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The cubic eigenparameter dependent discrete Dirac equations with principal functions

Year 2019, Volume: 68 Issue: 2, 1742 - 1760, 01.08.2019
https://doi.org/10.31801/cfsuasmas.454232

Abstract

Let us consider the Boundary Value Problem (BVP) for the discrete Dirac Equations

{<K1.1/>┊   #0.1

<K1.1 ilk="MATRIX" >
a_{n+1}y_{n+1}⁽²⁾+b_{n}y_{n}⁽²⁾+p_{n}y_{n}⁽¹⁾=λy_{n}⁽¹⁾
a_{n-1}y_{n-1}⁽¹⁾+b_{n}y_{n}⁽¹⁾+q_{n}y_{n}⁽²⁾=λy_{n}⁽²⁾ , n∈ℕ,
</K1.1>

(γ₀+γ₁λ+γ₂λ²+γ₃λ³)y₁⁽²⁾+(β₀+β₁λ+β₂λ²+β₃λ³)y₀⁽¹⁾=0,   #0.2

where (a_{n}), (b_{n}), (p_{n}) and (q_{n}), n∈ℕ are complex sequences, γ_{i}, β_{i}∈ℂ, i=0,1,2 and λ is a eigenparameter. Discussing the eigenvalues and the spectral singularities, we prove that the BVP (0.1), (0.2) has a finite number of eigenvalues and spectral singularities with a finite multiplicities, if

∑_{n=1}^{∞}exp(εn^{δ})(|1-a_{n}|+|1+b_{n}|+|p_{n}|+|q_{n}|)<∞,

holds, for some ε>0 and (1/2)≤δ≤1.

References

  • Agarwal, R. P. Difference equation and inequalities. 2000. Theory, Methods and Applications. Marcel Dekkar Inc., New York, Basel.
  • Agarwal, R. P. and Wong, P. J. Y. 1997. Advanced Topics in Difference Equations. Kluwer, Dordrecht.
  • Kelley, W. G. and Peterson, A. C. 2001. Difference Equations. An Introduction with Applications, Harcourt Academic Press.
  • Agarwal, R. P., Perera, K. and O'Regan, D. Multiple positive solutions of singular and nonsingular discrete problems via variational methods. Nonlinear Analysis, 2004, 58, 69-73.
  • Agarwal, R. P., Perera, K. and O'Regan, D. Multiple positive solutions of singular discrete p-Laplasian problems via variational methods. Advances in Difference Equations, 2005, 2005:2, 93-99.
  • Berezanski, Y. M. Integration of nonlinear difference equations by the inverse spectral problem method. Soviet Math. Dokl. 31, 1985, 264-267.
  • Toda, M. 1981. Theory of Nonlinear Lattices. Springer-Verlag, Berlin.
  • Bairamov, E. and Celebi, A. O. Spectrum and spectral expansion for the non-selfadjoint discrete Dirac operators. Quart. J. Math. Oxford, 1999, 50 (2), 371-384.
  • Bairamov, E., Cakar, O. and Krall, A. M. Non-selfadjoint difference operators and Jacobi matrices with spectral singularities. Math. Nachr., 2001, 229, 5-14.
  • Adivar, M. and Bairamov, E. Spectral properties of non-selfadjoint difference operators. J. Math. Anal. Appl., 2001, 261, 461-478.
  • Adivar, M. and Bairamov, E. Difference equations of second order with spectral singularities. J. Math. Anal. Appl., 2003, 277, 714-721.
  • Adivar, M. and Bohner, M. Spectral analysis of q-difference equations with spectral singularities. Math. Comput. Modelling, 2006, 43 (7-9), 695-703.
  • Adivar, M. and Bohner, M. Spectrum and principal vectors of second order q-difference equations. Indian J. Math., 2006, 48 (1),17-33.
  • Bairamov, E. and Coskun, C. Jost solutions and the spectrum of the system of difference equations. Appl. Math. Lett., 2004, 17, 1039-1045.
  • Bairamov, E. and Koprubasi, T. Eigenparameter dependent discrete Dirac equations with spectral singularities. Appl. Math. and Comp., 2010, 215, 4216-4220.
  • Aygar, Y., Olgun, M. and Koprubasi, T. Principal Functions of Nonselfadjoint Discrete Dirac Equations with Spectral Parameter in Boundary Conditions. Abstract and Applied Analysis, 2012, vol. 2012, ID 924628.
  • Koprubasi, T. Spectrum of the quadratic eigenparameter dependent discrete Dirac equations. Advances in Difference Equations, 2014, 2014:148.
  • Koprubasi, T. and Yokus, N. Quadratic eigenparameter dependent discrete Sturm Liouville equations with spectral singularities. Appl. Math. and Comp., 2014, 244, 57-62.
  • Levitan, B. M. and Sargsjan, I. S. Introduction to Spectral Theory. Translations of Mathematical Monographs, 1975, 39 .
  • Dolzhenko, E. P. Boundary value uniqueness theorems for analytic functions. Math. Notes, 1979, 26 (6), 437-442.
Year 2019, Volume: 68 Issue: 2, 1742 - 1760, 01.08.2019
https://doi.org/10.31801/cfsuasmas.454232

