Concept Of Fixed Point Research Articles

Bi-modal COVID-19 transmission with Caputo fractional derivative using statistical epidemic cases

The progression of the still ongoing COVID-19 epidemic must be studied in the world of differential operators other than those specified with integer-order temporal derivatives, according to ongoing scientific studies in the fields of fractional calculus, mathematical modeling, and epidemiology. Infectious diseases leave behind a historical footprint because of their long memory. With this in mind, the article below makes an effort to probe an epidemiological model using a Caputo differential operator with a singular kernel of power-law type. The ability of the Caputo operator to capture the evolution of complicated phenomena has been demonstrated in a number of studies, prompting us to conduct the analysis presented here. The analysis contains solid reasons while using the fractional operator for the COVID-19 epidemiological paradigm and presents the fixed point concept for the existence and uniqueness of its solutions. Hyers–Ulam–Rassias stability aids in finding model equilibrium, and the nonlinear least squares method yields the unknown parameters that also include the model’s fractional order. The actual cases of the infection support the superiority of the Caputo concept with evidence of smaller residuals. The numerical simulations are run to see how varying important parameters affect the disease’s spread.

Fractional order uncertain BAM neural networks with mixed time delays: An existence and Quasi-uniform stability analysis

This research investigates the presence of unique solutions and quasi-uniform stability for a class of fractional-order uncertain BAM neural networks utilizing the Banach fixed point concept, the contraction mapping principle, and analysis techniques. In order to guarantee the equilibrium point of fractional-order BAM neural networks with undetermined parameters, some new adequate criteria are devised, and both time delays result in quasi-uniform stability. The acquired results, which are simple to verify in practice, enhance and extend several earlier research works in some ways. Finally, two illustrative examples are provided to show the value of the suggested outcomes.

A fractal–fractional order model for exploring the dynamics of Monkeypox disease

This study explores the biological behaviour of the Monkeypox disease using a fractal–fractional operator. We discuss the existence and uniqueness of the solution of the model using the fixed-point concept. We further show that the Monkeypox fractal–fractional model is stable through the Hyers–Ulam and Hyers–Ulam Rassias stability criteria. The epidemiological threshold of the model is obtained. The numerical simulation for the proposed model is obtained using the Newton polynomial. For instance, the disease dies out at lower fractional values. We investigated the effects of some key parameters on the dynamics of the disease. The variation of the parameters shows that quarantine and isolation are effective approaches to managing, controlling, or eradicating the Monkeypox disease.

Iterative Approach for Solving Variational Inequality Problems using Fixed Point Concept

Iterative Approach for Solving Variational Inequality Problems using Fixed Point Concept

Stochastic dynamics of the fractal-fractional Ebola epidemic model combining a fear and environmental spreading mechanism

<abstract><p>Recent Ebola virus disease infections have been limited to human-to-human contact as well as the intricate linkages between the habitat, people and socioeconomic variables. The mechanisms of infection propagation can also occur as a consequence of variations in individual actions brought on by dread. This work studies the evolution of the Ebola virus disease by combining fear and environmental spread using a compartmental framework considering stochastic manipulation and a newly defined non-local fractal-fractional (F-F) derivative depending on the generalized Mittag-Leffler kernel. To determine the incidence of infection and person-to-person dissemination, we developed a fear-dependent interaction rate function. We begin by outlining several fundamental characteristics of the system, such as its fundamental reproducing value and equilibrium. Moreover, we examine the existence-uniqueness of non-negative solutions for the given randomized process. The ergodicity and stationary distribution of the infection are then demonstrated, along with the basic criteria for its eradication. Additionally, it has been studied how the suggested framework behaves under the F-F complexities of the Atangana-Baleanu derivative of fractional-order $ \rho $ and fractal-dimension $ \tau $. The developed scheme has also undergone phenomenological research in addition to the combination of nonlinear characterization by using the fixed point concept. The projected findings are demonstrated through numerical simulations. This research is anticipated to substantially increase the scientific underpinnings for understanding the patterns of infectious illnesses across the globe.</p></abstract>

Controllability and Observability Results of an Implicit Type Fractional Order Delay Dynamical System

