Fixed Point Theorem Research Articles

Chromatic fixed point theory and the Balmer spectrum for extraspecial 2-groups

abstract: In the early 1940s, P. A. Smith showed that if a finite $p$-group $G$ acts on a finite dimensional complex $X$ that is mod $p$ acyclic, then its space of fixed points, $X^G$, will also be mod $p$ acyclic. In their recent study of the Balmer spectrum of equivariant stable homotopy theory, Balmer and Sanders were led to study a question that can be shown to be equivalent to the following: if a $G$-space $X$ is a equivariant homotopy retract of the $p$-localization of a based finite $G$-C.W. complex, given $H<G$ and $n$, what is the smallest $r$ such that if $X^H$ is acyclic in the $(n+r)$th Morava $K$-theory, then $X^G$ must be acyclic in the $n$th Morava $K$-theory? Barthel et.~al. then answered this when $G$ is abelian, by finding general lower and upper bounds for these ``blue shift'' numbers which agree in the abelian case. In our paper, we first prove that these potential chromatic versions of Smith's theorem are equivalent to chromatic versions of a 1952 theorem of E. E. Floyd, which replaces acyclicity by bounds on dimensions of mod $p$ homology, and thus applies to all finite dimensional $G$-spaces. This unlocks new techniques and applications in chromatic fixed point theory. Applied to the problem of understanding blue shift numbers, we are able to use classic constructions and representation theory to search for lower bounds. We give a simple new proof of the known lower bound theorem, and then get the first results about nonabelian 2-groups that do not follow from previously known results. In particular, we are able to determine all blue shift numbers for extraspecial 2-groups. Applied in ways analogous to Smith's original applications, we prove new fixed point theorems for $K(n)_*$-homology disks and spheres. Finally, our methods offer a new way of using equivariant results to show the collapsing of certain Atiyah-Hirzebruch spectral sequences in certain cases. Our criterion appears to apply to the calculation of all 2-primary Morava $K$-theories of all real Grassmanians.

A new identification of Lagrange multipliers to study solutions of nonlinear Caputo–Fabrizio fractional problems

This research aims to provide a new identification of Lagrange multipliers using a novel hybrid method based on the Khalouta transform method and the variational iteration method and to study the solutions of nonlinear fractional problems. This method is called Khalouta variational iteration method (KHVIM). The fractional derivative is considered in Caputo–Fabrizio sense. The uniqueness and convergence results are investigated using Banach fixed point theorem. The results are obtained in the form of successive approximations corresponding to the proposed problem. To demonstrate the accuracy and effectiveness of the proposed method, three different nonlinear fractional partial differential equations are provided. Approximate solutions of the fractional equations were obtained. These solutions quickly converged to exact solutions with lower computational cost. Furthermore, the method used in this study is more generalized and allows our results to be more extensive and cover several new and existing fractional problems in the literature.

Existence and uniqueness of well-posed fractional boundary value problem.

In this paper, the existence and uniqueness of solution for a fractional differential model involving well-posed boundary conditions and implicit fractional differential equation is considered. The desired goals are achieved by using Banach contraction principle and Scheafer's fixed point theorem. To show the results applicability some examples are presented. The basic mathematical concept of well-posed fractional boundary value issues is investigated in this study. It dives into the existence and uniqueness of these difficulties, offering light on the conditions that allow for both practical and singular solutions. This study contributes to a better knowledge of fractional calculus and its applications in a variety of scientific and technical areas, giving significant insights for both scholars and practitioners.

Implementation of an effective transform technique for convergence analysis of the fractional enzyme kinetic and blood alcohol level models

In the current study, we solve the nonlinear time fractional‐order enzyme kinematic system and the linear time fractional blood alcohol level using an effective method called the Adomian natural decomposition method (ANDM), which is a combination of the Adomian decomposition method (ADM) and the natural transform method (NTM). On the basis of the Banach fixed point theorem, we offer proofs for the existence and uniqueness theorems as they apply to nonlinear ordinary differential equations. Using the current method, approximate analytical solutions of nonlinear time fractional‐order enzyme kinematic system and exact solutions to the linear time fractional‐order blood alcohol level problem have been obtained. The outcomes indicate that the ANDM is highly efficient and useful. The method outlined in this paper is intended to be applied in subsequent work to handle comparable nonlinear problems connected to fractional calculus.

