In this article, we prove some new uniqueness and Ulam-Hyers stability results of a nonlinear generalized fractional integro-differential equation in the frame of Caputo derivative involving a new kernel in terms of another function . Our approach is based on Babenko’s technique, Banach’s fixed point theorem, and Banach’s space of absolutely continuous functions. The obtained results are demonstrated by constructing numerical examples.

In this research work, we study two types of fractional boundary value problems for multi-term Langevin systems with generalized Caputo fractional operators of different orders. The existence and uniqueness results are acquired by applying Sadovskii’s and Banach’s fixed point theorems, whereas the guarantee of the existence of solutions is shown by Ulam–Hyer’s stability. Our reported results cover many outcomes as special cases. An example is provided to illustrate and validate our results.

In this paper, we consider a one-dimensional linear Bresse system with only one infinite memory acting in the second equation (the shear angle equation) of the system. We prove that the asymptotic stability of the system holds under some general condition imposed into the relaxation function, precisely, $ g^{\prime}(t)\le -\xi(t) G(g(t)). $ The proof is based on the multiplier method and makes use of convex functions and some inequalities. More specifically, we remove the constraint imposed on the boundedness condition on the initial data $ \eta{0x} $. This study generalizes and improves previous literature outcomes.

In this investigation, weighted $psi$-Caputo fractional derivatives are applied to analyze the solution of fractional pantograph problems with boundary conditions. We establish the existence of solutions to the indicated pantograph equations as well as their uniqueness. The study also takes into account the situation where $\psi \left(\mathsf{x}\right)=\mathsf{x}$. With the aid of Banach’s and Krasnoselskii’s classic fixed point results, we have established a the qualitative study. Different values of $\psi \left(\mathsf{x}\right)$ and $\mathsf{w}\left(\mathsf{x}\right)$ are discussed as special cases that are relevant to our current results. Additionally, in light of our findings, we provide a more general fractional system with the weighted $\psi$-Caputo-type that takes into account both the new problems and certain previously existing, related problems. Finally, we give two illustrations to support and validate the outcomes.