Recently, coupled systems of fractional differential equations play a central role in the modelling of many systems in e.g., financial economics, ecology, and many more. This study investigates the existence and uniqueness of solutions for a nonlinear coupled system of fractional differential equations involving Riemann-Liouville and Caputo derivatives with coupled Riemann-Stieltjes integro-multipoint boundary conditions. The main tools are known fixed point theorems, namely, Leray-Schauder alternative, Banach fixed point theorem, and the Krasnoselskii fixed point theorem. The new system, which can be considered as a generalized version of many previous fascinating systems, is where the article's novelty lies. Examples are presented to illustrate the results. In this way, we generalize several earlier results.

The study of Ulam stability for (functional, differential, difference, integral, integro-differential, and fractional differential) equations heavily relies on inequalities. In such scientific and engineering research, fixed point theorems (FPTs) are essential instruments. This article, on one hand, focused on the $\beta$-Ulam-Hyers stability ( $\beta$-UHS) of non-instantaneous impulsive fractional integro-differential equations (N-IIFIDEs) involving the Caputo-Fabrizio fractional derivatives (C-FFDs) in a Banach space. On the other hand, we established the existence and uniqueness (E-UR) of solutions by employing the Banach contraction mapping principle (BCMP) and Krasnoselskii's fixed point theorem (KFPT). To validate the theoretical insights, a carefully crafted example was introduced. In this way, we generalized recent interesting results.

This article focuses on the Ulam-Hyers stability (U-HS) of impulsive neutral fractional integro-differential equations (INFIDEs) involving the Atangana-Baleanu-Caputo (ABC) and Caputo-Fabrizio (C-F) fractional derivatives (FDs) in a Banach space. The Banach contraction mapping principle (BCMP) and Krasnoselskii's fixed point theorem (KFPT) are used to prove the existence and uniqueness of solutions (E-US). To highlight the usefulness of the theoretical insights, carefully crafted examples are introduced, enhancing and building upon prior scholarly contributions.

We investigated Hyers-Ulam stability (H-US) of the following generalization of the Hosszú equation (HFE): $\mathrm{{\rm Y}}(x_1+x_2-\alpha x_1{x}_2)+\mathrm{{\rm Y}}(\alpha x_1{x}_2)=\mathrm{{\rm Y}}(x_1)+\mathrm{{\rm Y}}(x_2)$, in the class of maps $\mathrm{{\rm Y}}$ from a quadratically closed field $\mathbb{K}$ into a linear space, where $\alpha \in \mathbb{K}$ is fixed. We considered this stability in cases where the linear space is equipped with either the $m$-norm or the classical norm. In this way, we extended some earlier stability outcomes obtained for maps from the set of reals $\mathbb{R}$ into a Banach space. We also proved some auxiliary stability results for the Cauchy additive equation $\psi (x+y)=\psi (x)+\psi (y)$ in $m$-Banach spaces. Finally, we discussed the symmetry issues that can be observed in these results.