Demiclosed principle and some fixed-point theorems for generalized nonexpansive mappings in Banach spaces

The aim of this paper is to discuss some results concerning the demiclosedness principle of generalized, nonexpansive mappings in uniformly convex spaces. Further, we present some new fixed-point theorems for generalized nonexpansive mappings in different settings of Banach spaces.

Two strains model of infectious diseases for mathematical analysis and simulations

ABSTRACT In this study, we study a two-strain nonlinear model for the transmission of COVID-19 with a vaccinated class. Here, it is remarkable that the model we consider contains two kinds of viruses known as Omicron and Delta variants denoted by A and B , respectively. Also, the uninfected population is denoted by S , the vaccinated class by V and the recovered individuals by R . In the presented study, we consider the proposed model under conformable fractional order derivatives. The fundamental reproductive number and equilibrium points are computed. Moreover, we determine the existence and uniqueness of the solution to the suggested model using fixed-point theory. Furthermore, we provide a suitable methodology by applying the Euler numerical method to calculate the approximate solution of each compartment of the proposed model. Additionally, using MATLAB-16, we simulate the given results graphically for a variety of fractional orders using some real values of the parameters and initial conditions.

A doubly history‐dependent quasivariational inequality arising in viscoelastic frictional contact problems with wear

AbstractWe aim here to investigate a new mathematical model that describes the contact between a viscoelastic body, accounting for long memory and wear effects, and an obstacle referred to as the foundation. The contact model is governed by a normal compliance condition, coupled with Coulomb's law of dry friction for sliding, and wear effects. We derive the variational formulation of the model, which involves coupling of a quasi‐variational inequality with a nonlinear equation. By pursuing the abstract history‐dependent quasi‐variational inequalities and leveraging the fixed point theorem, we establish results concerning both existence and uniqueness.

The Existence and Ulam Stability Analysis of a Multi-Term Implicit Fractional Differential Equation with Boundary Conditions

In this paper, we investigate a class of multi-term implicit fractional differential equation with boundary conditions. The application of the Schauder fixed point theorem and the Banach fixed point theorem allows us to establish the criterion for a solution that exists for the given equation, and the solution is unique. Afterwards, we give the criteria of Ulam–Hyers stability and Ulam–Hyers–Rassias stability. Additionally, we present an example to illustrate the practical application and effectiveness of the results.

Editorial Conclusion for the Special Issue “New Theory and Applications of Nonlinear Analysis, Fractional Calculus and Optimization”

Nonlinear analysis has widespread and significant applications in many areas at the core of many branches of pure and applied mathematics and modern science, including nonlinear ordinary and partial differential equations, critical point theory, functional analysis, fixed point theory, nonlinear optimization, fractional calculus, variational analysis, convex analysis, dynamical system theory, mathematical economics, data mining, signal processing, control theory, and many more [...]

On the displacement control performance of base-isolated structures incorporating nonlinear negative stiffness device

Low-frequency dominant far-field harmonic-like long period ground motions (FGMs) excitations have the potential to amplify the seismic response of base-isolated structures (BIS) and may result in isolator displacements exceeding their maximum limits. The application of negative stiffness devices (NSD) in isolated structures primarily focuses on SDOF systems, which fails to capture the displacement response of the superstructure on the isolation story. To address this issue, this study proposes the application of NSD to a 2DOF system in a BIS, with root mean square value as the metric for displacement transmissibility. Firstly, the displacement transmissibility of base-isolated structures with negative stiffness devices (BIS-NSD) and BIS is derived using the triple-harmonic balance (T-HB) method and the single-harmonic balance (S-HB) method, respectively. Subsequently, the applicability of the T-HB method and the S-HB method in obtaining analytical solution for the displacement transmissibility of BIS-NSD and BIS is revealed and validated through numerical solution. Parameters are selected based on fixed-point theorem for comparative analysis of the displacement and acceleration transmissibility between BIS-NSD and BIS. Finally, a comparative analysis between BIS-NSD and BIS under FGMs confirms the application advantages of NSD. The results indicate that compared to BIS, BIS-NSD significantly reduces the displacement transmissibility of the superstructure and the isolation story in the low-frequency range. As the primary resonance frequency decreases, NSD widens the vibration bandwidth of BIS. BIS-NSD effectively mitigates the increase in displacement caused by low-frequency ground motions. The acceleration transmissibility demonstrates that, without sacrificing the acceleration of the superstructure, it effectively controls the displacement of the isolation story.

