Existence and uniqueness of some Cauchy Type Problems in fractional q-difference calculus

TL;DR

This paper establishes conditions for the existence and uniqueness of solutions to certain nonlinear Cauchy type problems involving fractional q-derivatives, introducing new equivalences and operators in fractional q-difference calculus.

Contribution

It introduces a new equivalence between Cauchy type q-fractional problems and Volterra q-integral equations, and defines a q-analogue of the Hilfer fractional derivative with related results.

Findings

Proved existence and uniqueness conditions for q-fractional problems.

Established equivalence between q-fractional differential equations and integral equations.

Provided concrete examples illustrating the solutions.

Abstract

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.