Comparison theorems and asymptotic behavior of solutions of discrete fractional equations

• Published in: Electronic journal of qualitative theory of differential equations Vol. 2015; no. 89; pp. 1 - 18
• Main Authors: Jia, Baoguo, Erbe, Lynn, Peterson, Allan
• Format: Journal Article
• Language: English
• Published: University of Szeged 01.01.2015
• Subjects: discrete mittag-leffler function; nabla and delta fractional difference; rising and falling function
• ISSN: 1417-3875; 1417-3875
• DOI: 10.14232/ejqtde.2015.1.89

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 $\nabla^\nu_{\rho(a)}x(t)=cx(t),\ 0<\nu<1$, and $\Delta^\nu_{a+\nu-1}x(t)=cx(t+\nu-1),\ 0<\nu<1$, have similar asymptotic behavior with the solutions of the first order difference equations $\nabla x(t)=cx(t),\ |c|<1 $ and $\Delta x(t)=cx(t)$, $|c|<1 $, respectively.