Boundary value problems for nonlinear fractional differential equations with $\psi$-Caputo fractional

In this present paper, we will envisaged the existence and uniqueness of solutions for the following boundary value problem for a nonlinear fractional differential equation involving with $\psi$-Caputo fractional derivative. Our results are proved under Banach contraction principle and Krasnoselkii's fixed point theorem.

Some Boundary Problems, in Sobolev Spaces with Constant Exponents, Governed by the Nonlinear Operator of Plate Vibrations

In this paper, we aim to investigate certain nonlinear boundary problems within Sobolev spaces, where the exponents remain constant. We focus on the dynamically modified operator, incorporating a viscosity term into the nonlinear vibrations of plates. Vibrating plates have a broad range of applications. To address user requirements comprehensively, we've taken into account factors such as the geometric configuration, material density, plate thickness, and Poisson's ratio. After formulating the problems, our method involves converting them into hyperbolic-type nonlinear problems. In this study, we examine six boundary value problems, establishing existence and uniqueness theorems for each. Lastly, we establish the existence of a solution for the stationary problem by employing a variation of Brouwer's fixed point theorem.

Unveiling the Complexity of HIV Transmission: Integrating Multi-Level Infections via Fractal-Fractional Analysis

This article presents a non-linear deterministic mathematical model that captures the evolving dynamics of HIV disease spread, considering three levels of infection in a population. The model integrates fractal-fractional order derivatives using the Caputo operator and undergoes qualitative analysis to establish the existence and uniqueness of solutions via fixed-point theory. Ulam-Hyer stability is confirmed through nonlinear functional analysis, accounting for small perturbations. Numerical solutions are obtained using the fractional Adam-Bashforth iterative scheme and corroborated through MATLAB simulations. The results, plotted across various fractional orders and fractal dimensions, are compared with integer orders, revealing trends towards HIV disease-free equilibrium points for infective and recovered populations. Meanwhile, susceptible individuals decrease towards this equilibrium state, indicating stability in HIV exposure. The study emphasizes the critical role of controlling transmission rates to mitigate fatalities, curb HIV transmission, and enhance recovery rates. This proposed strategy offers a competitive advantage, enhancing comprehension of the model’s intricate dynamics.

A quasistatic electro-elastic contact problem with long memory and slip dependent coefficient of friction

In this paper we consider a mathematical model describing a quasistatic frictional contact problem between a deformable body and an obstacle, say a foundation. We assume that the behavior of the material is described by a linear electro-elastic constitutive law with long memory. The contact is modeled with a version of Coulomb's law of dry friction in which the normal stress is prescribed on the contact surface. Moreover, we consider a slip dependent coefficient of friction. We derive a variational formulation for the model, in the form of a coupled system for the displacements and the electric potential. Under a less assumption on the coefficient of friction, we prove the existence result of weak solutions of the model. We can show the uniqueness of solution by adding another condition (3.10). The proofs are based on arguments of time-dependent variational inequalities, differential equations and Banach fixed point theorem.

Global Dynamics of a Social Hierarchy-Stratified Malaria Model: Insight from Fractional Calculus

In this study, a mathematical model for the transmission dynamics of malaria among different socioeconomic groups in the human population interacting with a susceptible-infectious vector population is presented and analysed using a fractional-order derivative of the Caputo type. The total human population is stratified into two distinguished classes of lower and higher income individuals, with each class further subdivided into susceptible, infectious, and recovered populations. The socio hierachy-structured fractional-order malaria model is analyzed through the application of different dynamical system tools. The theory of positivity and boundedness based on the generalized mean value theorem is employed to investigate the basic properties of solutions of the model, while the Banach fixed point theory approach is used to prove the existence and uniqueness of the solution. Furthermore, unlike the existing related studies, comprehensive global asymptotic dynamics of the fractional-order malaria model around both disease-free and endemic equilibria are explored by generalizing the usual classical methods for establishing global asymptotic stability of the steady states. The asymptotic behavior of the trajectories of the system are graphically illustrated at different values of the fractional (noninteger) order.

