Generalized Fractional ( ρ , k , φ )‐Proportional Hilfer Derivatives and Some Properties

Building on previous work in fractional calculus, this paper introduces new definitions for the ( ρ , k , φ )‐proportional integral and ( ρ , k , φ )‐proportional H fractional derivative. This new approach retains the semigroup properties of traditional fractional integrals. A significant advantage of this fractional calculus is its compatibility with the majority of existing studies on fractional differential equations. Furthermore, we delve into the properties of the generalized fractional integrals and derivatives. We discuss, for instance, the mapping properties of the ( ρ , k , φ )‐proportional integral. To elucidate these concepts, we introduce a set of new weighted spaces. Additionally, we explore the generalized Laplace transform of both the ( ρ , k , φ )‐proportional integrals and ( ρ , k , φ )‐proportional H fractional derivatives. Also, examples concerning the linear ( ρ , k , φ )‐proportional H fractional equations are given to illustrate the main results.