Mild solution for \((\rho ,k,\Psi )\)-proportional Hilfer fractional Cauchy problem

Hilfer fractional derivative is an important and interesting operator in fractional calculus, and it can be applicable in pure theories and other fields. It yields to other notable definitions, Ψ-Hilfer, $(k,\mathrm{\Psi })$ $(k,\Psi )$ -Hilfer derivatives, etc. Motivated by the concepts of the proportional fractional derivative and $(k,\mathrm{\Psi })$ $(k,\Psi )$ -Hilfer fractional derivative, we first introduce new definitions of integral and derivative, termed the $(\rho ,k,\mathrm{\Psi })$ $(\rho ,k,\Psi )$ -proportional integral and $(\rho ,k,\mathrm{\Psi })$ $(\rho ,k,\Psi )$ -proportional Hilfer fractional derivative. This type of fractional derivative is advantageous as it aligns with earlier studies on fractional differential equations. Additionally, we present a more generalized version of the $(\rho ,\alpha ,\beta ,k,r)$ $(\rho ,\alpha ,\beta ,k,r)$ -resolvent family, followed by an exploration of its properties. By analyzing the generalized resolvent family, we examine the existence of mild solutions to the $(\rho ,k,\mathrm{\Psi })$ $(\rho ,k,\Psi )$ -proportional Hilfer fractional Cauchy problem, supported by an illustrative example to show the main result.