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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 |

### Keywords:

Crum-type transformations; weighted second-order difference equation; boundary value problems; eigenvalues### Software:

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\textit{S. Currie} and \textit{A. D. Love}, Bound. Value Probl. 2011, Article ID 743135, 27 p. (2011; Zbl 1216.39002)

### References:

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