To numerical explore a fractional implicit Q-differential equations with Hilfer type and via nonlocal conditions

This paper tries to show that there is only one solution for problem of fractional q-differential equations with  Hilfer type, and it does so by using a particular method known as Schaefer's fixed point theorem  and the Banach contraction principle. After that, we create a integral  type of the  problem for nonlocal condition. Next, we show that Ulam stability is true. The Gr"{o}wnwall rule for singular kernels of the equations helps to show our findings are correct. We confirm our findings by giving a few practical examples.

Keywords:

implicit q-differential equation , Hilfer fractional q-derivative , nonlinear analysis , existence , Ulam stability