QUALITATIVE ANALYSIS OF A PROPORTIONAL CAPUTO FRACTIONAL PANTOGRAPH DIFFERENTIAL EQUATION WITH MIXED NONLOCAL CONDITIONS
Bounmy Khaminsou, Chatthai Thaiprayoon, Weerawat Sudsutad, Sayooj Aby Jose
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.