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Asymptotic Behavior of Nabla Half Order\\$h$-Difference Equations
Keywords:Laplace Transformation; Mittag-Leffler function; Riemann-Liouville fractional $h$-difference
Abstract:
      Consider the nabla fractional difference equation \begin{equation}\label{al100} _{\rho(a)}\nabla^{0.5}_{h}x(t)=cx(t),\quad x(a)=A>0,\quad t\in(h\N)_{a+h}. \end{equation} where $_{\rho(a)}\nabla^{0.5}_hx(t)$ denotes Riemann-Liouville nabla half order $h$-difference of $x(t)$ on sets $(h\N)_{a}$. In this paper, we will discuss the asymptotic behaviors of the solutions of \eqref{al100}. The following Theorem is obtained. {\bf{Theorem A.}} Assume $c>0$, $ch^{0.5}\neq 1$. Then for all $A\ne 0$ the unique solution of the fractional initial value problem \eqref{al100} satisfies\(i) When $0<c<\frac{1}{h^{0.5}}$, $$\lim_{n\rightarrow\infty}x(a+nh)=+\infty.$$ (ii) When $\frac{1}{h^{0.5}}<c<\Big(\frac{2}{h}\Big)^{0.5}$, $$\limsup_{n\rightarrow\infty}x(a+nh)=+\infty,\quad\liminf_{n\rightarrow\infty}x(a+nh)=-\infty$$ (iii) When $c=\Big(\frac{2}{h}\Big)^{0.5}$, $$\limsup_{n\rightarrow\infty}x(a+nh)=2\sqrt{2}(\sqrt{2}-1)A,\quad\liminf_{ n\rightarrow\infty } x(a+nh)=-2\sqrt{2}(\sqrt{2}-1)A.$$ (iv) When $c>\Big(\frac{2}{h}\Big)^{0.5}$, $$\lim_{n\rightarrow\infty}x(a+nh)=0.$$ As an application, we get that\ {\bf{Corollary B.}} Assume $\frac{1}{h^{0.5}}<c\leq \Big(\frac{2}{h}\Big)^{0.5}$. Then every solution $x(t)$ of fractional difference equation $$_{\rho(a)}\nabla^{0.5}x(t)=cx(t), \quad t\in (h\N)_{a+h}$$ is oscillatory.