  In this paper we derive a sufficient condition for the existence of a unique
  solution of a Cauchy type q-fractional problem (involving the fractional
q-derivative of Riemann-Liouville type)
  for some nonlinear differential equations. The key technique is to first
prove that this Cauchy type q-fractional problem is equivalent to a
corresponding Volterra q-integral equation. Moreover, we define the
$q$-analogue of the Hilfer fractional derivative or composite fractional
derivative operator and prove some similar new equivalence, existence and
uniqueness results as above. Finally, some examples are presented to illustrate
our main results in cases where we can even give concrete formulas for these
unique solutions.