Abstract

References

  • Agarwal, R. P. Difference equation and inequalities. 2000. Theory, Methods and Applications. Marcel Dekkar Inc., New York, Basel.
  • Agarwal, R. P. and Wong, P. J. Y. 1997. Advanced Topics in Difference Equations. Kluwer, Dordrecht.
  • Kelley, W. G. and Peterson, A. C. 2001. Difference Equations. An Introduction with Applications, Harcourt Academic Press.
  • Agarwal, R. P., Perera, K. and O'Regan, D. Multiple positive solutions of singular and nonsingular discrete problems via variational methods. Nonlinear Analysis, 2004, 58, 69-73.
  • Agarwal, R. P., Perera, K. and O'Regan, D. Multiple positive solutions of singular discrete p-Laplasian problems via variational methods. Advances in Difference Equations, 2005, 2005:2, 93-99.
  • Berezanski, Y. M. Integration of nonlinear difference equations by the inverse spectral problem method. Soviet Math. Dokl. 31, 1985, 264-267.
  • Toda, M. 1981. Theory of Nonlinear Lattices. Springer-Verlag, Berlin.
  • Bairamov, E. and Celebi, A. O. Spectrum and spectral expansion for the non-selfadjoint discrete Dirac operators. Quart. J. Math. Oxford, 1999, 50 (2), 371-384.
  • Bairamov, E., Cakar, O. and Krall, A. M. Non-selfadjoint difference operators and Jacobi matrices with spectral singularities. Math. Nachr., 2001, 229, 5-14.
  • Adivar, M. and Bairamov, E. Spectral properties of non-selfadjoint difference operators. J. Math. Anal. Appl., 2001, 261, 461-478.
  • Adivar, M. and Bairamov, E. Difference equations of second order with spectral singularities. J. Math. Anal. Appl., 2003, 277, 714-721.
  • Adivar, M. and Bohner, M. Spectral analysis of q-difference equations with spectral singularities. Math. Comput. Modelling, 2006, 43 (7-9), 695-703.
  • Adivar, M. and Bohner, M. Spectrum and principal vectors of second order q-difference equations. Indian J. Math., 2006, 48 (1),17-33.
  • Bairamov, E. and Coskun, C. Jost solutions and the spectrum of the system of difference equations. Appl. Math. Lett., 2004, 17, 1039-1045.
  • Bairamov, E. and Koprubasi, T. Eigenparameter dependent discrete Dirac equations with spectral singularities. Appl. Math. and Comp., 2010, 215, 4216-4220.
  • Aygar, Y., Olgun, M. and Koprubasi, T. Principal Functions of Nonselfadjoint Discrete Dirac Equations with Spectral Parameter in Boundary Conditions. Abstract and Applied Analysis, 2012, vol. 2012, ID 924628.
  • Koprubasi, T. Spectrum of the quadratic eigenparameter dependent discrete Dirac equations. Advances in Difference Equations, 2014, 2014:148.
  • Koprubasi, T. and Yokus, N. Quadratic eigenparameter dependent discrete Sturm Liouville equations with spectral singularities. Appl. Math. and Comp., 2014, 244, 57-62.
  • Levitan, B. M. and Sargsjan, I. S. Introduction to Spectral Theory. Translations of Mathematical Monographs, 1975, 39 .
  • Dolzhenko, E. P. Boundary value uniqueness theorems for analytic functions. Math. Notes, 1979, 26 (6), 437-442.
There are 20 citations in total.

Details

Primary Language English
Journal Section Review Articles
Authors

Turhan Köprübaşı 0000-0003-1551-1527

Publication Date August 1, 2019
Submission Date August 17, 2018
Acceptance Date January 3, 2019
Published in Issue Year 2019 Volume: 68 Issue: 2

Cite

APA Köprübaşı, T. (2019). The cubic eigenparameter dependent discrete Dirac equations with principal functions. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 68(2), 1742-1760. https://doi.org/10.31801/cfsuasmas.454232
AMA Köprübaşı T. The cubic eigenparameter dependent discrete Dirac equations with principal functions. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. August 2019;68(2):1742-1760. doi:10.31801/cfsuasmas.454232
Chicago Köprübaşı, Turhan. “The Cubic Eigenparameter Dependent Discrete Dirac Equations With Principal Functions”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68, no. 2 (August 2019): 1742-60. https://doi.org/10.31801/cfsuasmas.454232.
EndNote Köprübaşı T (August 1, 2019) The cubic eigenparameter dependent discrete Dirac equations with principal functions. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68 2 1742–1760.
IEEE T. Köprübaşı, “The cubic eigenparameter dependent discrete Dirac equations with principal functions”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 68, no. 2, pp. 1742–1760, 2019, doi: 10.31801/cfsuasmas.454232.
ISNAD Köprübaşı, Turhan. “The Cubic Eigenparameter Dependent Discrete Dirac Equations With Principal Functions”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68/2 (August 2019), 1742-1760. https://doi.org/10.31801/cfsuasmas.454232.
JAMA Köprübaşı T. The cubic eigenparameter dependent discrete Dirac equations with principal functions. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2019;68:1742–1760.
MLA Köprübaşı, Turhan. “The Cubic Eigenparameter Dependent Discrete Dirac Equations With Principal Functions”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 68, no. 2, 2019, pp. 1742-60, doi:10.31801/cfsuasmas.454232.
Vancouver Köprübaşı T. The cubic eigenparameter dependent discrete Dirac equations with principal functions. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2019;68(2):1742-60.

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics.

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