Turkish Journal of Mathematics
Abstract
Let $L$ denote the selfadjoint difference operator of second order with boundary and impulsive conditions generated in $\ell _{2}\left( %TCIMACRO{\U{2115} } %BeginExpansion \mathbb{N} %EndExpansion \right) $ by \begin{equation*} a_{n-1}y_{n-1}+b_{n}y_{n}+a_{n}y_{n+1}=\left( 2\cosh z\right) y_{n}\text{ },% \text{ }n\in %TCIMACRO{\U{2115} } %BeginExpansion \mathbb{N} %EndExpansion \setminus \left\{ k-1,k,k+1\right\} , \end{equation*}% \begin{equation*} \begin{array}{c} y_{0}=0\text{ }, \\ \left\{ \begin{array}{c} y_{k+1}=\theta _{1}y_{k-1} \\ \bigtriangleup y_{k+1}=\theta _{2}\bigtriangledown y_{k-1} \end{array}% \right. ,\text{ }\theta _{1},\theta _{2}\in %TCIMACRO{\U{211d}}% %BeginExpansion \mathbb{R}, %EndExpansion \end{array}% \end{equation*} where $\left\{ a_{n}\right\} _{n\in %TCIMACRO{\U{2115} } %BeginExpansion \mathbb{N} %EndExpansion },$ $\left\{ b_{n}\right\} _{n\in %TCIMACRO{\U{2115} } %BeginExpansion \mathbb{N} %EndExpansion }$ are real sequences and $\bigtriangleup ,\bigtriangledown $ are respectively forward and backward operators. In this paper, the spectral properties of $L$ such as the resolvent operator, the spectrum, the eigenvalues, the scattering function and their properties are investigated. Moreover, an example about the scattering function and the existence of eigenvalues is given in the special cases, if \begin{equation*} \sum\limits_{n=1}^{\infty }n\left( \left\vert 1-a_{n}\right\vert +\left\vert b_{n}\right\vert \right)
DOI
10.3906/mat-2104-97
Keywords
Discrete equations, impulsive condition, hyperbolic eigenparameter, spectral analysis, scattering function, resolvent operator, eigenvalues
First Page
377
Last Page
396
Recommended Citation
KÖPRÜBAŞI, TURHAN and KÜÇÜKEVCİLİOĞLU, YELDA AYGAR
(2022)
"Discrete impulsive Sturm-Liouville equation with hyperboliceigenparameter,"
Turkish Journal of Mathematics: Vol. 46:
No.
2, Article 4.
https://doi.org/10.3906/mat-2104-97
Available at:
https://journals.tubitak.gov.tr/math/vol46/iss2/4
