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

Summary: 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