Some relations between the Caputo fractional difference operators and integer-order differences

Abstract

In this article, we are concerned with the relationships between the sign of Caputo fractional differences and integer nabla differences. In particular, we show that if $N-1<\nu<N$, $f:\mathbb{N}_{a-N+1}\to\mathbb{R}$, $\nabla^\nu_{a^*}f(t)\geq 0$, for $t\in\mathbb{N}_{a+1}$ and $\nabla^{N-1}f(a)\geq 0$, then $\nabla^{N-1}f(t)\geq 0$ for $t\in\mathbb{N}_a$.

Keywords

• Caputo fractional difference
• monotonicity
• Taylor monomial