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Oscillatory properties of certain nonlinear fractional nabla difference equations
Weinian Li,Weihong Sheng,Pingping Zhang
Keywords:oscillation, nonlinear fractional nabla difference equation, Riemann--Liouville fractional nabla difference operator
Abstract:
      In this paper, we investigate oscillation of the following nonlinear fractional nabla difference equations of the form$$ \left\{ \begin{array}{lll}\nabla(\nabla_a^{\alpha}x(t))+q(t)f(x(t))=g(t), \ \ t\in \mathbb{N}_a, \\nabla_a^{-(1-\alpha)}x(t)\big|_{t=a}=c, \end{array}\right.$$ where $\nabla f(t)=f(t)-f(t-1)$, $c$ and $\alpha $ are constants, $0<\alpha<1$, $\nabla_a^{\alpha}x$ is the Riemann--Liouville fractional nabla difference operator of order $\alpha$ of $x$, $a\geq 0$ is a real number, and $\mathbb{N}_a=\{a,a+1,a+2,\cdots\}$. Some oscillation criteria are established.