Two asymptotic results of solutions for nabla fractional $(q,h)$-difference equations

Authors: FEIFEI DU, LYNN ERBE, BAOGUO JIA, ALLAN PETERSON

Abstract: In this paper we study the Caputo and Riemann--Liouville nabla $(q,h)$-fractional difference equation and obtain the following two main results: Assume $0<\alpha<1$ and there is a constant $b$ such that $c(t)\leq b<0,$ for $t\in\tilde{\mathbb{T}}_{(q,h)}^{\sigma(a)}.$ Then any solution, $x(t),$ of the nabla Caputo $(q,h)$-fractional difference equation \begin{equation}\label{bianxishu11} ^{C}_{a}\nabla^{\alpha}_{(q,h)}x(t)=c(t)x(t),\quad\quad t\in\tilde{\mathbb{T}}_{(q,h)}^{\sigma(a)} \end{equation} with $x(a)>0$ satisfies $$\lim_{t\rightarrow\infty}x(t)=0.$$ Assume $0<\alpha< 1$ , $c(t)\leq0$, $ t\in\tilde{\mathbb{T}}^{\sigma^{2}(a)}_{(q,h)}$, and $x(t)$ is a solution of the equation \begin{equation}\label{bianxishu22} _{a}\nabla^{\alpha}_{(q,h)}x(t)=c(t)x(t),\quad \quad t\in\tilde{\mathbb{T}}^{\sigma^{2}(a)}_{(q,h)}, \end{equation} satisfying $x(\sigma(a))>0$. Then $x(t)>0$, $t\in\tilde{\mathbb{T}}^{\sigma(a)}_{(q,h)}$ and $$\lim_{t\rightarrow\infty}x(t)=0.$$ Theorem A and Theorem B extend the results in other recent works of the authors.

Keywords: Nabla fractional difference, $(q,h)$-calculus, monotonicity, asymptotic behavior

Full Text: PDF