Aryan Ali Mohammed

Department of Mathematics, School of Science, Faculty of Science and Science Education, University of Sulaimani, Sulaimanyah,

Iraq.

[1] Aigounv G.A, Karwan H.F. Jwamer, 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., 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] DARWISH, A. A., “On a non-self adjoint singular boundary value problem”, Kyungpook Math. J Taeygu, Vol. (33), No. 1, pp. 1-11, (1993).

[4] HINTON, D.B., “An expansion theorem for an eigenvalue problem with eigenvalue parameter in the boundary condition”, Quart. J. Math. Oxford Vol. (30), No. 2, pp. 33-42, (1979).

[5] Jwamer .K.H and Aigounv. G. A., “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).

[6] Karwan H.F. Jwamer and Aryan A.M, “ 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).

[7] Karwan H.F. Jwamer, Khelan. H. Qadr, “Estimation of Normalized Eigenfunctions of Spectral Problem with Smooth Coefficients”, The proceeding of 7th international conference on Theory and Applications in Mathematics and informatics, Special issue: Journal of Acta Universitatis Apulensis, Romania, pp. 113-132, (2011).

[8] NAIMARK, M. A., “Linear differential operators”, Frederick Ungar Publishing Co., Inc., London, (1968).

[9] SCHNEIDER, A., “A note on eigenvalue problems with eigenvalue parameter in the boundary Conditions”, Math. Z. Vol. (136), pp. 163-167, (1974).

[10] Simon J. A. Malham, “An introduction to asymptotic analysis”, (2005). [11] SHKALUKOV, A.A., “Boundary value problem for ordinary differential equation with parameter in the boundary condition”, Trudy cemunara Um. U.G. Petrovskovo 9, Moscow (Russ), (1983).

[12] TIKHNOW, A.H. and SAMARSKU, A.A., “Equations of Mathematical Physics”, Moscow, pp. 146-152, (1953).

[13] WALTER, J., “Regular eigenvalue problem with eigenvalue parameter in the boundary conditions”, Math. Z. Vol. (133), pp. 301-312, (1973).

Iraq.

**Abstract**

In this work, we derived the general solution for nth order linear ordinary differential

equations of the form for and are continuous functions of via the method of variation of

parameters. Likewise we found the eigenfunctions for the given second boundary value

problem [1]-[3] as well as the boundedness of eigenfunctions, and the sign of real part of

the eigenvalues have been specified through the sign of one parameter in the boundary

conditions. Finally, the asymptotic behavior of eigenvalues to the given problem was

studied.

**Key Words:**

Eigenparameters, eigenfunctions, boundary conditions, asymptotic behavior.

References

References

[1] Aigounv G.A, Karwan H.F. Jwamer, 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., 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] DARWISH, A. A., “On a non-self adjoint singular boundary value problem”, Kyungpook Math. J Taeygu, Vol. (33), No. 1, pp. 1-11, (1993).

[4] HINTON, D.B., “An expansion theorem for an eigenvalue problem with eigenvalue parameter in the boundary condition”, Quart. J. Math. Oxford Vol. (30), No. 2, pp. 33-42, (1979).

[5] Jwamer .K.H and Aigounv. G. A., “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).

[6] Karwan H.F. Jwamer and Aryan A.M, “ 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).

[7] Karwan H.F. Jwamer, Khelan. H. Qadr, “Estimation of Normalized Eigenfunctions of Spectral Problem with Smooth Coefficients”, The proceeding of 7th international conference on Theory and Applications in Mathematics and informatics, Special issue: Journal of Acta Universitatis Apulensis, Romania, pp. 113-132, (2011).

[8] NAIMARK, M. A., “Linear differential operators”, Frederick Ungar Publishing Co., Inc., London, (1968).

[9] SCHNEIDER, A., “A note on eigenvalue problems with eigenvalue parameter in the boundary Conditions”, Math. Z. Vol. (136), pp. 163-167, (1974).

[10] Simon J. A. Malham, “An introduction to asymptotic analysis”, (2005). [11] SHKALUKOV, A.A., “Boundary value problem for ordinary differential equation with parameter in the boundary condition”, Trudy cemunara Um. U.G. Petrovskovo 9, Moscow (Russ), (1983).

[12] TIKHNOW, A.H. and SAMARSKU, A.A., “Equations of Mathematical Physics”, Moscow, pp. 146-152, (1953).

[13] WALTER, J., “Regular eigenvalue problem with eigenvalue parameter in the boundary conditions”, Math. Z. Vol. (133), pp. 301-312, (1973).