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Zaman Skalaları Üzerinde Parametreye Bağlı Dirac Sistemleri

Year 2018, Volume: 39 Issue: 4, 864 - 870, 24.12.2018

Ahmet Sinan Özkan

https://doi.org/10.17776/csj.471958
Cited By: 4

Abstract

Bu çalışmada bir zaman skalası üzerinde iki farklı genelleştirilmiş
Dirac sistemi ve parametreye bağlı sınır koşulları ile üretilen bir sınır değer
problemi ele alınmıştır.  Sistemlerin
eşleniksiz (disconjugate) olması için yeterli koşullar ve problemin özdeğerlerinin
sayısı ile ilgili bir formül elde edilmiştir.

Keywords

Zaman Skalası Dinamik denklemler Dirac sistemi parametreye bağlı sınır koşulları eşleniksizlik

References

  • [1]. Hilger, S., Analysis on measure chains – a unified approach to continuous and discrete calculus. Results in Math., 18 (1990) 18-56.
  • [2]. Erbe, L., Hilger, S., Sturmian theory on measure chains, Differ. Equ. Dyn. Syst., 1 (1993) 223-244.
  • [3]. Agarwal, R.P., Bohner, M., Wong, P.J.Y., Sturm-Liouville eigenvalue problems on time scales, Appl. Math. Comput., 99 (1999) 153-166.
  • [4]. Allahverdiev, B.P., Eryilmaz, A., Tuna, H., Dissipative Sturm-Liouville operators with a spectral parameter in the boundary condition on bounded time scales, Electronic Journal of Differential Equations, 95 (2017) 1–13.
  • [5]. Amster, P., De Napoli, P., Pinasco, J.P., Eigenvalue distribution of second-order dynamic equations on time scales considered as fractals, J. Math. Anal. Appl., 343 (2008) 573-584.
  • [6]. Amster, P., De Napoli, P., Pinasco, J.P., Detailed asymptotic of eigenvalues on time scales, J. Differ. Equ. Appl., 15 (2009) 225-231.
  • [7]. Davidson, F.A., Rynne, B.P., Global bifurcation on time scales, J. Math. Anal. Appl., 267 (2002) 345-360.
  • [8]. Davidson, F.A., Rynne, B.P., Self-adjoint boundary value problems on time scales, Electron. J. Differ. Equ., 175 (2007) 1-10.
  • [9]. Davidson, F.A., Rynne, B.P., Eigenfunction expansions in L2 spaces for boundary value problems on time-scales, J. Math. Anal. Appl., 335 (2007) 1038-1051.
  • [10]. Erbe, L., Peterson, A., Eigenvalue conditions and positive solutions, J. Differ. Equ. Appl., 6 (2000) 165-191.
  • [11]. Guseinov, G.S., Eigenfunction expansions for a Sturm-Liouville problem on time scales, Int. J. Differ. Equ., 2 (2007) 93-104.
  • [12]. Guseinov, G.S., An expansion theorem for a Sturm-Liouville operator on semi-unbounded time scales, Adv. Dyn. Syst. Appl., 3 (2008) 147-160.
  • [13]. Hilscher, R.S., Zemanek, P., Weyl-Titchmarsh theory for time scale symplectic systems on half line, Abstr. Appl. Anal., Art. ID 738520 (2011) 41 pp.
  • [14]. Huseynov, A., Limit point and limit circle cases for dynamic equations on time scales, Hacet. J. Math. Stat., 39 (2010) 379-392.
  • [15]. Huseynov, A., Bairamov, E., On expansions in eigenfunctions for second order dynamic equations on time scales, Nonlinear Dyn. Syst. Theory, 9 (2009) 7-88.
  • [16]. Kong, Q., Sturm-Liouville problems on time scales with separated boundary conditions, Results Math., 52 (2008) 111-121.
  • [17]. Rynne, B.P., L2 spaces and boundary value problems on time-scales, J. Math. Anal. Appl., 328 (2007) 1217-1236.
  • [18]. Sun, S., Bohner, M., Chen,S., Weyl-Titchmarsh theory for Hamiltonian dynamic systems. Abstr. Appl. Anal. Art., ID 514760 (2010) 18 pp.
  • [19]. Bohner, M., Peterson, A., Dynamic Equations on Time Scales, Birkhauser, Boston, MA, 2001.
  • [20]. Bohner, M., Peterson, A., Advances in Dynamic Equations on Time Scales, Birkhauser, Boston, MA, 2003.
  • [21]. Gulsen, T, Yilmaz, E, Spectral theory of Dirac system on time scales, Appl. Anal., 96-16 (2017) 2684-2694.
  • [22]. Gulsen, T, Yilmaz, E, Goktas, S, Conformable fractional Dirac system on time scales, J. Inequal. Appl, 161-1 (2017) 1-10.

Parameter-Dependent Dirac Systems on Time Scales

Year 2018, Volume: 39 Issue: 4, 864 - 870, 24.12.2018

Ahmet Sinan Özkan

https://doi.org/10.17776/csj.471958
Cited By: 4

Abstract

In this study, we consider two generalized Dirac systems on a time scale
and a boundary-value problem with boundary conditions depending on the spectral
parameter. We give some sufficient conditions for disconjugacy of the systems
and obtain a formula about the number of eigenvalues of the problem.

