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# Erratum to: Hierarchies of difference boundary value problems

- Sonja Currie
^{1}Email author and - Anne D Love
^{1}

**2012**:66

https://doi.org/10.1186/1687-2770-2012-66

© Currie and Love; licensee Springer 2012

**Received:**25 May 2012**Accepted:**30 May 2012**Published:**28 June 2012

## Keywords

- Differential Equation
- Partial Differential Equation
- Ordinary Differential Equation
- Functional Equation
- Spectral Parameter

## Erratum to: Boundary value problems, Volume 2011, Article ID 743135

- (1)
The following paragraph needs to be inserted immediately after Theorem 4.2:

- (2)
In Corollary 4.4 and its proof, there were typographical errors as well as notation that was not apparent. These should read as follows:

**Corollary 4.4**

*If*${\lambda}_{1},\dots ,{\lambda}_{s+l+m+1}$

*are the eigenvalues of any one of the original boundary value problems*(1)-(9),

*in Theorem*4.2,

*with corresponding eigenfunctions*${u}_{1}(n),\dots ,{u}_{s+l+m+1}(n)$,

*then*

- (i)
${\lambda}_{0}=0$, ${\lambda}_{1},\dots ,{\lambda}_{s+l+m+1}$

*are the eigenvalues of the corresponding transformed boundary value problems*(1)-(3),*in Theorem*4.2,*with corresponding eigenfunctions*$\frac{1}{z(n-1)c(n-1)}$, ${u}_{1}(n),\dots ,{u}_{s+l+m+1}(n)$; - (ii)
${\lambda}_{1},\dots ,{\lambda}_{s+l+m+1}$

*are the eigenvalues of the corresponding transformed boundary value problems*(4)-(9),*in Theorem*4.2,*with corresponding eigenfunctions*${u}_{1}(n),\dots ,{u}_{s+l+m+1}(n)$.

*Also*, *if*${\lambda}_{0}=0$, ${\lambda}_{1},\dots ,{\lambda}_{s+l+m}$*are the eigenvalues of any one of the original boundary value problems* (10)-(12), *in Theorem* 4.2, *with corresponding eigenfunctions*$z(n)$, ${u}_{1}(n),\dots ,{u}_{s+l+m}(n)$, *then*${\lambda}_{1},\dots ,{\lambda}_{s+l+m}$*are the eigenvalues of the corresponding transformed boundary value problems* (10)-(12), *in Theorem* 4.2, *with corresponding eigenfunctions*${u}_{1}(n),\dots ,{u}_{s+l+m}(n)$.

*Proof*By Theorems 2.1, 3.2, 3.3, 3.4 we have that (2.1) transforms eigenfunctions of the original boundary value problems (1)-(9) to eigenfunctions of the corresponding transformed boundary value problems. In particular, if ${\lambda}_{1},\dots ,{\lambda}_{s+l+m+1}$ are the eigenvalues of one of the original boundary value problems, (1)-(9), with eigenfunctions ${u}_{1}(n),\dots ,{u}_{s+l+m+1}(n)$, then:

- (i)
$\frac{1}{z(n-1)c(n-1)}$, ${u}_{1}(n),\dots ,{u}_{s+l+m+1}(n)$ are the eigenfunctions of the corresponding transformed boundary value problem, (1)-(3), with eigenvalues ${\lambda}_{0}=0$, ${\lambda}_{1},\dots ,{\lambda}_{s+l+m+1}$. Since the transformed boundary value problems, (1)-(3), have $s+l+m+2$ eigenvalues, it follows that ${\lambda}_{0}=0$, ${\lambda}_{1},\dots ,{\lambda}_{s+l+m+1}$ constitute all the eigenvalues of the transformed boundary value problem;

- (ii)
${u}_{1}(n),\dots ,{u}_{s+l+m+1}(n)$ are the eigenfunctions of the corresponding transformed boundary value problem, (4)-(9), with eigenvalues ${\lambda}_{1},\dots ,{\lambda}_{s+l+m+1}$. Since the transformed boundary value problems, (4)-(9), have $s+l+m+1$ eigenvalues, it follows that ${\lambda}_{1},\dots ,{\lambda}_{s+l+m+1}$ constitute all the eigenvalues of the transformed boundary value problem.

Also, again by Theorems 2.1, 3.2, 3.3, 3.4 we have that (2.1) transforms eigenfunctions of the original boundary value problems (10)-(12) to eigenfunctions of the corresponding transformed boundary value problems. In particular, if ${\lambda}_{0}=0$, ${\lambda}_{1},\dots ,{\lambda}_{s+l+m}$ are the eigenvalues of one of the original boundary value problems, (10)-(12), with eigenfunctions $z(n)$, ${u}_{1}(n),\dots ,{u}_{s+l+m}(n)$, then ${u}_{1}(n),\dots ,{u}_{s+l+m}(n)$ are the eigenfunctions of the corresponding transformed boundary value problem, (10)-(12), with eigenvalues ${\lambda}_{1},\dots ,{\lambda}_{s+l+m}$. Since the transformed boundary value problems, (10)-(12), have $s+l+m$ eigenvalues, it follows that ${\lambda}_{1},\dots ,{\lambda}_{s+l+m}$ constitute all the eigenvalues of the transformed boundary value problem. □

## Declarations

### Acknowledgements

SC was supported by NRF grant no. IFR2011040100017.

## Authors’ Affiliations

## Copyright

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