| Summary: | Abstract In this paper, sufficient conditions are established for the oscillation of solutions of q-fractional difference equations of the form { ∇ 0 α q x ( t ) + f 1 ( t , x ) = r ( t ) + f 2 ( t , x ) , t > 0 , lim t → 0 + q I 0 j − α x ( t ) = b j ( j = 1 , 2 , … , m ) , $$ \left \{ \textstyle\begin{array}{l} {}_{q}\nabla_{0}^{\alpha}x(t)+f_{1}(t,x)=r(t)+f_{2}(t,x), \quad t>0 ,\\ \lim_{t \to0^{+}}{{}_{q}I_{0}^{j-\alpha}x(t)}=b_{j} \quad(j=1,2,\ldots,m), \end{array}\displaystyle \right . $$ where m = ⌈ α ⌉ $m=\lceil\alpha\rceil$ , ∇ 0 α q ${}_{q}\nabla_{0}^{\alpha}$ is the Riemann-Liouville q-differential operator and I 0 m − α q ${}_{q}I_{0}^{m-\alpha}$ is the q-fractional integral. The results are also obtained when the Riemann-Liouville q-differential operator is replaced by Caputo q-fractional difference. Examples are provided to demonstrate the effectiveness of the main result.
|
|---|