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| Oscillation Theory of $h$-fractional Difference Equations |
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| KeyWord:$h$-deference equations Oscillation Fractional |
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| Abstract: |
| In this paper, we initiate the oscillation theory for $h$-fractional difference equations of the form
\begin{equation*}
\begin{cases}
_{a}\Delta^{\alpha}_{h}x(t)+r(t)x(t)=e(t)+f(t,x(t)),\ \ \ t\in\mathbb{T}_{h}^{a},\ \ 1<\alpha<2,\x(a)=c_{0},\ \ \Delta_{h}x(a)=c_{1},\ \ \ c_{0}, c_{1}\in\mathbb{R},
\end{cases}
\end{equation*}
where $_{a}\Delta^{\alpha}_{h}$ is the Riemann-Liouville $h$-fractional difference of order $\alpha,$ $\mathbb{T}_{h}^{a}:=\{a+kh, k\in\mathbb{Z^{+}}\cup\{0\}\},$ and $a\geqslant0,$ $h>0.$
We study the oscillation of $h$-fractional difference equations
with Riemann-Liouville derivative, and obtain some sufficient
conditions for oscillation of every solution. Finally, we give an
example to illustrate our main results. |
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