## Mathematics, Department of

#### Date of this Version

12-5-2015

#### Citation

2015, No. 89, 1–18; doi: 10.14232/ejqtde.2015.1.89 http://www.math.u-szeged.hu/ejqtde/

#### Abstract

Consider the following n-th order nabla and delta fractional difference equations

rn r (a)x(t) = c(t)x(t), t 2 Na+1, x(a) > 0.

and

Va+v-1x(t) = c(t)x(t + v - 1), t 2 Na, x(a + n - 1) > 0

We establish comparison theorems by which we compare the solutions x(t) of (*) and (**) with the solutions of the equations rn r(a)x(t) = bx(t) and Dn a+v-1x(t) = bx(t + v -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 vp(a)x(t) = cx(t), 0 < n < 1, and Dn a+v-1x(t) = cx(t + v - 1), 0 < n < 1, have similar asymptotic behavior with the solutions of the first order difference equations rx(t) = cx(t), jcj < 1 and Dx(t) = cx(t), jcj < 1, respectively.