963.

S. B. Norkin, Second Order Differential Equations with a Delay Argument, Moscow, Nauka, 1965.

J. Hale, Theory of Functional-Differential Equations, Springer-Verlag, New York, 1977.

V. V. Vlasov, On the solvability and properties of solutions of functional-differential equations in a Hilbert space, Mat. Sb., 186(1995), no.8, 67--92 (Russian); English transl. in Sb. Math., 186 (1995), no. 8, 1147--1172.

J. B. Conway, Functions of One Complex Variable, 2nd ed., vol.I, Springer-Verlag, New York, 1995.