On the Boundary Value Problems of {\Psi} -Hilfer Fractional Differential Equations

In the current paper, we derive the comparison results for the homogeneous and non-homogeneous linear initial value problem (IVP) for $\Psi$-Hilfer fractional differential equations. In the presence of upper and lower solutions, the obtained comparison results and the location of roots theorem utilized to prove the existence and uniqueness of the solution for the linear $\Psi$-Hilfer boundary value problem (BVP) through the linear non-homogeneous $\Psi$-Hilfer IVP. Assuming the existence of lower solution $w_0$ and upper solution $z_0$, we establish the existence of minimal and maximal solutions for the nonlinear $\Psi$-Hilfer BVP in the line segment $[w_0,\,z_0]$ of the weighted space $C_{1-\,\gamma ;\, \Psi }\left( J,\,\R\right)$. Further, it demonstrated that the iterative Picard type sequences that began with lower and upper solutions respectively converges to a minimal and maximal solutions, and that started with any point on a line segment converge to the exact solution of nonlinear $\Psi$-Hilfer BVP. Finally, an example is provided in support of the main results we acquired.