Aryan Ali Mohammed

Department of Mathematics, College of Science, University of Sulaimani, Sulaimanyah, Iraq.

DOI: https://doi.org/10.17656/jzs.10625

[1] Aigounv, G.A., Jwamer, K. H. and Djalaeva, G.A. "Estimates for the eigenfunctions of the Regge Problem", Matemaicheskie Zametki, Vol. 92, Issue 1, pp. 141-144, (2012), Moscow. (Translated: Mathematical Notes, Springer, Vol. 92, No.7, pp. 127-130, (2012).

[2] Aigunov, G. A. and Jwamer, K.H. "Asymptotic behaviour of orthonormal eigenfunctions for a problem of Regge type with integrable positive weight function", (Russian Math. Surveys Vol. 64, No. 6, pp. 1131-1132, (2009)), Uspekhi Mat. Nauk, Vol. 64, No. 6, pp. 169-170, (2009).

[3] Aigounv, G. A. and Jwamer, .K.H. "About Uniform Limitation of Normalized Eigenfunctions of T. Regge Problem in the Case of Weight Functions Satisfying to Lipschitz Condition", Gen. Math. Notes, Vol. 1, No. 2, pp. 115-129, December (2010).

[4] Fulton, C. T. "Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions", Proceedings of the Royal Society of Edinburgh, Vol. 77, No. 3-4, pp. 293–308, (1977).

[5] Hermann,.B. "Das Eigenwertproblem der selbstadjungierten linearen Differential gleichung vierter Ordnung", Mathematische Zeitschrift, Vol. 34, No. 1, pp. 293–319, (1932).

[6] Jamel, B. A. and Shkalikov, A. A. "The Sturm-Liouville problem with physical and spectral parameters in the boundary condition", Mathematical Notes, Vol. 66, pp. 127–134, (1999).

[7] Jwamer, K. H. and Mohammed, A. A. "Boundedness of Normalized Eigenfunctions of the Spectral Problem in the Case of Weight Function Satisfying the Lipschitz Condition", Journal of Zankoy Sulaimani – Part A (JZS-A), Vol. 15, No.1, pp. 79-94, (2013).

[8] Mamedov, Kh. R. "On a Basic Problem for Second Order Differential Equation with a Discontinuous Coefficient and a Spectral Parameter in Boundary Conditions", Seventh International Conference on Geometry, Integrability and Quantization, Varna, Bulgaria, June 2-10, (2005).

[9] Mamedov, Kh. R. and Menken, H. "The Levinson-type Formula for a Boundary Value Problem with a Spectral Parameter in the Boundary Condition", Arab. J. Sci., Eng. Sect. A Sci., Vol. 34, No. 1, pp. 219-226, (2009).

[10] Mamedov, Kh. R and Cetinkaya, F. A. "A Uniqueness Theorem for a Sturm-Liouville Equation with Spectral Parameter in Boundary Conditions", Appl. Math. Inf. Sci. Vol. 9, No. 2, pp. 981-988, (2015).

[11] Maurice, R. "Vibrations in Mechanical Systems", Springer, Berlin, Germany, (1987).

[12] Menken, H. and Mamedov, Kh. R. "Basis Property in of the Root Functions Corresponding to a Boundary Value Problem", J. Appl. Funct. Anal., Vol. 5, No. 4, pp. 351-356, (2010).

[13] Naimark, M. A. "Investigation of the spectrum and the expansion in eigenfunctions of a non-self adjoint operator of second order on a semi-axis", Am. Math. Soc. Transl. Vol. 16, No. 2, pp. 103-193, (1960).

[14] Nazim, B. K. and Ziyatkhan, S. A. "The oscillation properties of the boundary value problem with spectral parameter in the boundary Condition", Transactions of National Academy of Sciences of Azerbaijan. Series of Physical-Technical and Mathematical Sciences, Vol. 25, No. 7, pp. 61–68, (2005).

[15] Nazim, B. K. and Ziyatkhan, S. A. "On the basis property of the system of eigenfunctions of a spectral problem with spectral parameter in the boundary condition", Differential Equations, Vol. 43, No. 7, pp. 905–915, (2007).

[16] Nihal, Y. and Turhan, K. "Spectrum of the Sturm-Liouville operators with boundary conditions polynomially dependent on the spectral parameter", A Springer Open Journal. Journal of Inequalities and Applications, Vol. 42, pp. 1-7, (2015).

[17] Nikolai, Y. K. and Evgenii, I. M. "The basis property in Lp of the systems of eigenfunctions corresponding to two problems with a spectral parameter in the boundary condition", Differential Equations, Vol. 36, No. 10, pp. 1498–1501, (2000).

[18] Nikolai, Y. K. and Evgenii, I. M. "On the singularities of the root space of one spectral problem with a spectral parameter in the boundary condition", Doklady Mathematics, Vol. 66, No. 1, pp. 14–18, (2002).

[19] Tamarkin, Y. D. "On some general problems of theory of ordinary linear differential equations and about decomposition of arbitrary functions in series", Petrograd, (1917).

Department of Mathematics, College of Science, University of Sulaimani, Sulaimanyah, Iraq.

