<abstract><p>In this manuscript, we analyze the existence, uniqueness and Ulam's stability for Caputo proportional fractional integro-differential equation involving mixed nonlocal conditions with respect to another function. The uniqueness result is proved via Banach's fixed point theorem and the existence results are established by using the Leray-Schauder nonlinear alternative and Krasnoselskii's fixed point theorem. Furthermore, by using the nonlinear analysis techniques, we investigate appropriate conditions and results to study various different types of Ulam's stability. In addition, numerical examples are also constructed to demonstrate the application of the main results.</p></abstract>

Results reported in this paper study the existence and stability of a class of implicit generalized proportional fractional integro-differential Langevin equations with nonlocal fractional integral conditions. The main theorems are proved with the help of Banach’s, Krasnoselskii’s, and Schaefer’s fixed point theorems and Ulam’s approach. Finally, an example is given to demonstrate the applicability of our theoretical findings.

This manuscript investigates an extended boundary value problem for a fractional pantograph differential equation with instantaneous impulses under the Caputo proportional fractional derivative with respect to another function. The solution of the proposed problem is obtained using Mittag–Leffler functions. The existence and uniqueness results of the proposed problem are established by combining the well-known fixed point theorems of Banach and Krasnoselskii with nonlinear functional techniques. In addition, numerical examples are presented to demonstrate our theoretical analysis.

In this paper, we propose a generalized Gronwall inequality in the context of the ψ-Hilfer proportional fractional derivative. Using Picard’s successive approximation and the definition of Mittag–Leffler functions, we construct the representation formula of the solution for the ψ-Hilfer proportional fractional differential equation with constant coefficient in the form of the Mittag–Leffler kernel. The uniqueness result is proved by using Banach’s fixed-point theorem with some properties of the Mittag–Leffler kernel. Additionally, Ulam–Hyers–Mittag–Leffler stability results are analyzed. Finally, numerical examples are provided to demonstrate the theory’s application.