Spectral Properties of the Anti-Periodic Boundary-Value-Transition Problems
This work is concerned with the boundary-value-transition problem consisting of a
two-interval Sturm-Liouville equation
Lu ≔ −u′′(x) + q(x)u(x) = λu(x) , x ∈ [−1,0) ∪ (0,1]
together with anti-periodic boundary conditions, given by
u(−1) = −u(1)
u′(−1) = −u′(1)
and transition conditions at the interior point x = 0, given by
u(+0) = Ku(−0)
u′(+0) =1/Ku′(−0)
where q(x) is a continuous function in the intervals [−1,0) and (0,1] with finite limit values q(±0) ,
K ≠ 0 is the real number and λ is the complex eigenvalue parameter. In this study we shall investigate
some properties of the eigenvalues and eigenfunctions of the considered problem.
Keywords:
Anti-periodic Sturm-Liouville problem, , eigenvalue, eigenfunction, , transition condition,
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- [1] G. D. Birkhoff, On the Asymptotic Character of the Solution of the Certain Linear Differential Equations (1908).
- [2] J. D. Tamarkin, Some General Problems of The Theory of Ordinary Linear Differential Equations And Expansions of An Arbitary Function in Series of Fundamental Functions, Math. Z. 27 (1928) 1-54.
- [3] J. W. Lee, Spectral Properties and Oscillation Theorems for Periodic Boundary-Value Problems of Sturm Liouville Type, Journal of Differential Equations 11 (1972) 592-606.
- [4] G. V. Berghe, M. V. Daele, H. D. Meyer, A modified difference scheme for periodic and semiperiodic Sturm-Liouville problems, Applied Numerical Mathematics 18 (1995) 69-78.
- [5] Y. Liu, Periodic Boundary Value Problems for Higher Order Impulsive Functional Differential Equations. SDÜ Fen Edebiyat Fakültesi Fen Dergisi (E-dergi) 2 (2007) 253-272.
- [6] D. B. Wang, Periodic Boundary Value Problems for Nonlinear First-Order Impulsive Dynamic Equations on Time Scales, Advances in Difference Equations 12 (2012).
- [7] V. Malathi, B. S. Mohamed, B. T. Bachok, Computing Eigenvalues Of Periodic Sturm-Liouville Problems Using Shooting Technique And Direct Integration Method, International Journal of Computer Mathematics, 68 (1996) 119-132.
- [8] K. Aydemir, O. Sh. Mukhtarov, Completeness Of One Two-Interval Boundary Value Problem With Transmission Conditions, Miskolc Mathematical Notes 15 (2014) 293-303.
- [9] K. Aydemir, O. Sh. Mukhtarov, Class of Sturm-Liouville problems with eigen-parameter dependent transmission conditions, Numerical Functional Analysis and Optimization 38(10) (2017) 1260-1275.
- [10] M. Kandemir, O. Sh. Mukhtarov, Nonlocal Sturm-Liouville problems with integral terms in the boundary conditions, Electronic Journal of Differential Equations 11 (2017) 112.
- [11] O. Sh. Mukhtarov, H. Olğar, K. Aydemir, Resolvent Operator and Spectrum of New Type Boundary Value Problems, Filomat 29 (2015) 1671–1680.
- [12] O. Sh. Mukhtarov, H. Olğar, K. Aydemir, I. Jabbarov, Operator-Pencil Realization Of One Sturm- Liouville Problem With Transmission Conditions, Applied And Computational Mathematics 17(2)(2018) 284-294.
- [13] E. C. Titchmars, Eigenfunctions Expansion Associated with Second Order Differential Equations I, second edn. Oxford Univ. press, London 1962.
- Yayın Aralığı: 4
- Başlangıç: 2014
- Yayıncı: Naim Çağman