Discrete impulsive Sturm-Liouville equation with hyperbolic eigenparameter

Authors: TURHAN KÖPRÜBAŞI, YELDA AYGAR KÜÇÜKEVCİLİOĞLU

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 $$\begin{equation*} \sum\limits_{n=1}^{\infty }n\left( \left\vert 1-a_{n}\right\vert +\left\vert b_{n}\right\vert \right) <\infty . \end{equation*}$$

Keywords: Discrete equations, impulsive condition, hyperbolic eigenparameter, spectral analysis, scattering function, resolvent operator, eigenvalues

Full Text: PDF