Fixed Point Theorem Research Articles

Solvability Criterion for a System Arising from Monge–Ampère Equations with Two Parameters

Monge–Ampère equations have important research significance in many fields such as geometry, convex geometry and mathematical physics. In this paper, under some superlinear and sublinear conditions, the existence of nontrivial solutions for a system arising from Monge–Ampère equations with two parameters is investigated based on the Guo–Krasnosel’skii fixed point theorem. In the end, two examples are given to illustrate our theoretical results.

Fixed point theorem and iterated function system in φ-metric modular space

We introduce and study the concept of φ-metric modular space and, then define φ-α-Meir-Keeler contraction on it and explore its fixed point. Further, we define the Hausdorff distance between two non-empty compact subsets of the considered space. Some topological properties of φ-metric modular space are also explored. Additionally, we prove the existence of the attractor (fractal) of the IFS consisting of φ-α-Meir-Keeler contractions.

Existence of periodic solutions and stability for a nonlinear system of neutral differential equations

In this article, we study the existence and uniqueness of periodic solutions, and stability of the zero solution to the nonlinear neutral system $$ \frac{d}{dt}x(t)=A(t)h\big(x(t-\tau_1(t))\big)+\frac{d}{dt}Q\big(t,x(t-\tau_2(t))\big) +G\big(t,x(t),x(t-\tau_2(t))\big). $$ We use integrating factors to transform the neutral differential equation into an equivalent integral equation. Then we construct appropriate mappings and employ Krasnoselskii's fixed point theorem to show the existence of a periodic solution. We also use the contraction mapping principle to show the existence of a unique periodic solution and the asymptotic stability of the zero solution. Our results generalize the corresponding results in the existing literature. An example is given to illustrate our results.For more information see https://ejde.math.txstate.edu/Volumes/2024/21/abstr.html

On Suzuki-Proinov Type Contractions in Modular $b$-Metric Spaces with an Application

In this paper, by taking ${{\mathcal C}_\mathcal{A}}-$simulation function and Proinov type function into account, we set up a new contraction mapping called Suzuki$-$Proinov $\mathpzc{Z^*}_{\aE^*}^{\aR}(\alpha)-$contraction, including both rational expressions that possess quadratic terms and $\aE-$type contractions. Furthermore, we demonstrate a common fixed point theorem through the mappings endowed with triangular $\alpha-$admissibility in the setting of modular $b-$metric spaces. Besides that, we achieve some new outcomes that contribute to the current ones in the literature through the main theorem, and, as an application, we examine the existence of solutions to a class of functional equations emerging in dynamic programming.

Fixed point theorems for continuous single-valued and upper semicontinuous set-valued mappings in $p$-vector and locally $p$-convex spaces

The goal of this paper is to establish a general fixed point theorem for compact single-valued continuous mappings in Hausdorff $p$-vector spaces, and a fixed point theorem for upper semicontinuous set-valued mappings in locally $p$-convex spaces for $p\in (0, 1]$. These results not only provide a solution to Schauder conjecture in the affirmative under the setting of $p$-vector spaces for compact single-valued continuous mappings, but also show the existence of fixed points for upper semicontinuous set-valued mappings defined on $s$-convex subsets in Hausdorff locally $p$-convex spaces, which would be fundamental for nonlinear functional analysis, where $s, p \in (0, 1]$.

Existence of positive solutions in the space of Hölder functions for a class of nonlinear fractional differential equations with integral boundary conditions

We study the existence of positive solutions for a fractional differential equation with integral boundary conditions. Our solutions are placed in the space of Hölder functions and the main tools used in the proof of the results are the classical Schauder fixed point theorem and a sufficient condition about the relative compactness in Hölder spaces. Moreover, some examples are shown illustrating the results.

A fixed point theorem for nonself nonlinear contractions in length spaces

In 1988 N.A. Assad showed that a nonself nonlinear contraction taking a closed subset of a complete metrically convex space into the space so that the boundary of this subset is mapped back into the subset itself has a unique fixed point. In the present paper we extend this result by replacing the complete metrically convex space with a complete metric space which is a length space.

