Some Results for Generalized Proportional Fractional Integro‐Differential Equations With Multi‐Point Boundary Conditions

ABSTRACT In this article, we investigate some new results about the existence, uniqueness and different types of Ulam stability of the solutions for a class of integro‐differential equations involving Generalized Proportional Fractional (GPF) derivative of Reimann–Liouville‐type with m‐point boundary conditions utilizing fixed point theorems with the help of the lower regularized incomplete gamma function and the maximum value of the integral of the Green's function. In addition, by using inequality techniques, we establish new Lyapunov‐type and Hartman–Wintner‐type inequalities for m‐point boundary value problems with GPF derivative that generalizes a recently‐obtained result. Moreover, a sharper lower bound of eigenvalues for a Sturm–Liouville eigenvalue problem is obtained. In this respect, we improve some previous results for Mittag–Leffler functions. Finally, we apply the obtained results on a Volterra integro‐differential equation involving GPF derivative and GPF integral, and then to Mittag–Leffler functions.