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

Jia, Baoguo ; Erbe, Lynn ; Peterson, Allan

Electronic journal of differential equations, Tome 2015 (2015)

Résumé

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 $N-1\nu$, for $t\in\mathbb{N}_{a+1}$ and $\nabla^{N-1}f(a)\geq 0$, then $\nabla^{N-1}f(t)\geq 0$ for $t\in\mathbb{N}_a$. Conversely, if $N-1\nu$, and $\nabla^{N}f(t)\geq 0$ for $t\in\mathbb{N}_{a+1}$, then $\nabla^{\nu}_{a^*}f(t)\geq 0$, for each $t\in\mathbb{N}_{a+1}$. As applications of these two results, we get that if $1\nu2, f:\mathbb{N}_{a-1}\to\mathbb{R}, \nabla^\nu_{a^*}f(t)\geq 0$ for $t\in\mathbb{N}_{a+1}$ and $f(a)\geq f(a-1)$, then $f(t)$ is an increasing function for $t\in \mathbb{N}_{a-1}$. Conversely if $0\nu1, f:\mathbb{N}_{a-1}\to\mathbb{R}$ and $f$ is an increasing function for $t\in\mathbb{N}_{a}$, then $\nabla^\nu_{a^*}f(t)\geq 0$, for each $t\in\mathbb{N}_{a+1}$. We also give a counterexample to show that the above assumption $f(a)\geq f(a-1)$ in the last result is essential. These results demonstrate that, in some sense, the positivity of the $\nu$-th order Caputo fractional difference has a strong connection to the monotonicity of $f(t)$.

Détail

Classification : 39A12, 39A70
Keywords: Caputo fractional difference, monotonicity, Taylor monomial