A $\beta$-Sturm Liouville problem associated with the general quantum operator

Abstract

Let $\,I\subseteq\mathbb{R}\,$ be an interval and $\,\beta:\,I\rightarrow\,I\,$ a strictly increasing and continuous function with a unique fixed point $\,s_0\in I\,$ which satisfies $\,(s_0-t)(\beta(t)-t)\geq 0\,$ for all $\,t\in I$, where the equality holds only when $\,t=s_0$. The general quantum operator defined in 2015 by Hamza et al., $\,D_{\beta}[f](t):=\displaystyle\frac{f\big(\beta(t)\big)-f(t)}{\beta(t)-t}\,$ if $\,t\neq s_0\,$ and $\,D_{\beta}[f](s_0):=f^{\prime}(s_0)\,$ if $\,t=s_0,$ generalizes the Jackson $\,q$-operator $\,D_{q}\,$ and also the Hahn $\,(q,\omega)$-operator, $\,D_{q,\omega}$. Regarding a $\beta-$Sturm Liouville eigenvalue problem associated with the above operator $\,D_{\beta}\,$, we construct the $\beta-$Lagrange's identity, show that it is self-adjoint in $\,\mathscr{L}_{\beta}^2([a,b]),$ and exhibit some properties for the corresponding eigenvalues and eigenfunctions.