On generalized \((k,\psi )\)-Hilfer proportional fractional operator and its applications to the higher-order Cauchy problem

In this work, we introduce a novel idea of generalized $(k,\psi )$ $({{k}},\psi )$ -Hilfer proportional fractional operators. The proposed operator combines the $(k,\psi )$ $({{k}},\psi )$ -Riemann–Liouville and $(k,\psi )$ $({{k}},\psi )$ -Caputo proportional fractional operators. Some properties and auxiliary results of the proposed operators are investigated. The ψ-Laplace transform and its properties of the proposed operators are established and utilized to solve Cauchy-type problems. Furthermore, the uniqueness result for a higher-order initial value problem under $(k,\psi )$ $({{k}},\psi )$ -Hilfer proportional fractional operators is proved by using Picard’s iterative technique. At the end, examples are provided to present the theoretical results. This new type of proposed operator can help other researchers who are still working on real-world problems.