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Discontinuous Sturm-Liouville Problems Containing Eigenparameter in the Boundary Conditions

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Acta Mathematica Sinica,English Series 2006年第5期22卷 1519-1528页

作者： M.KADAKAL O.sh.mukhtarov Ondokuz Mayis University Science and Arts Faculty Department of MathematicsKurupelit 55139 Samsun Turkey Gaziosmanpasa University Science and Arts Faculty Department of MathematicsTokat 60000 Turkey

In this paper, discontinuous Sturm-Liouville problems, which contain eigenvalue parameters both in the equation and in one of the boundary conditions, are investigated. By using an operatortheoretic interpretation we extend some classic results for regular Sturm-Liouville problems and obtain asymptotic approximate formulae for eigenvalues and normalized eigenfunctions. We modify some techniques of [Fulton, C. T., Proc. Roy. Soc. Edin. 77 （A）, 293-308 （1977）], [Walter, J., Math. Z., 133, 301-312 （1973）] and [Titchmarsh, E. C., Eigenfunctions Expansion Associated with Second Order Differential Equations I, 2nd edn., Oxford Univ. Pres, London, 1962], then by using these techniques we obtain asymptotic formulae for eigenelement norms and normalized eigenfunctions.

关键词： Sturm-Liouville problem discontinuous boundary-value problem eigenvalue and eigenfunction eigenelement normalized eigenfunctions

STURM-LIOUVILLE PROBLEMS WITH EIGENDEPENDENT BOUNDARY AND TRANSMISSIONS CONDITIONS

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Acta Mathematica Scientia 2005年第4期25卷 731-740页

作者： Z.Akdog■an M.Demirci O.sh.mukhtarov Department of Mathematics Faculty of Science-Art Gaziosmanpas■a University 60100 Tokat TurkeyDepartment of Mathematics Faculty of Science-Art Gaziosmanpas■a University 60100 Tokat TurkeyDepartment of Mathematics Faculty of Science-Art Gaziosmanpas■a University 60100 Tokat Turkey

The purpose of this paper is to extend some fundamental spectral properties of regular Sturm-Liouville problems to special kind discontinuous boundary value problem, which consist of a Sturm-Liouville equation with piecewise continuous potential together with eigenvalue parameter on the boundary and transmission conditions. The authors suggest their own approach for finding asymptotic approximations formulas for eigenvalues and eigenfunctions of such discontinuous problems.

关键词： Sturm-Liouville problems transmission conditions asymptotic of eigenvalues and eigenfunctions discontinuous boundary-value problems

ASYMPTOTIC BEHAVIOUR OF EIGENVALUES FOR THE DISCONTINUOUS BOUNDARY-VALUE PROBLEM WITH FUNCTIONAL-TRANSMISSION CONDITIONS

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Acta Mathematica Scientia 2002年第3期22卷 335-345页

作者： O.sh.mukhtarov Department of Mathematics, Science and Arts Faculty, Gaziosmanpasa University, Tokat, TurkeyMustafa Kandemir Department of Mathematics, Faculty of A mas y a Education, Ondokuz Mayis University, Amasya, Turkey Gaziosmanpasa Univ Sci & Arts Fac Dept Math Tokat Turkey Ondokuz Mayis Univ Fac Amasya Educ Dept Math Amasya Turkey

In this study, the boundary-value problem with eigenvalue parameter generated by the differential equation with discontinuous coefficients and boundary conditions which contains not only endpoints of the considered interval, but also point of discontinuity and linear functionals is investigated. So, the problem is not pure boundary-value. The authors single out a class of linear functionals and find simple algebraic conditions on coefficients, which garantee the existence of infinit number eigenvalues. Also the asymptotic formulas for eigenvalues are found.

关键词： Asymptotic behaviour of eigenvalues boundary-value problems functional-conditions discontinuous coefficients