On the nonlinear (k, ψ)-Hilfer fractional differential equations

In the current paper, we present the most generalized variant of the Hilfer derivative so-called (k, psi)-Hilfer fractional derivative operator. The (k, psi)-Riemann-Liouville and (k, psi)-Caputo fractional deriva-tives are obtained as a special case of ( k, psi)-Hilfer fractional derivative. We demonstrate a few proper-ties of (k, psi)-Riemann-Liouville fractional integral and derivative that expected to build up the calculus of (k, psi)-Hilfer fractional derivative operator. We present some significant outcomes about (k, psi)-Hilfer fractional derivative operator that require to derive the equivalent fractional integral equation to nonlin-ear (k, psi)-Hilfer fractional differential equation. We prove the existence and uniqueness for the solution of nonlinear (k, psi)-Hilfer fractional differential equation. In the conclusion section, we list the various k-fractional derivatives that are specific cases of (k, psi)-Hilfer fractional derivative. (C) 2021 Published by Elsevier Ltd.