Electron. J. Diff. Equ., Vol. 2015 (2015), No. 163, pp. 1-7.

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

Baoguo Jia, Lynn Erbe, Allan Peterson

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 $t\in\mathbb{N}_a$. Conversely, if $N-1<\nu<N$, $f:\mathbb{N}_{a-N+1}\to\mathbb{R}$, 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 $1<\nu<2$, $f:\mathbb{N}_{a-1}\to\mathbb{R}$, $\nabla^\nu_{a^*}f(t)\geq 0$ 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 $0<\nu<1$, $f:\mathbb{N}_{a-1}\to\mathbb{R}$ 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 $f(t)$.

Submitted May 30, 2015. Published June 17, 2015.
Math Subject Classifications: 39A12, 39A70.
Key Words: Caputo fractional difference; monotonicity; Taylor monomial.

Show me the PDF file (185 KB), TEX file, and other files for this article.

Baoguo Jia
School of Mathematics and Computer Science
Sun Yat-Sen University
Guangzhou 510275, China
email: mcsjbg@mail.sysu.edu.cn
Lynn Erbe
Department of Mathematics
University of Nebraska-Lincoln
Lincoln, NE 68588-0130, USA
email: lerbe2@math.unl.edu
Allan Peterson
Department of Mathematics
University of Nebraska-Lincoln
Lincoln, NE 68588-0130, USA
email: apeterson1@math.unl.edu

Return to the EJDE web page