Schauder Fixed Point Theorem Research Articles

Existence and Uniqueness of Second-Order Impulsive Delay Differential Systems

In this paper, we study the existence and uniqueness of second-order impulsive delay differential systems. Firstly, we define cosine-type and sine-type delay matrix functions, which are used to derive the solutions of the impulsive delay differential systems. Secondly, based on the Schauder and Banach fixed-point theorems, we establish sufficient conditions that guarantee the existence and uniqueness of solutions to nonlinear impulsive delay differential systems. Finally, several examples are given to illustrate our theoretical results.

A Novel Fractional Model Combined with Numerical Simulation to Examine the Impact of Lockdown on COVID-19 Spread

One of the best procedures for preventing the spread of the coronavirus is a lockdown, if it is implemented correctly. In order to assess how well lockdowns prevent the virus’s propagation, this paper presents a fractional-order mathematical model constructed by the proportional-Caputo operator. This model consists of five nonlinear fractional-order differential equations. The solution’s existence and uniqueness are investigated using the Schauder and Banach fixed-point theorems. Also, this study produces a stability analysis utilizing Ulam–Hyers and modified Ulam–Hyers criteria. Furthermore, the Adams–Bashforth–Moulton approach is used to implement numerical simulations that show how the model behaves with different parameter combinations and to validate the theoretical results.

A noncompact Schauder fixed point theorem in random normed modules and its applications

A noncompact Schauder fixed point theorem in random normed modules and its applications

The Study of Fixed Point Theorem over Modular F-metric Spaces

One of the most significant discoveries in the annals of mathematical history is Schauder's fixed-point theorem, which is generally recognized to be among the most important discoveries. The fact that the Brüwer fixed point hypothesis cannot be used in dimensions of space that are infinitely large is one of the most important discoveries in the history of mathematics. The facts that have been supplied make it feasible to claim that the great majority of them are of a topological nature. In 2019, N. Manav and D. Turkoglu introduced a new class of generalized metric space called modular F metric space as a generalization of metric space. It is generally agreed upon that this is one of the most significant discoveries that has been made in the subject of mathematics that has ever been produced. In this article, we study the basic structure of modular F-metric spaces and the concept of an equivalent relationship between modular F metric space and modular F metric bounded space. Moreover, we study the fixed point theorem (New version of Banach Contraction Principle) over modular F metric spaces. This is something that one would be able to predict occurring given the circumstances that are present. These results extend, broaden, and integrate many previously published results.

Existence and Uniqueness of Solution of Fractional Differential Equation Using Non-local Operator

Objectives : To investigate the existence and uniqueness of a solution to the non-local Cauchy problem of order using the Atangana Baleanu (AB) fractional derivative operator. Methods : Using Schauder's fixed point theorem and the Arzela-Ascoli theorem, the study proved the existence of a solution to the given problem. Further, it obtains results for the uniqueness of the solution. Findings : This study proves the existence of a solution to the non-local Cauchy problem under the given conditions. Results are also provided for the uniqueness of the solution. Novelty : A novel approach to fractional differential equations is represented by applying Atangana-Baleanu fractional derivative operator to the non-local Cauchy problem. Schauder's fixed point theorem and Arzela-Ascoli's theorem, are used to show existence and uniqueness. Further, one detailed example has been solved. Keywords: Existence, Uniqueness, Non-local operator, Schauder fixed point theorem, Arzela-Ascoli theorem, Cauchy problem

Positive Solutions to a System of Coupled Hadamard Fractional Boundary Value Problems

We explore the existence, uniqueness, and multiplicity of positive solutions to a system of Hadamard fractional differential equations that contain fractional integral terms. Defined on a finite interval, this system is subject to general coupled nonlocal boundary conditions encompassing Riemann–Stieltjes integrals and Hadamard fractional derivatives. To establish the main results, we employ several fixed-point theorems, namely the Banach contraction mapping principle, the Schauder fixed-point theorem, the Leggett–Williams fixed-point theorem, and the Guo–Krasnosel’skii fixed-point theorem.

