AASCIT - Journal - All Issues

ISSN: 2375-3927

International Journal of Mathematical Analysis and Applications

The Journal

Manuscript Information

Investigations of the Green Function and Discreteness of Spectrum of Higher Order Operator-Differential Equations on Semi-Axis

International Journal of Mathematical Analysis and Applications
Vol.4 , No. 6, Publication Date: Dec. 5, 2017, Page: 52-58

774 Views Since December 5, 2017, 318 Downloads Since Dec. 5, 2017

Paper in PDF (276K)

Follow on

Authors

[1]	Gamidulla Aslanov, Department of Functional Analysis, Institute of Mathematics and Mechanics of Azerbaijan National Academy of Sciences, Baku, Azerbaijan.
[2]	Novrasta Abdullayeva, Department of Differential Equations, Faculty of Natural Sciences, Sumqayit State University, Sumqayit, Azerbaijan.

Abstract

In the present paper the Green's function and the spectrum of even higher order operator-differential equations are studied. As first the Green's function of the principal part of the equation with frozen coefficients is constructed. By using the Levy method, the integral equation is obtained for the Green's function coefficients. In the Banach spaces of operator valued functions the solution of the obtained integral equations is studied. The uniform estimation of the Green function from which in particular the discreteness of the spectrum is derived.

Keywords

Hilbert Space, Banach Space, Operator-Differential Equation, Green's Function, Resolvent, Eigenvalue, Spectrum

Reference

[01]	Levitan B. M. Investigation of the Green function of Sturm-Liouville equation with on operator coefficient. Mat. Sb., 1968, 76 (118), pp. 239-270 (Russian)
[02]	Kostyuchenko A. G., Levitan B. M. On asymptotic behavior of eigen values of Sturm-Liouville operator problem. Funktsional. Anal. i Prilozhen., 1967, 1 (1), p. 86-96 (Russia).
[03]	Abdukadirov E. On the Green function of Sturm-Liouville equation with operator coefficients. Dokl. Akad. Nauk SSSR, 1970, 195 (3), p. 519-522 (Russian).
[04]	Bayramoglu M. Asymptotics of the number of eigen values of ordinary differential equations with operator coefficients. Funktsional. Anal. i Prilozhen. Baku: "Elm", 1971, p. 33-62 (in Russian).
[05]	Aslanov G. I. Asymptotics of the number of eigen values of ordinary differential equations with operator coefficients on a semi-axis. Dokl. Nats. Akad. Nauk Azerb. SSR, 1976, XXXII (3), p. 3-6 (in Russian).
[06]	Abudov A. A., Aslanov G. I. Distribution of eigen values of operator-differential equations of 2n order. Izv. Nats. Akad. Nauk Azerb. SSR, ser.fiz.-tekh. i mat. nauk. 1980, (1), p. 9-14 (in Russian).
[07]	Aliyev B. I., Bayramoglu M. The Green function of higher order ordinary differential operators with operator coefficients on a semi-axis Izv. Nats. Akad. Nauk Azerb. SSR, ser.fiz.-tekh. i mat. nauk. 1981, 2 (4), p. 33-38 (in Russian)
[08]	Aslanov G. I., Kasumova G. I. On the distribution of the eigenvalues of high order operator-differential equation on half-axis. International Scientific conference "Mathematical analysis, differential equations and their applications" Uzhqorod, Ukraine, 18-23 September, 2006, Abstracts, p. 138-139.
[09]	Aslanov G. I., Abdullayeva N. S. Discreteness spectrum and asymptotic distribution of eigenvalues of operator differential equations of higher order on semi-axis. Trans. NAS Azerb., 2016, 36 (4), p. 47-53.
[10]	Aslanov G. I., Badalova K. G. Weighted trace of higher order operator-differential equations on the semi-axis. Proc. Inst. Math. Mech., 2012, 36 (2), p. 17-24.
[11]	Aliev A. R., Eyvazov E. H. On discreteness of the spectrum of a higher order differential operator in multidimensional case. Proc. Inst. Math. Mech., 2014, 40 (1), p. 28-35.
[12]	Aslanova N. M., Movsumova H. F. On asymptotics of eigenvalues for second order differential-operator equation. Caspian J. Appl. Math., Ecol. Econ., 2015, 3 (2), p. 96-105.
[13]	Aslanova N. M., Movsumova H. F. Investigation of the Eigenvalue Distribution and Trace Formula of the Bessel Operator Equation. Asian Res. J. Math., 2016, 1 (3), p. 1-23.
[14]	Aslanova N. M., Bayramoglu M., Aslanov Kh. M. On spectrum and trace formula for one class of singular problems. An. Ştiint. Univ. Al. I. Cuza Iasi Mat., 2016, 3 (2), p. 725-735.
[15]	Şen E., Bayramov A., Orucoğlu K. Regularized trace formula for higher order differential operators with unbounded coefficients. Electron. Journal Differ. Equations (EJDE). 2016, 31, p. 1-12.
[16]	Şen E., Bayramov A., Oruçoğlu K. The regularized trace formula for a differential operator with unbounded operator coefficient. Advanced Studies in Contemporary Mathematics. 2015; 25 (4), p. 583-590.
[17]	Zhang M. Z. Oscillation criteria and spectrum of self-adjoint even order two-term differential operators. Applied Mech. Mat., 2015, 751, p. 331-336.
[18]	Tinoco A. F., Das Lacava J. C., Pereira Filho O. M. C. On the calculation of spectral Green's functions of cylindrical multilayer structures. IEEE Latin America Trans., 2015, 13 (4), p. 935-942.