Variational Principles in Quaternionic Analysis with Applications to the Stationary MHD Equations

In this paper we aim to combine tools from variational calculus with modern techniques from quaternionic analysis that involve Dirac type operators and related hypercomplex integral operators. The aim is to develop new methods for showing geometry independent explicit global existence and uniqueness criteria as well as new computational methods with special focus to the stationary incompressible viscous magnetohydrodynamic equations. We first show how to specifically apply variational calculus in the quaternionic setting. To this end we explain how the mountain pass theorem can be successfully applied to guarantee the existence of (weak) solutions. To achieve this, the quaternionic integral operator calculus serves as a key ingredient allowing us to apply Schauder’s fixed point theorem. The advantage of the approach using Schauder’s fixed point theorem is that it is also applicable to large data since it does not require any kind of contraction property. These considerations will allow us to provide explicit iterative algorithms for its numerical solution. Finally to obtain more precise a-priori estimates one can use in the situations dealing with small data the Banach fixed point theorem which then also grants the uniqueness.

Controllability of psi-Hilfer fractional differential equations with infinite delay via measure of noncompactness

In this article, we study the controllability of ψ-Hilfer fractional differential equations with infinite delay. Sufficient conditions for controllability results are obtained by using the notion of the measure of noncompactness and the Mönch fixed point theorem. The novel feature of this study is to inquire into the controllability notion by using ψ-Hilfer fractional derivative, the generalized variant of the Hilfer derivative. Finally, we provide a numerical example to illustrate our main result.

Remarks and a Generalization of the Hegedüs–Szilágyi Fixed Point Theorem

Remarks and a Generalization of the Hegedüs–Szilágyi Fixed Point Theorem

On a New Proof of the Key Step in the Proof of Brouwer’s Fixed Point Theorem

On a New Proof of the Key Step in the Proof of Brouwer’s Fixed Point Theorem

On a (k;chi)-Hilfer fractional system with coupled nonlocal boundary conditions including various fractional derivatives and Riemann–Stieltjes integrals

In the present research, we investigate the existence and uniqueness of solutions for a system of (k; χ)-Hilfer fractional differential equations, subject to coupled nonlocal boundary conditions, which contain various fractional derivatives and Riemann–Stieltjes integrals. The uniqueness result relies on the Banach contraction mapping principle, while the existence results depend on the Leray–Schauder alternative and Krasnosel’skiĭ fixed point theorem. Examples are also constructed to illustrate the obtained results.

On the unique weak solvability of second-order unconditionally stable difference scheme for the system of sine-Gordon equations

In the present paper, a nonlinear system of sine-Gordon equations that describes the DNA dynamics is considered. A novel unconditionally stable second-order accuracy difference scheme corresponding to the system of sine-Gordon equations is presented. In this work, for the first time in the literature, weak solution of this difference scheme is studied. The existence and uniqueness of the weak solution for the difference scheme are proved in the space of distributions, and the methods of variational calculus are applied. The finite-difference method and the fixed point theory are used in combination to perform numerical experiments that verify the theoretical statements.

Numerical analysis of COVID-19 model with Caputo fractional order derivative

This paper focuses on the numerical solutions of a six-compartment fractional model with Caputo derivative. In this model, we obtain non-negative and bounded solutions, equilibrium points, and the basic reproduction number and analyze the stability of disease free equilibrium point. The existence and uniqueness of the solution are proven by employing the Picard–Lindelof approach and fixed point theory. The product–integral trapezoidal rule is employed to simulate the system of FODEs (fractional ordinary differential equations). The numerical results are presented in the form of graphs for each compartment. Finally, the sensitivity of the most important parameter (β) and its impact on COVID-19 dynamics and the basic reproduction number are reported.

Cosmic strings arising in a self-dual Abelian Higgs model

In this note we construct self-dual cosmic strings from an Abelian Higgs model in two-dimension with a polynomial formation of the potential energy density. By integrating the Einstein equations, we obtain an equivalent form to the sources, which is a nonlinear elliptic equation with singularities and complicated exponential terms. We prove the existence of a solution governing strings in the broken symmetry category on the whole plane, and the multiple string solutions are valid under a sufficient condition imposed only on the total number of strings. The technique of upper–lower solutions and the method of regularization are employed to show the existence of a solution when there are at least two distant string centers. When all the string centers are identical, a fixed-point theorem is used to study the properties of the nonlinear elliptic equation. Finally, we establish the sharp asymptotic estimate for the solutions at infinity and derive the dependence of the total gravitational curvature on the string number.

Positive solutions for Hadamard-type fractional differential equations with nonlocal conditions on an infinite interval

The purpose of this paper is to analyse the local existence and uniqueness of positive solutions for a Hadamard-type fractional differential equation with nonlocal boundary conditions on an infinite interval. The technique used to arrive our results depends on two fixed point theorems of a sum operator in partial ordering Banach spaces. The local existence and uniqueness of positive solution is given, and we can make iterative sequences to approximate the unique positive solution. For the illustration of the main results, we list two concrete examples in the last section.

