The cubic eigenparameter dependent discrete Dirac equations with principal functions

Year 2019, Volume: 68 Issue: 2, 1742 - 1760, 01.08.2019

Turhan Köprübaşı

https://doi.org/10.31801/cfsuasmas.454232

Cited By: 5

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.

Keywords

Discrete Dirac equations , Eigenparameter , Spectral analysis , Spectrum , Principal functions

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.