Recently, several research articles have investigated the existence of solutions for dynamical systems with fractional order and their controllability. Nevertheless, very little attention has been given to the observability of such dynamical systems. In the present work, we explore the outcomes of controllability and observability regarding a differential system of fractional order with input delay. Laplace and inverse Laplace transforms, along with the Mittage–Leffler matrix function, are applied to the proposed dynamical system in Caputo’s sense, and a general solution is obtained in the form of an integral equation. Then, we set out conditions for the controllability of the underlying model, regarding the linear case. We then expound controllability conditions for the nonlinear case by utilizing the fixed point result of Schaefer and the Arzola–Ascoli theorem. Using the fixed point concept, we prove the observability of the linear case using the observability Grammian matrix. The necessary and sufficient conditions for the nonlinear case are investigated with the help of the Banach contraction mapping theorem. Finally, we add some examples to elaborate on our work.

Contributions of the fixed point technique to solve the 2D Volterra integral equations, Riemann\u2013Liouville fractional integrals, and Atangana\u2013Baleanu integral operators

In this manuscript, some fixed point results for generalized contractive type mappings under mild conditions in the setting of double controlled metric spaces (in short, eta _{gimel }^{nu }-metric spaces) are obtained. Moreover, some related consequences dealing with a common fixed point concept and nontrivial examples to support our results are presented. Ultimately, we use the theoretical results to discuss the existence and uniqueness of solutions of 2D Volterra integral equations, Riemann–Liouville integrals and Atangana–Baleanu integral operators are given.

Spectral Fixed Point Method for Nonlinear Oscillation Equation with Periodic Solution

Based on the fixed point concept in functional analysis, an improvement on the traditional spectral method is proposed for nonlinear oscillation equations with periodic solution. The key idea of this new approach (namely, the spectral fixed point method, SFPM) is to construct a contractive map to replace the nonlinear oscillation equation into a series of linear oscillation equations. Usually the series of linear oscillation equations can be solved relatively easily. Different from other existing numerical methods, such as the well-known Runge-Kutta method, SFPM can directly obtain the Fourier series solution of the nonlinear oscillation without resorting to the Fast Fourier Transform (FFT) algorithm. In the meanwhile, the steepest descent seeking algorithm is proposed in the framework of SFPM to improve the computational efficiency. Finally, some typical cases are investigated by SFPM and the comparison with the Runge-Kutta method shows that the present method is of high accuracy and efficiency.

Fixed Point Analytical Method for Nonlinear Differential Equations

Abstract In this paper, a new analytical method, which is based on the fixed point concept in functional analysis (namely, the fixed point analytical method (FPM)), is proposed to acquire the explicit analytical solutions of nonlinear differential equations. The key idea of this approach is to construct a contractive map which replaces the nonlinear differential equation into a series of linear differential equations. Usually, the series of linear equations can be solved relatively easily and have explicit analytical solutions. The FPM is different from all existing analytical methods, such as the well-known perturbation technique applied in weakly nonlinear problems, because it is independent of any small physical or artificial parameters at all; thus, it can handle more nonlinear problems, including strongly nonlinear ones. Two typical cases are investigated by FPM in detail and the comparison with the numerical results shows that the present method is one of high accuracy and efficiency.

Discrete Fixed Points: Models, Complexities, and Applications

We study three discrete fixed point concept (SPERNER, DPZP, BROUWER) under two different models: the polynomial-time function model and the oracle function model. We fully characterize the computational complexities of these three problems. The computational complexity unification of the above problems gives us more choices in the study of different applications. As an example, by a reduction from DPZP, we derive asymptotically equal lower and upper bound for TUCKER in the oracle model. The same reduction also allows us to derive a single proof for the PPAD-completeness of TUCKER in any constant dimension, which is significantly simpler than the recent proofs.

A New Approach to Breaking Down Chaotic Secure Communication

In this paper, we propose the time-varying stable fixed point concept to advance the generalized synchronization and present a general approach to calculating the value of chaotic system parameters according to some signal data. It provides a new approach to studying an unknown chaotic attractor and unmask the chaotic communication. We break down two chaotic communication schemes as examples with this approach.