Non-linear biphasic mixture model: Existence and uniqueness results

Abstract This paper is concerned with the development and analysis of a mathematical model that is motivated by interstitial hydrodynamics and tissue deformation mechanics (poro-elasto-hydrodynamics) within an in-vitro solid tumour. The classical mixture theory is adopted for mass and momentum balance equations for a two-phase system. A main contribution of this study is we treat the physiological transport parameter (i.e., hydraulic resistivity) as anisotropic and heterogeneous, thus the governing system is strongly coupled and non-linear. We derived a weak formulation and then formulated the equivalent fixed-point problem. This enabled us to use the Galerkin method, and the classical results on monotone operators combined with the well-known Schauder and Banach fixed-point theorems to prove the existence and uniqueness of results.

Existence result for a nonlinear mixed boundary value problem for the heat equation

In this paper we study the existence of solutions in parabolic Schauder space of a nonlinear mixed boundary value problem for the heat equation in a perforated domain. From a given regular open set Ω⊆Rn we remove a cavity ω⊆Ω. On the exterior boundary of Ω∖ω‾ we prescribe a Neumann boundary condition, while on the interior boundary we set a nonlinear Robin-type condition. Under suitable assumptions on the data and by means of Leray Schauder Fixed-Point Theorem, we prove the existence of (at least) one solution u∈C01+α2;1+α([0,T]×(Ω‾∖ω)).

Propagation dynamics of a nonlocal dispersal Zika transmission model with general incidence

In this paper, we are interested in propagation dynamics of a nonlocal dispersal Zika transmission model with general incidence. When the threshold is greater than one, we prove that there is a wave speed such that the model has a traveling wave solution with speed , and there is no traveling wave solution with speed less than . When the threshold is less than or equal to one, we show that there is no nontrivial traveling wave solution. The approaches we use here are Schauder's fixed point theorem with an explicit construction of a pair of upper and lower solutions, the contradictory approach, and the two‐sided Laplace transform.

Periodic solution for neutral-type differential equation with piecewise impulses on time scales

In this paper, we establish the existence and stability of periodic solutions for neutral-type differential equations with piecewise impulses on time scales. We first obtain some sufficient conditions for the existence of a unique periodic solution by using the Banach contraction mapping principle. We also prove the existence of at least one periodic solution using the Schauder fixed point theorem. In addition, we establish the stability results based on the existence of periodic solutions. It is worth noting that the results of this paper are based on time scales, so that they are applicable to continuous, discrete, and other types of systems.

System of Nonlinear (<i>k</i>, ψ)-Hilfer Fractional Order Hybrid Boundary Value Problems

This paper focuses on the existence and uniqueness of solutions for a new class of coupled fractional differential equations subject to specific boundary conditions. We introduce a novel approach that incorporates the (k, ψ)-Hilfer fractional derivative, a recent advancement in fractional calculus. By leveraging the properties of this operator, we develop a comprehensive framework to analyze the proposed system. Our findings, established through the application of Schauder's fixed point theorem and the Banach contraction principle, contribute to the understanding of complex phenomena modeled by fractional differential equations. This research expands the boundaries of fractional calculus and offers potential applications in various scientific and engineering domains.

Coupled wave equations with monotone nonlinearity: existence of solution and controllability results

In this manuscript, we investigate a coupled semilinear system characterised by two wave equations in which the nonlinear function satisfies the monotonicity condition. Initially, the existence of a solution is established through the application of Schauder's fixed point theorem. Thereafter, the results pertaining to the exact controllability are demonstrated for two different cases by establishing a relation between the reachable set of the semilinear system and its linear counterpart. Additionally, an illustrative example is provided to showcase the practical relevance of our theoretical findings.

The study of nonlinear fractional boundary value problems involving the p-Laplacian operator

The p-Laplacian has attracted considerable attention in numerous fields such as mechanics, image processing and game theory. It is a nonlinear operator which has been used in the modelling and qualitative aspects in numerous problems. In this research work, we propose a new nonlinear fractional differential equation involving the p-Laplacian, which include the generalized Caputo fractional derivatives. We investigate the existence and uniqueness of solutions to our proposed problem through the application using the Banach and Schauder’s fixed-point theorems. Additionally, we illustrate the practical applicability of our findings by applying them to a specific example, thereby validating their efficacy.

