QUALITATIVE ANALYSIS OF A PROPORTIONAL CAPUTO FRACTIONAL PANTOGRAPH DIFFERENTIAL EQUATION WITH MIXED NONLOCAL CONDITIONS

Abstract

In this paper, we investigate existence, uniqueness and four different types of Ulam’s stability, that is, Ulam-Hyers stability, generalized Ulam-Hyers stability, Ulam-Hyers- Rassias stability and generalized Ulam-Hyers-Rassias stability of the solution for a class of nonlinear fractional Pantograph differential equation in term of a proportional Caputo fractional derivative with mixed nonlocal conditions. We construct sufficient conditions for the existence and uniqueness of solutions by utilizing well-known classical fixed point theorems such as Banach contraction principle, Leray-Schauder nonlinear alternative and Krasnosel’ski ̆i’s fixed point theorem. Finally, two examples are also given to point out the applicability of our main results.