Principal functions of impulsive difference operators on semi axis

Abstract

In this paper, we investigate the continuous spectrum and resolvent operator of a second-order difference operator with an impulsive condition. Then, under certain conditions, we prove finiteness of eigenvalues, spectral singularities. At last, we present principal functions of corresponding impulsive operator.

Keywords

• impulsive condition,difference operator,spectrum,principal functions

References

1. Samoilenko, A. M. and Perestyuk, N. A., Impulsive differential equations, World Scientific, Singapore, 1995.
2. Lakshmikantham, V., Bainov, D. D. and Simeonov, P. S., Theory of impulsive differential equations, World Scientific, Singapore, 1998.
3. Bainov, D. D. and Simeonov, P. S., Oscillation theory of impulsive differential equations, Int. Publ., Orlando, 1998.
4. Uğurlu, E. and Bairamov, E., Spectral analysis of eigenparameter dependent boundary value transmission problems, J. Math. Anal. Appl., 413, 1, (2014), 482--494.
5. Mostafazadeh, A., Spectral singularities of a general point interaction, J. Phys. A. Math. Theory, 44, 375302, (2011), (9pp).
6. He, Z. M. and Zhang, X. M. , Monoton iterative technique for first order impulsive difference equations with periodic boundary conditions, Appl. Math. Comput., 156, 3, (2004), 605--620.
7. Wang, P. and Wang, W., Boundary value problems for first order impulsive difference equations, Int. Journal of Difference Equations, 1, (2006), 249--259.
8. Zhang, Q., On a linear delay difference equations with impulses., Annals of Differential Equations, 18, 2, (2002), 197--204.

9. Naimark, M. A., Investigation of the spectrum and the expansion in eigenfunctions of a non-selfadjoint operators of second order on a semi-axis, AMS Transl. (2), 16, (1960), 103-193.
10. Marchenko, V. A., Sturm-Liouville operators and applications, Birkhauser Verlag, Basel, 1986.
11. Levitan B. M. and Sargsjan I. S., Sturm-Liouville and Dirac operators, Kluwer Academic Publishers Group, Dordrecht, 1991.
12. Agarwal, R. P., Difference equations and inequalities, in: Theory, Methods and Applications, Marcel Dekkar Inc., New York, Basel, 2000.
13. Kelley, W. G and Peterson, A. C., Difference equations: an introduction with applications, Harcourt Academic Press, 2001.
14. Krall, A.M., Bairamov, E. and Cakar, O., Spectral analysis of a non-selfadjoint discrete Schrödinger operators with spectral singularities, Math. Nachr., 231, (2001), 89--104.
15. Bairamov, E., Cakar, O. and Krall, A.M., Non-Selfadjoint Difference Operators and Jacobi Matrices with Spectral Singularities, Math. Nachr., 229, (2001), 5--14.
16. Adıvar, M. and Bairamov, E., Spectral Properties of Non-Selfadjoint Difference Operators, J. Math. Anal. and Appl., 261, (2001), 461--478.
17. Adıvar, M. and Bairamov, E., Difference Equations of Second Order with Spectral Singularities, J. Math. Anal. Appl., 277, (2003), 714--721.
18. Guseinov, G. Sh., The inverse problem of scattering theory for a second order difference equation, Sov. Math., Dokl., 230, (1976), 1045-1048.
19. Olgun, M., Köprübaşı, T. and Aygar, Y., Principal functions of non-selfadjoint difference operator with spectral parameter in boundary conditions, Abstr. Appl. Anal., Art. ID 608329, 10, (2011).
20. Erdal, I., Yardımcı, S., Eigenvalues and Scattering Properties of Difference Operators with Impulsive Condition, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 68, 1, (2019), 663--671.
21. Lusternik, L.A, Sobolev V. I., Elements of Functional Analysis, Halsted Press, New York, 1974.
22. Glazman I. M., Direct Methods of Qualitative Spectral Anaysis of Singular Differential Operators, Jerusalem, 1965.
23. Dolzhenko E. P., Boundary value uniqueness theorems for analytic functions, Math. Notes, 26, (1979), 437-442.