Implicit fractional differential equations with Φ–Caputo fractional derivative in Banach spaces

Abstract

In the present paper, we consider a class of initial value problem for nonlinear implicit fractional differential equations involving the Φ–Caputo fractional derivative in Banach spaces. A new minimal condition on the parameters and the relationship between them is used to show the existence, uniqueness, and Mittag-Leffler–Ulam–Hyers stability of solutions by using the Mönch’s fixed point theorem combined with a classical technique for measures of noncompactness. Moreover, the Mittag-Leffler–Ulam–Hyers stability of proposed problem is discussed. Finally, two examples are presented to illustrate the results.