The main goal of this study is to demonstrate an existence result of a coupled implicit Riemann-Liouville fractional integral equation. First, we prove a new fixed point theorem in spaces with an extended norm structure. That theorem generalized Darbo’s theorem associated with the vector Kuratowski measure of noncompactness. Second, we employ our obtained fixed point theorem to investigate the existence of solutions to the coupled implicit fractional integral equation on the generalized Banach space $\mathcal{C}\left(\right[0,1],\mathbb{R})\times \mathcal{C}\left(\right[0,1],\mathbb{R})$.

This paper is devoted to studying the existence and uniqueness of a system of coupled fractional differential equations involving a Riemann–Liouville derivative in the Cartesian product of fractional Sobolev spaces $\mathcal{E}=W_{a^{+}}^{{\gamma }_1,1}(a,b)\times W_{a^{+}}^{{\gamma }_2,1}(a,b)$. Our strategy is to endow the space $\mathcal{E}$ with a vector-valued norm and apply the Perov fixed point theorem. An example is given to show the usefulness of our main results.

Smoking is associated with various detrimental health conditions, including cancer, heart disease, stroke, lung illnesses, diabetes, and fatal diseases. Motivated by the application of fractional calculus in epidemiological modeling and the exploration of memory and nonlocal effects, this paper introduces a mathematical model that captures the dynamics of relapse in a smoking cessation context and presents the dynamic behavior of the proposed model utilizing Caputo fractional derivatives. The model incorporates four compartments representing potential, persistent (heavy), temporally recovered, and permanently recovered smokers. The basic reproduction number $R_0$ is computed, and the local and global dynamic behaviors of the free equilibrium smoking point (${\mathcal{Y}}_0$) and the smoking-present equilibrium point (${\mathcal{Y}}^{*}$) are analyzed. It is demonstrated that the free equilibrium smoking point (${\mathcal{Y}}_0$) exhibits global asymptotic stability when $R_0\le 1$, while the smoking-present equilibrium point (${\mathcal{Y}}^{*}$) is globally asymptotically stable when $R_0>1$. Additionally, analytical results are validated through a numerical simulation using the predictor–corrector PECE method for fractional differential equations in Matlab software.

The propose of this manuscript is to study some user-friendly versions of fixed point theorem of Schauder and Krasnoselskii for 2 × 2 block operator matrix on product of two Banach spaces furnished with its norm and weak topology, in the setting that the input operators are not necessarily (weakly) compact and (weakly) continuous, by using the technique of Measure of (Weak) Noncompactness.

The severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) is responsible for coronavirus disease-19 (COVID-19). This virus has caused a global pandemic, marked by several mutations leading to multiple waves of infection. This paper proposes a comprehensive and integrative mathematical approach to the third wave of COVID-19 (Omicron) in the Kingdom of Saudi Arabia (KSA) for the period between 16 December 2022 and 8 February 2023. It may help to implement a better response in the next waves. For this purpose, in this article, we generate a new mathematical transmission model for coronavirus, particularly during the third wave in the KSA caused by the Omicron variant, factoring in the impact of vaccination. We developed this model using a fractal-fractional derivative approach. It categorizes the total population into six segments: susceptible, vaccinated, exposed, asymptomatic infected, symptomatic infected, and recovered individuals. The conventional least-squares method is used for estimating the model parameters. The Perov fixed point theorem is utilized to demonstrate the solution’s uniqueness and existence. Moreover, we investigate the Ulam–Hyers stability of this fractal–fractional model. Our numerical approach involves a two-step Newton polynomial approximation. We present simulation results that vary according to the fractional orders ($\gamma$) and fractal dimensions ($\theta$), providing detailed analysis and discussion. Our graphical analysis shows that the fractal-fractional derivative model offers more biologically realistic results than traditional integer-order and other fractional models.