Existence of mild solutions for a class of $(\rho ,k,\phi )$⁃proportional Hilfer fractional evolution equation

The existence， uniqueness and continuous dependence of solutions for a class of $(\rho ,k,\phi )$⁃proportional Hilfer fractional Cauchy problems were investigated. The probability density function， properties of the $(\rho ,k,\phi )$⁃proportional Hilfer fractional derivative and semigroup theory were utilized to define mild solutions. A proper weighted space was introduced， and within this space， the Banach contraction principle was applied to discuss the uniqueness of the solutions. The continuous dependence of the data on the Cauchy problem was proven by constructing a generalized Gronwall inequality.