Traveling waves for a nonlocal delayed reaction-diffusion SIR epidemic model with demography effects

Considering the importance of time delay phenomenon in disease transmission and the interdependence of time delay and spatial location, a reaction-diffusion SIR epidemic model with nonlocal time delay (or time-space time delay) is proposed and the travelling wave solutions are discussed. Specifically, we define the basic reproduction number R 0 and the critical wave speed c ∗ . For every wave speed c ≥ c ∗ , the existence of travelling wave solutions is studied by using the upper–lower solutions, the fixed-point theorem and some limit techniques when R 0 > 1 . The nonexistence of travelling waves when R 0 > 1 for any 0 < c < c ∗ or R 0 < 1 for any c>0 is also proved. Finally, the influence of time-delay on disease propagation, especially the critical wave speed, is discussed through analysis and simulations.

Optimal control problem for higher-order non-instantaneous impulsive fractional system with non-local conditions

The purpose of this article is to study the optimal control problem of higher-order non-instantaneous impulsive fractional system with non-local conditions by virtue of cosine operator theory in the Hilbert space. Initially, the sufficient conditions for the existence of a mild solution of the proposed system are verified by applying the fixed point theorem in infinite dimensional space. Further, we formulate and prove the existence results for optimal control of the proposed system with the corresponding cost functional by using Balder's theorem. Finally, an example is provided to illustrate the main results.

On Implicit Atangana–Baleanu–Caputo Fractional Integro-Differential Equations with Delay and Impulses

In this paper, we study the existence and uniqueness of solutions for impulsive Atangana-Baleanu-Caputo ABC fractional integro-differential equations with boundary conditions. Schaefer’s fixed point theorem and Banach contraction principle are used to prove the existence and uniqueness results. An example is presented to illustrate the results.

On qualitative analysis of a fractional hybrid Langevin differential equation with novel boundary conditions

A hybrid system interacts with the discrete and continuous dynamics of a physical dynamical system. The notion of a hybrid system gives embedded control systems a great advantage. The Langevin differential equation can accurately depict many physical phenomena and help researchers effectively represent anomalous diffusion. This paper considers a fractional hybrid Langevin differential equation, including the ψ-Caputo fractional operator. Furthermore, some novel boundaries selected are considered to be a problem. We used the Schauder and Banach fixed-point theorems to prove the existence and uniqueness of solutions to the considered problem. Additionally, the Ulam-Hyer stability is evaluated. Finally, we present a representative example to verify the theoretical outcomes of our findings.

Analytical design of non-grounded tuned mass-damper-inerter for base-excited structures

This research contributes new analytical formulae for H∞ optimal tuning of non-grounded tuned mass-damper-inerter (TMDI) for minimizing free-end displacement of base-excited cantilevered structures. The derivation relies on the fixed-point theory, making use of single-mode modelling of the cantilevered structure while accommodating arbitrary TMDI placement along the structural height. Optimal TMDI tuning parameters are derived for given dominant mode shape of structure and TMDI inertial properties (i.e., secondary mass and inertance) under two different types of harmonic excitations with frequency-independent displacement and acceleration amplitudes. The applicability of the derived TMDI tuning formulae for response mitigation of lightly damped cantilevered structures under stationary broadband support excitation is established through comparisons with numerically optimal TMDI properties for a wide range of TMDI inertial properties. Moreover, the analytical TMDI tuning formulae derived in this study achieve enhanced structural performance for base-excited structures compared to those from the literature, derived under various modelling assumptions and optimality criteria. Lastly, the potential of the proposed TMDI tuning for structural response mitigation is numerically evaluated by examining displacement, acceleration, and energy dissipation response history data of an experimentally identified reinforced concrete bridge pier model subjected to 100 earthquake ground motion (GM) records. Overall, reported results demonstrate that the derived analytical formulae can significantly extend the practical application of TMDI as a bona fide dynamic vibration absorber for base-excited structures by circumventing the need for computationally demanding numerical TMDI tuning optimization.