DOI: https://doi.org/10.17656/jzs.10625

Original: 18 October 2016, Revised: 29 December 2016, Accepted: 9 January 2017, Published online: 20 June 2017

**Abstract**

**Key Words:**Eigenfunctions, Eigenvalues, Boundary Value Problems, Asymptotic Behaviour of Eigenvalues.

**References**

[1] Aigounv, G.A., Jwamer, K. H. and Djalaeva, G.A. "Estimates for the eigenfunctions of the Regge Problem", Matemaicheskie Zametki, Vol. 92, Issue 1, pp. 141-144, (2012), Moscow. (Translated: Mathematical Notes, Springer, Vol. 92, No.7, pp. 127-130, (2012).

[2] Aigunov, G. A. and Jwamer, K.H. "Asymptotic behaviour of orthonormal eigenfunctions for a problem of Regge type with integrable positive weight function", (Russian Math. Surveys Vol. 64, No. 6, pp. 1131-1132, (2009)), Uspekhi Mat. Nauk, Vol. 64, No. 6, pp. 169-170, (2009).

[3] Aigounv, G. A. and Jwamer, .K.H. "About Uniform Limitation of Normalized Eigenfunctions of T. Regge Problem in the Case of Weight Functions Satisfying to Lipschitz Condition", Gen. Math. Notes, Vol. 1, No. 2, pp. 115-129, December (2010).

[4] Fulton, C. T. "Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions", Proceedings of the Royal Society of Edinburgh, Vol. 77, No. 3-4, pp. 293–308, (1977).

[5] Hermann,.B. "Das Eigenwertproblem der selbstadjungierten linearen Differential gleichung vierter Ordnung", Mathematische Zeitschrift, Vol. 34, No. 1, pp. 293–319, (1932).

[6] Jamel, B. A. and Shkalikov, A. A. "The Sturm-Liouville problem with physical and spectral parameters in the boundary condition", Mathematical Notes, Vol. 66, pp. 127–134, (1999).

[7] Jwamer, K. H. and Mohammed, A. A. "Boundedness of Normalized Eigenfunctions of the Spectral Problem in the Case of Weight Function Satisfying the Lipschitz Condition", Journal of Zankoy Sulaimani – Part A (JZS-A), Vol. 15, No.1, pp. 79-94, (2013).

[8] Mamedov, Kh. R. "On a Basic Problem for Second Order Differential Equation with a Discontinuous Coefficient and a Spectral Parameter in Boundary Conditions", Seventh International Conference on Geometry, Integrability and Quantization, Varna, Bulgaria, June 2-10, (2005).

[9] Mamedov, Kh. R. and Menken, H. "The Levinson-type Formula for a Boundary Value Problem with a Spectral Parameter in the Boundary Condition", Arab. J. Sci., Eng. Sect. A Sci., Vol. 34, No. 1, pp. 219-226, (2009).

[10] Mamedov, Kh. R and Cetinkaya, F. A. "A Uniqueness Theorem for a Sturm-Liouville Equation with Spectral Parameter in Boundary Conditions", Appl. Math. Inf. Sci. Vol. 9, No. 2, pp. 981-988, (2015).

[11] Maurice, R. "Vibrations in Mechanical Systems", Springer, Berlin, Germany, (1987).

[12] Menken, H. and Mamedov, Kh. R. "Basis Property in of the Root Functions Corresponding to a Boundary Value Problem", J. Appl. Funct. Anal., Vol. 5, No. 4, pp. 351-356, (2010).

[13] Naimark, M. A. "Investigation of the spectrum and the expansion in eigenfunctions of a non-self adjoint operator of second order on a semi-axis", Am. Math. Soc. Transl. Vol. 16, No. 2, pp. 103-193, (1960).

[14] Nazim, B. K. and Ziyatkhan, S. A. "The oscillation properties of the boundary value problem with spectral parameter in the boundary Condition", Transactions of National Academy of Sciences of Azerbaijan. Series of Physical-Technical and Mathematical Sciences, Vol. 25, No. 7, pp. 61–68, (2005).

[15] Nazim, B. K. and Ziyatkhan, S. A. "On the basis property of the system of eigenfunctions of a spectral problem with spectral parameter in the boundary condition", Differential Equations, Vol. 43, No. 7, pp. 905–915, (2007).

[16] Nihal, Y. and Turhan, K. "Spectrum of the Sturm-Liouville operators with boundary conditions polynomially dependent on the spectral parameter", A Springer Open Journal. Journal of Inequalities and Applications, Vol. 42, pp. 1-7, (2015).

[17] Nikolai, Y. K. and Evgenii, I. M. "The basis property in Lp of the systems of eigenfunctions corresponding to two problems with a spectral parameter in the boundary condition", Differential Equations, Vol. 36, No. 10, pp. 1498–1501, (2000).

[18] Nikolai, Y. K. and Evgenii, I. M. "On the singularities of the root space of one spectral problem with a spectral parameter in the boundary condition", Doklady Mathematics, Vol. 66, No. 1, pp. 14–18, (2002).

[19] Tamarkin, Y. D. "On some general problems of theory of ordinary linear differential equations and about decomposition of arbitrary functions in series", Petrograd, (1917).