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Consider the following $\nu$-th order nabla and delta fractional difference equations \begin{equation} \begin{aligned} \nabla^\nu_{\rho(a)}x(t)&=c(t)x(t),\quad \quad t\in\mathbb{N}_{a+1},\\ x(a)&>0. \end{aligned}\tag{$\ast$} \end{equation} and \begin{equation} \begin{aligned} \Delta^\nu_{a+\nu-1}x(t)&=c(t)x(t+\nu-1),\quad \quad t\in\mathbb{N}_{a},\\ x(a+\nu-1)&>0. \end{aligned}\tag{$\ast\ast$} \end{equation} We establish comparison theorems by which we compare the solutions $x(t)$ of ($\ast$) and ($\ast\ast$) with the solutions of the equations $\nabla^\nu_{\rho(a)}x(t)=bx(t)$ and $\Delta^\nu_{a+\nu-1}x(t)=bx(t+\nu-1),$ respectively, where $b$ is a constant. We obtain four asymptotic results, one of them extends the recent result [F. M. Atici, P. W. Eloe, Rocky Mountain J. Math. 41(2011) 353--370]. These results show that the solutions of two fractional difference equations $\nabla^\nu_{\rho(a)}x(t)=cx(t),\ 0 |
