Monotonicity and non-monotonicity results for sequential fractional delta differences of mixed order

Abstract

We consider the discrete fractional sequential difference $\Delta _{1+a-\mu }^{\nu }\Delta _a^{\mu }f(t)$, where $t\in \mathbb {N}_{3-\mu -\nu +a}$, in two separate cases, where in each case we require that $\mu +\nu \in (1,2)$. In the first case, we show that when $\mu \in (0,1)$ and $\nu \in (1,2)$ it follows that the condition $\Delta _{1+a-\mu }^{\nu }\Delta _a^{\mu }f(t)\ge 0$ implies that f is an increasing map when we impose that $f(a)\ge 0$, $\Delta f(a)\ge 0$, and $\Delta f(a+1)\ge 0$. On the other hand, when $\mu \in (1,2)$ and $\nu \in (0,1)$ we demonstrate that the situation is very different and that this type of monotonicity result only holds when restricted to a proper subregion of the $(\mu ,\nu )$-parameter space coupled with some additional auxiliary conditions.