EXISTENCE AND UNIQUENESS OF SOME CAUCHY TYPE PROBLEMS IN FRACTIONAL q-DIFFERENCE CALCULUS

Abstract

In this paper we derive a sucient 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 dierential equations.
The key technique is to rst prove that this Cauchy type q-fractional problem is
equivalent to a corresponding Volterra q-integral equation. Moreover, we dene
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.