Impulsive Dirac system on time scales

Bilender P.  Allahverdiev; Hüseyin Tuna

doi:10.37863/umzh.v75i6.7120

Bilender P. Allahverdiev Department of Mathematics, Khazar University, Baku, Azerbaijan and Research Center of Econophysics, UNEC-Azerbaijan State University of Economics, Baku, Azerbaijan
Hüseyin Tuna Department of Mathematics, Mehmet Akif Ersoy University, Burdur, Turkey

DOI: https://doi.org/10.37863/umzh.v75i6.7120

Keywords: Impulsive dynamic Dirac system, time scale, eigenfunction expansion, spectrum, self-adjoint operators.

Abstract

UDC 517.9

We consider an impulsive Dirac system on Sturmian time scales. An existence theorem is given for this system. А maximal, minimal and self-adjoint operators generated by the impulsive dynamic Dirac system are constructed. We also construct the Green function for this problem. Finally, an eigenfunction expansion is obtained.

References

C. Ahlbrandt, M. Bohner, T. Voepel, Variable change for Sturm–Liouville differential expressions on time scales, J. Different. Equat. and Appl., 9, № 1, 93–107 (2003). DOI: https://doi.org/10.1080/10236190290015371

B. P. Allahverdiev, H. Tuna, Titchmarsh–Weyl theory for Dirac systems with transmission conditions, Mediterr. J. Math., 15, № 4 (2018). DOI: https://doi.org/10.1007/s00009-018-1197-6

B. P. Allahverdiev, H. Tuna, Spectral expansion for the singular Dirac system with impulsive conditions, Turkish J. Math., 42, 2527–2545 (2018). DOI: https://doi.org/10.3906/mat-1803-79

B. P. Allahverdiev, H. Tuna, One dimensional Dirac operators on time scales, Casp. J. Math. Sci., 10, № 2, 195–209 (2021).

D. R. Anderson, G. Sh. Guseinov, J. Hoffacker, Higher-order self-adjoint boundary-value problems on time scales, J. Comput. and Appl. Math., 194, № 2, 309–342 (2006). DOI: https://doi.org/10.1016/j.cam.2005.07.020

D. R. Anderson, Titchmarsh–Sims–Weyl theory for complex Hamiltonian systems on Sturmian time scales, J. Math. Anal. and Appl., 373, № 2, 709–725 (2011). DOI: https://doi.org/10.1016/j.jmaa.2010.08.023

F. Atici Merdivenci, G. Sh. Guseinov, On Green's functions and positive solutions for boundary-value problems on time scales, J. Comput. and Appl. Math., 141, № 1–2, 75–99 (2002). DOI: https://doi.org/10.1016/S0377-0427(01)00437-X

E. Bairamov, Ş. Solmaz, Scattering theory of Dirac operator with the impulsive condition on whole axis, Math. Methods. Appl. Sci., 44, № 9, 7732–7746 (2021). DOI: https://doi.org/10.1002/mma.6645

E. Bairamov, Ş. Solmaz, Spectrum and scattering function of the impulsive discrete Dirac systems, Turkish J. Math., 42, 3182–3194 (2018). DOI: https://doi.org/10.3906/mat-1806-5

M. Benchohra M., S. K. Ntouyas, A. Ouahab, Existence results for second order boundary-value problem of impulsive dynamic equations on time scales, J. Math. Anal. and Appl., 296, 69–73 (2004). DOI: https://doi.org/10.1016/j.jmaa.2004.02.057

M. Bohner, A. Peterson, Dynamic equations on time scales, Birkhäuser, Boston (2001). DOI: https://doi.org/10.1007/978-1-4612-0201-1

M. Bohner, A. Peterson (Eds.), Advances in dynamic equations on time scales, Birkhäuser, Boston (2003). DOI: https://doi.org/10.1007/978-0-8176-8230-9

T. Gulsen, S. M. Sian, E. Yilmaz, H. Koyunbakan, Impulsive diffusion equation on time scales, Int. J. Anal. Appl., 16, № 1, 137–148 (2018).

T. Gulsen, E. Yılmaz, Spectral theory of Dirac system on time scales, Appl. Anal., 96, № 16, 2684–2694 (2017). DOI: https://doi.org/10.1080/00036811.2016.1236923

S. Hilger, Ein Maßtkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten, Ph.D. Thesis, Univ. Würzburg, Germany (1988).

