Mild Solutions of the (ρ₁,ρ₂,k₁,k₂,φ)-Proportional Hilfer–Cauchy Problem

Inspired by prior research on fractional calculus, we introduce new fractional integral and derivative operators: the $({\rho }_1,{\rho }_2,k_1,k_2,\phi )$-proportional integral and the $({\rho }_1,{\rho }_2,k_1,k_2,\phi )$-proportional Hilfer fractional derivative. Numerous previous studied fractional integrals and derivatives can be considered as particular instances of the novel operators introduced above. Some properties of the $({\rho }_1,{\rho }_2,k_1,k_2,\phi )$-proportional integral are discussed, including mapping properties, the generalized Laplace transform of the $({\rho }_1,{\rho }_2,k_1,k_2,\phi )$-proportional integral and $({\rho }_1,{\rho }_2,k_1,k_2,\phi )$-proportional Hilfer fractional derivative. The results obtained suggest that the most comprehensive formulation of this fractional calculus has been achieved. Under the guidance of the findings from earlier sections, we investigate the existence of mild solutions for the $({\rho }_1,{\rho }_2,k_1,k_2,\phi )$-proportional Hilfer fractional Cauchy problem. An illustrative example is provided to demonstrate the main results.