Keywords

Time scale Dynamic Equations Dirac-System parameter-dependent boundary conditions disconjugacy

References

  • [1]. Hilger, S., Analysis on measure chains – a unified approach to continuous and discrete calculus. Results in Math., 18 (1990) 18-56.
  • [2]. Erbe, L., Hilger, S., Sturmian theory on measure chains, Differ. Equ. Dyn. Syst., 1 (1993) 223-244.
  • [3]. Agarwal, R.P., Bohner, M., Wong, P.J.Y., Sturm-Liouville eigenvalue problems on time scales, Appl. Math. Comput., 99 (1999) 153-166.
  • [4]. Allahverdiev, B.P., Eryilmaz, A., Tuna, H., Dissipative Sturm-Liouville operators with a spectral parameter in the boundary condition on bounded time scales, Electronic Journal of Differential Equations, 95 (2017) 1–13.
  • [5]. Amster, P., De Napoli, P., Pinasco, J.P., Eigenvalue distribution of second-order dynamic equations on time scales considered as fractals, J. Math. Anal. Appl., 343 (2008) 573-584.
  • [6]. Amster, P., De Napoli, P., Pinasco, J.P., Detailed asymptotic of eigenvalues on time scales, J. Differ. Equ. Appl., 15 (2009) 225-231.
  • [7]. Davidson, F.A., Rynne, B.P., Global bifurcation on time scales, J. Math. Anal. Appl., 267 (2002) 345-360.
  • [8]. Davidson, F.A., Rynne, B.P., Self-adjoint boundary value problems on time scales, Electron. J. Differ. Equ., 175 (2007) 1-10.
  • [9]. Davidson, F.A., Rynne, B.P., Eigenfunction expansions in L2 spaces for boundary value problems on time-scales, J. Math. Anal. Appl., 335 (2007) 1038-1051.
  • [10]. Erbe, L., Peterson, A., Eigenvalue conditions and positive solutions, J. Differ. Equ. Appl., 6 (2000) 165-191.
  • [11]. Guseinov, G.S., Eigenfunction expansions for a Sturm-Liouville problem on time scales, Int. J. Differ. Equ., 2 (2007) 93-104.
  • [12]. Guseinov, G.S., An expansion theorem for a Sturm-Liouville operator on semi-unbounded time scales, Adv. Dyn. Syst. Appl., 3 (2008) 147-160.
  • [13]. Hilscher, R.S., Zemanek, P., Weyl-Titchmarsh theory for time scale symplectic systems on half line, Abstr. Appl. Anal., Art. ID 738520 (2011) 41 pp.
  • [14]. Huseynov, A., Limit point and limit circle cases for dynamic equations on time scales, Hacet. J. Math. Stat., 39 (2010) 379-392.
  • [15]. Huseynov, A., Bairamov, E., On expansions in eigenfunctions for second order dynamic equations on time scales, Nonlinear Dyn. Syst. Theory, 9 (2009) 7-88.
  • [16]. Kong, Q., Sturm-Liouville problems on time scales with separated boundary conditions, Results Math., 52 (2008) 111-121.
  • [17]. Rynne, B.P., L2 spaces and boundary value problems on time-scales, J. Math. Anal. Appl., 328 (2007) 1217-1236.
  • [18]. Sun, S., Bohner, M., Chen,S., Weyl-Titchmarsh theory for Hamiltonian dynamic systems. Abstr. Appl. Anal. Art., ID 514760 (2010) 18 pp.
  • [19]. Bohner, M., Peterson, A., Dynamic Equations on Time Scales, Birkhauser, Boston, MA, 2001.
  • [20]. Bohner, M., Peterson, A., Advances in Dynamic Equations on Time Scales, Birkhauser, Boston, MA, 2003.
  • [21]. Gulsen, T, Yilmaz, E, Spectral theory of Dirac system on time scales, Appl. Anal., 96-16 (2017) 2684-2694.
  • [22]. Gulsen, T, Yilmaz, E, Goktas, S, Conformable fractional Dirac system on time scales, J. Inequal. Appl, 161-1 (2017) 1-10.
There are 22 citations in total.

Details

Primary Language English
Journal Section Natural Sciences
Authors

Ahmet Sinan Özkan 0000-0002-9703-8982

Publication Date December 24, 2018
Submission Date October 18, 2018
Acceptance Date November 20, 2018
Published in Issue Year 2018Volume: 39 Issue: 4

Cite

APA Özkan, A. S. (2018). Parameter-Dependent Dirac Systems on Time Scales. Cumhuriyet Science Journal, 39(4), 864-870. https://doi.org/10.17776/csj.471958

Cited By

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On Expansion in Eigenfunction for Dirac Systems on the Unbounded Time Scales
Differential Equations and Dynamical Systems
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On the number of eigenvalues for parameter‐dependent diffusion problem on time scales
Mathematical Methods in the Applied Sciences
https://doi.org/10.1002/mma.6805
Dissipative Dirac operator with general boundary conditions on time scales
Ukrains’kyi Matematychnyi Zhurnal
B. P. Allahverdiev
https://doi.org/10.37863/umzh.v72i5.546