The study investigates a system which uses two kinds of hybrid fractional differential equations (HFDEs) that connect through the application of the \(\psi\) -Caputo fractional derivative (CFD). The system which includes impulse effects and non-local initial conditions presents a mathematical challenge which scientists can use to study actual systems that exhibit memory and sudden changes in behavior. The researchers established solution existence and uniqueness through their method which transformed the original problem into a fractional integral formulation. The analysis conducted their analyses employing two conventional fixed-point approaches. The analysis established specific requirements for the nonlinear components and impulse elements and non-local conditions to make their methods operational. The research team used practical examples to prove their theoretical results and show how their method works. The findings establish new results about the qualitative behavior of \(\psi\) -Caputo fractional systems which contain impulse effects and build on previous research conducted in this area.