G. Hovhannisyan, On Dirac equation on a time scale, J. Math. Phys., 52, № 10, 1–17 (2011). DOI: https://doi.org/10.1063/1.3644343

B. Keskin, Inverse problems for impulsive Dirac operators with spectral parameters contained in the boundary and multitransfer conditions, Math. Methods. Appl. Sci., 38, № 15, 3339–3345 (2015). DOI: https://doi.org/10.1002/mma.3299

A. N. Kolmogorov, S. V. Fomin, Introductory real analysis, Translated by R. A. Silverman, Dover Publ., New York (1970).

Q. Kong, Q. R. Wang, Using time scales to study multi-interval Sturm–Liouville problems with interface conditions, Results Math., 63, 451–465 (2013). DOI: https://doi.org/10.1007/s00025-011-0208-8

T. KöprübaŞı, A study of impulsive discrete Dirac system with hyperbolic eigenparameter, Turkish J. Math., 45, № 1, 540–548 (2021). DOI: https://doi.org/10.3906/mat-2010-29

B. M. Levitan, I. S. Sargsjan, Sturm–Liouville and Dirac operators, Mathematics and its Applications (Soviet Series), Kluwer Acad. Publ. Group, Dordrecht (1991) (translated from the Russian). DOI: https://doi.org/10.1007/978-94-011-3748-5

V. Lakshmikantham, D. D. Bainov, P. S. Simeonov, Theory of impulsive differential equations, vol. 6, Ser. Modern Appl. Math., World Sci., Teaneck, NJ, USA (1989). DOI: https://doi.org/10.1142/0906

K. R. Mamedov, On an inverse scattering problem for a discontinuous Sturm–Liouville equation with a spectral parameter in the boundary condition, Bound.-Value Probl., 2010, Article ID 171967 (2010). DOI: https://doi.org/10.1155/2010/171967

K. R. Mamedov, N. Palamut, On a direct problem of scattering theory for a class of Sturm–Liouville operator with discontinuous coefficient, Proc. Jangjeon Math. Soc., 12, № 2, 243–251 (2009).

O. Sh. Mukhtarov, Discontinuous boundary-value problem with spectral parameter in boundary conditions, Turkish J. Math., 18, 183–192 (1994).

O. Sh. Mukhtarov, K. Aydemir, The eigenvalue problem with interaction conditions at one interior singular point, Filomat, 31, № 17, 5411–5420 (2017). DOI: https://doi.org/10.2298/FIL1717411M

O. Sh. Mukhtarov, H. Olǧar, K. Aydemir, Resolvent operator and spectrum of new type boundary-value problems, Filomat, 29, № 7, 1671–1680 (2015). DOI: https://doi.org/10.2298/FIL1507671M

M. A. Naimark, Linear differential operators, 2nd ed., Nauka, Moscow (1969).

H. Olǧar, O. Sh. Mukhtarov, Weak eigenfunctions of two-interval Sturm–Liouville problems together with interaction conditions, J. Math. Phys., 58, Article 042201 (2017); DOI: 10.1063/1.4979615. DOI: https://doi.org/10.1063/1.4979615

A. S. Ozkan, I. Adalar, Half-inverse Sturm–Liouville problem on a time scale, Inverse Probl., 36, № 2, Article ID 025015 (2020). DOI: https://doi.org/10.1088/1361-6420/ab2a21

A. S. Ozkan, Ambarzumyan-type theorems on a time scale, J. Inverse Ill-Posed Probl., 26, № 5, 633–637 (2018). DOI: https://doi.org/10.1515/jiip-2017-0124

S. özkan, R. Kh. Amirov, An interior inverse problem for the impulsive Dirac operator, Tamkang J. Math., 42, № 3, 259–263 (2011). DOI: https://doi.org/10.5556/j.tkjm.42.2011.824

A. M. Samoilenko, N. A. Perestyuk, Impulsive differential equations, vol. 14, World Sci. Ser. Nonlinear Sci., Ser. A Monogr. and Treatises, World Sci., River Edge, NJ, USA (1995). DOI: https://doi.org/10.1142/2892

B. Thaller, The dirac equation, Springer (1992). DOI: https://doi.org/10.1007/978-3-662-02753-0

J. Weidmann, Spectral theory of ordinary differential operators, Lect. Notes Math., 1258, Springer, Berlin (1987). DOI: https://doi.org/10.1007/BFb0077960