Fractional Cauchy problems associated with the bi-ordinal Hilfer fractional $q$-derivative

To study the existence and uniqueness of solutions to Cauchy-type problems for fractional q-difference equations with the bi-ordinal Hilfer fractional q-derivative which is an extension of the Hilfer fractional q-derivative. An approach is based on the equivalence of the nonlinear Cauchy-type problem with a nonlinear Volterra q-integral equation of the second kind. Applying an analog of Banach fixed point theorem we prove the uniqueness and the existence of the solution. Moreover, we present an explicit solution to the q-analog of the Cauchy problem for the linear case.