Hilfer proportional nonlocal fractional integro-multipoint boundary value problems

Abstract

In this article, we introduce and study a boundary value problem for (k, χ ¯ ∗) \left(k,{\bar{\chi }}_{∗ }) -Hilfer generalized proportional fractional differential equation of order in an interval (1, 2], equipped with integro-multipoint nonlocal boundary conditions. In the scalar case setting, the existence results are proved via Leray-Schauder nonlinear alternative and Krasnosel'skia's fixed point theorem, while the existence of a unique solution is established by applying Banach's contraction mapping principle. In Banach's space setting, an existence result is proved via Mönch's fixed point theorem and the measure of noncompactness. Finally, the obtained theoretical results are well illustrated by constructed examples.