Stability criteria for volterra type linear nabla fractional difference equations

In this study, we give some necessary and sufficient conditions on the stability for Volterra type linear nabla fractional difference equations of the form ∇-1vx(t)=λx(t),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla _{-1}^{v}x(t)=\lambda x(t),$$\end{document}t∈N1,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\in \mathbb {N}_{1},$$\end{document} with initial condition ∇-1v-1x(t)t=0=x0.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla _{-1}^{v-1}x(t)\left| _{t=0}=x_{0}.\right. $$\end{document}For this, first of all we show that the above equation is a convolution-type Volterra equation, then give the stability conditions by using the stability analysis methods of the convolution type Volterra equations. Also we give some examples to illustrate our theoretic results.