In this paper, motivated by Hilfer and Hilfer-Katugampola fractional
derivatives, we introduce new Hilfer-generalized proportional
derivatives which interpolate the classical fractional derivatives of
Hilfer, Riemann-Liouville, Caputo and generalized proportional
fractional derivatives. We also present some important properties of the
proposed derivatives. Furthermore, as an application, we show that this
equation is equivalent to the Volterra integral equation and prove the
existence, uniqueness of the solution to the Cauchy problem with the
nonlocal initial condition. Finally, two examples were given to
illustrate the results.