This manuscript investigates the existence, uniqueness, and different forms of Ulam stability for a system of three coupled differential equations involving the Riemann–Liouville (RL) fractional operator. The Leray–Schauder alternative is employed to confirm the existence of solutions, while the Banach contraction principle is used to establish their uniqueness. Stability conditions are derived utilizing classical nonlinear functional analysis techniques. Theoretical findings are illustrated with an example. The proposed system generalizes third-order ordinary differential equations (ODEs) with different boundary conditions (BCs).

This paper investigates the dynamics of the SIR infectious disease model, with a specific emphasis on utilizing a harmonic mean-type incidence rate. It thoroughly analyzes the model’s equilibrium points, computes the basic reproductive rate, and evaluates the stability of the model at disease-free and endemic equilibrium states, both locally and globally. Additionally, sensitivity analysis is carried out. A sophisticated stability theory, primarily focusing on the characteristics of the Volterra–Lyapunov (V-L) matrices, is developed to examine the overall trajectory of the model globally. In addition to that, we describe the transmission of infectious disease through a mathematical model using fractal-fractional differential operators. We prove the existence and uniqueness of solutions in the SIR model framework with a harmonic mean-type incidence rate by using the Banach contraction approach. Functional analysis is used together with the Ulam–Hyers (UH) stability approach to perform stability analysis. We simulate the numerical results by using a computational scheme with the help of MATLAB. This study advances our knowledge of the dynamics of epidemic dissemination and facilitates the development of disease prevention and mitigation tactics.

This article examines the commutativity of rings with antiautomorphisms, specifically when they are equipped with derivations that satisfy certain algebraic identities. Moreover, we present examples to demonstrate the necessity of the various restrictions imposed in the hypotheses of our theorems.

Let ℧ be a prime ring of char(℧) $\ne 2$ with its center $Z.$ This article introduces new classes of endomorphisms and investigates how they relate to antiautomorphisms of prime rings and the commutativity of prime rings. Additionally, we fully describe and classify some of these endomorphisms. We also give examples to prove the necessity of the numerous restrictions included in the hypotheses of our results.