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

TL;DR

This paper investigates the existence and uniqueness of solutions for fractional q-difference equations involving the bi-ordinal Hilfer fractional q-derivative, extending previous derivatives and providing explicit solutions for linear cases.

Contribution

It introduces the bi-ordinal Hilfer fractional q-derivative and establishes existence and uniqueness results using fixed point theory, along with explicit solutions for linear problems.

Findings

Proved existence and uniqueness of solutions for nonlinear problems.

Established equivalence with Volterra q-integral equations.

Provided explicit solutions for linear fractional q-difference equations.

Abstract

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.