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Hierarchies of difference boundary value problems. (English) Zbl 1216.39002

Bound. Value Probl. 2011, Article ID 743135, 27 p. (2011); erratum ibid. 2012, Article ID 66, 2 p. (2012).
Summary: This paper generalises the work done in earlier work of the authors [Adv. Difference Equ. 2010, Article ID 623508, 23 p. (2010; Zbl 1202.39001); Adv. Difference Equ. 2010, Article ID 947058, 22 p. (2010; Zbl 1198.39001)], where we studied the effect of applying two Crum-type transformations to a weighted second-order difference equation with various combinations of Dirichlet, non-Dirichlet, and affine \(\lambda\)-dependent boundary conditions at the end points, where \(\lambda\) is the eigenparameter. We now consider general \(\lambda\)-dependent boundary conditions. In particular we show, using one of the Crum-type transformations, that it is possible to go up and down a hierarchy of boundary value problems keeping the form of the second-order difference equation constant but possibly increasing or decreasing the dependence on \(\lambda\) of the boundary conditions at each step. In addition, we show that the transformed boundary value problem either gains or loses an eigenvalue, or the number of eigenvalues remains the same as we step up or down the hierarchy.

MSC:

39A12 Discrete version of topics in analysis
39A10 Additive difference equations
34B15 Nonlinear boundary value problems for ordinary differential equations

Software:

OPQ
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References:

[1] Binding, PA; Browne, PJ; Watson, BA, Spectral isomorphisms between generalized Sturm-Liouville problems, No. 130, 135-152, (2002), Basel, Switzerland · Zbl 1038.34027
[2] Browne, PJ; Nillsen, RV, On difference operators and their factorization, Canadian Journal of Mathematics, 35, 873-897, (1983) · Zbl 0505.39003
[3] Binding, PA; Browne, PJ; Watson, BA, Sturm-Liouville problems with boundary conditions rationally dependent on the eigenparameter. II, Journal of Computational and Applied Mathematics, 148, 147-168, (2002) · Zbl 1019.34028
[4] Binding, PA; Browne, PJ; Watson, BA, Sturm-Liouville problems with reducible boundary conditions, Proceedings of the Edinburgh Mathematical Society. Series II, 49, 593-608, (2006) · Zbl 1126.34021
[5] Binding, PA; Browne, PJ; Watson, BA, Transformations between Sturm-Liouville problems with eigenvalue dependent and independent boundary conditions, Bulletin of the London Mathematical Society, 33, 749-757, (2001) · Zbl 1030.34027
[6] Ding, H-Y; Sun, Y-P; Xue, F-C, A hierarchy of differential-difference equations, conservation laws and new integrable coupling system, Communications in Nonlinear Science and Numerical Simulation, 15, 2037-2043, (2010) · Zbl 1222.37077
[7] Luo, L; Fan, E-G, A hierarchy of differential-difference equations and their integrable couplings, Chinese Physics Letters, 24, 1444-1447, (2007)
[8] Clarkson, PA; Hone, ANW; Joshi, N, Hierarchies of difference equations and Bäcklund transformations, Journal of Nonlinear Mathematical Physics, 10, 13-26, (2003) · Zbl 1362.39006
[9] Wu, Y; Geng, X, A new hierarchy of integrable differential-difference equations and Darboux transformation, Journal of Physics A, 31, l677-l684, (1998) · Zbl 0931.35190
[10] Currie, S; Love, AD, Transformations of difference equations I, No. 2010, 22, (2010) · Zbl 1198.39001
[11] Currie, S; Love, AD, Transformations of difference equations II, No. 2010, 23, (2010) · Zbl 1202.39001
[12] Binding, PA; Browne, PJ; Watson, BA, Sturm-Liouville problems with boundary conditions rationally dependent on the eigenparameter. I, Proceedings of the Edinburgh Mathematical Society. Series II, 45, 631-645, (2002) · Zbl 1019.34027
[13] Gautschi W: Orthogonal Polynomials: Computation and Approximation, Numerical Mathematics and Scientific Computation. Oxford University Press, New York, NY, USA; 2004:x+301. · Zbl 1130.42300
[14] Miller KS: Linear Difference Equations. W. A. Benjamin, New York, NY, USA; 1968:x+105. · Zbl 0162.40201
[15] Miller KS: An Introduction to the Calculus of Finite Differences and Difference Equations. Dover, New York, NY, USA; 1966:viii+167. · Zbl 0141.27701
[16] Atkinson FV: Discrete and Continuous Boundary Problems, Mathematics in Science and Engineering. Volume 8. Academic Press, New York, NY, USA; 1964:xiv+570.
[17] Teschl G: Jacobi Operators and Completely Integrable Nonlinear Lattices, Mathematical Surveys and Monographs. Volume 72. American Mathematical Society, Providence, RI, USA; 2000:xvii+351.
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