On existence of solutions for \(\Psi \)-Hilfer type fractional BVP of generalized higher order

This research examines generalized boundary value problems of the \(\Psi \) $\mathrm{\Psi }$ -Hilfer fractional differential equations of order \(2<\alpha <3\) $2<\alpha <3$ and \(n-1< \alpha <n\) $n-1<\alpha <n$ \((n\ge 4)\) $(n\ge 4)$ , type \(\beta \in [0,1]\) $\beta \in [0,1]$ under two different initial conditions. Through equivalent fractional integral equations, the existence and uniqueness of a solution have been demonstrated. This study relies on a variety of fractional calculus tools as well as nonlinear theorems due to Leray–Schauder and Schaefer. To conclude, examples are provided to illustrate how the key results might be justified.