On the existence of solutions for nonlocal sequential boundary fractional differential equations via ψ-Riemann–Liouville derivative

The purpose of this paper is to study a generalized Riemann–Liouville fractional differential equation and system with nonlocal boundary conditions. Firstly, some properties of the Green function are presented and then Lyapunov-type inequalities for a sequential ψ-Riemann–Liouville fractional boundary value problem are established. Also, the existence and uniqueness of solutions are proved by using Banach and Schauder fixed-point theorems. Furthermore, the existence and uniqueness of solutions to a sequential nonlinear differential system is established by means of Schauder’s and Perov’s fixed-point theorems. Examples are given to validate the theoretical results.

The Existence and Uniqueness of Radial Solutions for Biharmonic Elliptic Equations in an Annulus

This paper concerns with the existence of radial solutions of the biharmonic elliptic equation ▵2u=f(|x|,u,|∇u|,▵u) in an annular domain Ω={x∈RN:r1<|x|<r2}(N≥2) with the boundary conditions u|∂Ω=0 and ▵u|∂Ω=0, where f:[r1,r2]×R×R+×R→R is continuous. Under certain inequality conditions on f involving the principal eigenvalue λ1 of the Laplace operator −▵ with boundary condition u|∂Ω=0, an existence result and a uniqueness result are obtained. The inequality conditions allow for f(r,ξ,ζ,η) to be a superlinear growth on ξ,ζ,η as |(ξ,ζ,η)|→∞. Our discussion is based on the Leray–Schauder fixed point theorem, spectral theory of linear operators and technique of prior estimates.

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.

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.

New technique for controllability results of Hilfer fractional hybrid Langevin dynamical system

Our work introduces a novel type of hybrid Langevin equations that include both Riemann and Caputo fractional order derivatives. While the measure of non-compactness has become significant to fixed point theory, we apply the measure of non-compactness approach as an essential aspect for arriving at the controllability the result. The Schauder fixed point theorem is then used in a generalized version to make use of the contemporary analytic technique. To improve the comprehensibility of our findings, which we provided a numerical example.

Exact solitary wave propagations for the stochastic Burgers’ equation under the influence of white noise and its comparison with computational scheme

In this manuscript, the well-known stochastic Burgers’ equation in under investigation numerically and analytically. The stochastic Burgers’ equation plays an important role in the fields of applied mathematics such as fluid dynamics, gas dynamics, traffic flow, and nonlinear acoustics. This study is presented the existence, approximate, and exact stochastic solitary wave results. The existence of results is shown by the help of Schauder fixed point theorem. For the approximate results the proposed stochastic finite difference scheme is constructed. The analysis of the proposed scheme is analyzed by presented the consistency and stability of scheme. The consistency is checked under the mean square sense while the stability condition is gained by the help of Von-Neumann criteria. Meanwhile, the stochastic exact solutions are constructed by using the generalized exponential rational function method. These exact stochastic solutions are obtained in the form of hyperbolic, trigonometric and exponential functions. Mainly, the comparison of both numerical and exact solutions are analyzed via simulations. The unique physical problems are constructed from the newly constructed soliton solutions to compare the numerical results with exact solutions under the presence of randomness. The 3D and line plots are dispatched that are shown the similar behavior by choosing the different values of parameters. These results are the main innovation of this study under the noise effects.

On solutions of a semilinear measure‐driven evolution equation with nonlocal conditions on infinite interval

AbstractThis paper studies the existence and asymptotic properties of solutions for a semilinear measure‐driven evolution equation with nonlocal conditions on an infinite interval. The existence result of the solutions for the considered equation is established by Schauder's fixed point theorem. Then, the asymptotic stability of solutions is further proved to show that all the solutions may converge to the unique solution of the corresponding Cauchy problem. In addition, under some conditions the existence of global attracting sets and quasi‐invariant sets of mild solutions is investigated as well. Finally, an example is provided to illustrate the applications of the obtained results.