Fixed Point Results Under Hausdorff Distance in the Fractal Spaces

We establish a novel notion of Hausdorff distance and explore some of its topological characteristics by using an extended modular metric. We prove a fixed point theorem on generalized modular fractal space from the concept of iterated function system (IFS) and contraction.

The Existence of Solutions for w-Weighted ψ-Hilfer Fractional Differential Inclusions of Order μ ∈ (1, 2) with Non-Instantaneous Impulses in Banach Spaces

In this research, we obtain the sufficient conditions that guarantee that the set of solutions for an impulsive fractional differential inclusion involving a w-weighted ψ-Hilfer fractional derivative, D0,tσ,v,ψ,w,of order μ∈(1,2), in infinite dimensional Banach spaces that are not empty and compact. We demonstrate the exact relation between a differential equation involving D0,tσ,v,ψ,w of order μ ∈(1,2) in the presence of non-instantaneous impulses and its corresponding fractional integral equation. Then, we derive the formula for the solution for the considered problem. The desired results are achieved using the properties of the w-weighted ψ-Hilfer fractional derivative and appropriate fixed-point theorems for multivalued functions. Since the operator D0,tσ,v,ψ,w includes many types of well-known fractional differential operators, our results generalize several results recently published in the literature. We give an example that illustrates and supports our theoretical results.

Modeling and investigating the spread of COVID-19 dynamics with Atangana-Baleanu fractional derivative: a numerical prospective

Fractional-order models have been used in the study of COVID-19 to incorporate memory and hereditary properties into the systems Moira and Xu (2003) Respirology 8 S9â14. These models have been applied to analyze the dynamics and behavior of the novel coronavirus. Various fractional-order models have been proposed, including the SIR and SEIR models, with the addition of compartments such as asymptomatic classes and virus repositories. Overall, fractional-order models have proven to be effective in studying the dynamics and spread of COVID-19. In this paper, we propose a nonlinear fractional model employing the Atangana-Baleanu derivative to describe COVID-19. To offer a clearer perspective, our investigation incorporates two distinct quarantine stages within the population. The first quarantine class consists of individuals who have not yet contracted the virus but have chosen to self-isolate at home. The second quarantine class encompasses individuals who are infected and are undergoing quarantine in hospitals. Additionally, we introduce a vaccination class, consisting of the portion of the population that has received the COVID-19 vaccination and is now at a reduced risk of infection. Fixed point theorems are employed to prove the existence and uniqueness of the solutions. The model’s threshold parameter R0 is calculated to investigate the pandemic’s future dynamics. The Toufik-Atangana scheme is applied to obtain numerical solutions for the fractional model. Further, we analyzed the data to assess how vaccination impacts the spread of the virus.The model includes two novel parameters to determine the speed and effectiveness of vaccination measures. To evaluate the accuracy of our model, we simulate the graphical results of this stage and compared them with integer-order derivative. The mathematical model for COVID-19 shows that both equilibrium points are locally stable. Moreover, to gain a deeper understanding of the disease, we conduct a sensitivity analysis to examine the effect of parameters on R0 . The study recommends continuing hospitalization and home isolation for infected individuals until virus transmission reduces sufficiently after vaccination. The given model provides useful insights into the pandemic’s dynamics and suggests measures for controlling its spread.

Boundary Value Problem of Nonlinear Fractional Differential Equations of Mixed Volterra-Fredohlm Integral Equations in Banach Space

This paper is dedicated to study the existence and uniqueness of solutions for a boundary value problem of nonlinear fractional differential equations of mixed Volterra-Fredohlm integral equations in Banach space, the recent researches considered the study of differential equations of mixed Volterra-Fredholm integral equations with classical order and the study of existence and uniqueness of solutions using approched numerical methodes, the objective of this paper is the study of the existence and uniqueness of fractional order of differential equations with mixed Volterra-Fredholm integral equations using fixed point theory. This work have two important results, the first result was the discussion of the existence of solutions using the Krasnoselskii fixed point theorem after transforming the problem into integral equation firstly then into operator problem suitable for the fixed point theory. The second result will be the existence and uniqueness of solution, this result was obtained by the use of Banach fixed point theorem after the same transformation used in the first one. This work give as conclusion that the boundary value problem of nonlinear fractional differential equations of mixed Volterra-Fredholm integral equation has a unique solution in Banach space. Finally, this work was ended with an example to illustrate the results obtained.