The concept of fixed points serves as an effective and essential tool in analyzing nonlinear phenomena. This study investigates the existence and uniqueness of solutions for a class of Basset-type fractional differential equations with boundary conditions involving the Caputo–Fabrizio fractional derivative. These equations emerge from the generalized Basset force describing the motion of a sphere settling in a viscous fluid. Darbo’s fixed point theorem, combined with the measure of noncompactness, is applied to establish the existence of solutions. Uniqueness is ensured via Banach’s fixed point theorem. Additionally, stability analysis is performed using Ulam–Hyers and Ulam–Hyers–Rassias concepts. An illustrative example, supported by tables and figures, demonstrates the applicability of the theoretical results.

The research is focused on establishing the existence and uniqueness of solutions for a specific set of Caputo–Fabrizio fractional differential equations under periodic boundary conditions (PBCs). To achieve this, the study employs the fractional derivative within the Caputo–Fabrizio framework and utilizes the proposed variation of the parameter method. This method is utilized to simplify the second-order fractional differential equation, transforming it into a second-order nonlinear ordinary differential equation. The research findings are rooted in the fixed points of the Schauder fixed point and Banach fixed point theorems. Furthermore, the study delves into stability analysis using the Hyers–Ulam stability concept. Theoretical examples are included to illustrate and demonstrate the implications of the established theorems.

This manuscript aims to study the effectiveness of two different levels of fractional orders in the frame of Caputo-Hadamard (ℂℍ)-derivatives on a special type class of delay problem supplemented by Dirichlet boundary conditions. The corresponding Hadamard fractional integral equation is derived for a proposed ℂℍ-fractional pantograph system. The Banach, Schaefer, and Krasnoselskii fixed point theorems (𝔽ℙ𝕋𝑠), are used to investigate sufficient conditions of the existence and uniqueness theorems for the proposed system. Furthermore, the Green functions properties are investigated and used to discuss the Ulam-Hyers (𝕌ℍ)stability and its generalized by utilizing nonlinear analysis topics. Finally, three mathematical examples are provided with numerical results and figures by using Matlab software to illustrate the validity of our findings.

The nonlinear fractional differential equation (FDE) is discussed in this study. First, we will investigate the existence and uniqueness solution of the nonlinear differential equation to the Atangana-Baleanu fractional derivative in the sense of Caputo with the initial periodic condition and integral boundary condition by Krasnoselskii’s and Banach fixed point theorems. Then, we will study the Hyers-Ulam stability of our problem. Finally, we presented an example to demonstrate the use of our main theorems.