Abstract
In this article, we formulate nabla fractional sums and differences of order
0
<
α
≤
1
on the time scale
h
Z
, where
0
<
h
≤
1
. Then, we prove that if the nabla
h
-Riemann–Liouville (RL) fractional difference operator
(
a
∇
h
α
y
)
(
t
)
>
0
, then
y
(
t
)
is
α
-increasing. Conversely, if
y
(
t
)
is
α
-increasing and
y
(
a
)
>
0
, then
(
a
∇
h
α
y
)
(
t
)
>
0
. The monotonicity results for the nabla
h
-Caputo fractional difference operator are also concluded by using the relation between
h
-nabla RL and Caputo fractional difference operators. It is observed that the reported monotonicity coefficient is not affected by the step
h
. We formulate a nabla
h
-fractional difference initial value problem as well. Finally, we furniture our results by proving a fractional difference version of the Mean Value Theorem (MVT) on
h
Z
.