Abstract

In the current paper, we present the most generalized variant of the Hilfer derivative so-called (k, Ψ)-Hilfer fractional derivative operator. The (k, Ψ)-Riemann-Liouville and (k, Ψ)-Caputo fractional derivatives are obtained as special case of (k, Ψ)-Hilfer fractional derivative. We demonstrate a few properties of (k, Ψ)-Riemann-Liouville fractional integral and derivative that expected to build up the calculus of (k, Ψ)-Hilfer fractional derivative operator. We present some significant outcomes about (k, Ψ)-Hilfer fractional derivative operator that require to derive the equivalent fractional integral equation to nonlinear (k, Ψ)-Hilfer fractional differential equation. We prove the existence and uniqueness for the solution of nonlinear (k, Ψ)-Hilfer fractional differential equation. In the conclusion section, we list the various k-fractional derivatives that are specific cases of (k, Ψ)-Hilfer fractional derivative.