Contraction Mapping Principle Research Articles

Highly Non-contractive Iterated Function Systems on Euclidean Space Can Have an Attractor

AbstractIterated function systems (IFSs) and their attractors have been central to the theory of fractal geometry almost from its inception. Moreover, contractivity of the functions in the IFS has been central to the theory of iterated functions systems. If the functions in the IFS are contractions, then the IFS is guaranteed to have a unique attractor. The converse question, does the existence of an attractor imply that the IFS is contractive, originates in a 1959 work by Bessaga which proves a converse to the contraction mapping theorem. Although a converse is true in that case, it is known that it does not always hold for an IFS. In general, there do exist IFSs with attractors and which are not contractive. However, in the context of IFSs in Euclidean space, this question has been open. In this paper we show that a highly non-contractive iterated function system in Euclidean space can have an attractor. In order to do that, we introduce the concept of an L-expansive map, i.e., a map that has Lipschitz constant strictly greater than one under any remetrization. This is necessitated by the absence of positively expansive maps on the interval.

Weyl almost periodic solutions in distribution to a mean-field stochastic differential equation driven by fractional Brownian motion

In this paper, we consider a class of mean-field stochastic differential equations driven by both Brownian motion and fractional Brownian motion. Using the Banach fixed point theorem and inequality technique, we obtain sufficient conditions for the existence and uniqueness of Weyl almost periodic solutions in distribution of the equations under consideration.

Higher-ordered hybrid fractional differential equations with fractional boundary conditions: Stability analysis and existence theory

In the present article, the p-Laplacian operator is applied for examining hybrid fractional differential equations (HFDEs). Establishing the existence and uniqueness (EU) results and analyzing the Hyers–Ulam (HU) stability for HFDEs incorporating fractional derivatives of different orders with the p-Laplacian operator are the primary objectives of this research. With an understanding of the Green function, we will transform the provided HFDE into a corresponding integral form of hybrid FDEs for EU results. A fixed point theorem is employed to examine the existence of solution (ES), and the Banach contraction mapping principle technique is employed to determine the uniqueness of solution. The result of the HU stability study is examined using dynamical systems and functional analysis approaches. Additionally, an application is presented to demonstrate the findings.

On the fractional perturbed neutral integro-differential systems via deformable derivatives: an existence study

In this paper, we provide some appropriate conditions for the existence of solutions for a perturbed fractional neutral integro-differential system under the deformable derivative in a Banach space. Using the Banach contraction principle and Krasnoselskii’s fixed point theorem, we establish some new existence theorems. Moreover, we provide two numerical examples to demonstrate the applicability of the theoretical results

On the optimal controllability for a class of Katugampola fractional systems

This study is centered on the optimal controllability of differential equations involving fractional derivatives of Katugampola. We derive both necessary and sufficient conditions for optimal controllability by extending Gronwall’s inequality with singular kernels. Furthermore, we establish conditions ensuring the existence and uniqueness of mild solutions using the Banach fixed-point theorem and the generalized Laplace transform. To underscore the practical relevance of our findings, we provide an illustrative example.

On a fully coupled nonlocal multipoint boundary value problem for a dual hybrid system of nonlinear q-fractional differential equations

A new class of nonlocal multipoint boundary value problems involving a dual hybrid system of nonlinear Riemann–Liouville-type [Formula: see text]-fractional differential equations is studied in this paper. Existence and uniqueness results for the given problem are derived by applying the Leray–Schauder nonlinear alternative and the Banach contraction mapping principle. Examples are presented for illustrating the obtained results. The work established in this paper is a useful contribution to the existing literature on [Formula: see text]-fractional differential equations. Some interesting special cases are also discussed.

Generalized solutions for time $\psi$-fractional evolution equations

This paper focuses on the time fractional wave problem with the use of a new fractional derivative in Colombeau algebra. Using Banach's fixed point theorem and Laplace transforms, we give and prove the integral solution of the problem. In Colombeau's algebra, The existence and uniqueness of the solution are demonstrated using the Gronwall lemma.

Application of Fixed Point Theorems to Differential Equation in b-Multiplicative Metric Spaces

Objectives: In this manuscript, some fixed point theorems have been provided in b-multiplicative metric spaces. Methods: We proved the unique fixed point theorems using the Banach contraction principle and generalized Lipschitz contractive mappings. Findings: We established the P Property and the T-Stability of Picard’s iteration in b-multiplicative metric spaces. We also provide an example to demonstrate the result. Novelty: Using our results, we obtained the existence and uniqueness of solutions for ordinary multiplicative differential equations with initial value problems. 2020 Mathematics Subject Classification: 47H10, 54H25. Keywords: Fixed Point, Generalized Lipschitz Contractive, Picard’s iteration, b-Multiplicative Metric Space (b − MMS), T -Stability, Differential equation

Mathematical analysis of intrahost spread and control of Dengue Virus: Unraveling the crucial role of antigenic immunity

Studying viral infections within the host is crucial for understanding healthy cell responses and immunity. This research employed a reaction-diffusion Caputo fractional order model and a modified homotopy perturbation method to explore how immune cell memory and mobility affect the dynamic interactions of cytotoxic T-lymphocyte cells (CTLs), B-cell antibodies, uninfected monocytes, infected monocytes, and Dengue Virus (DENV) particle concentration within the host. The model guaranteed a positive solution, with the existence and uniqueness of the solution assured by the Banach fixed point theorem and a general stability supported by the Hyers-Ulam theorem. Sensitivity analysis highlighted that the DENV invasion rate speeds the spread of infected cells, while simulation results and fractional order analysis revealed that cell-mediated memory from past exposure will enhance the activation of CTLs and B-cell antibodies to strengthen immunity against dengue virus.

Mild and classical solutions to fractional Cauchy problem on time scales

In this manuscript, aims to examine the uniqueness and existence of impulsive non-local conditions, as well as classical and mild solutions to the fractional Cauchy problem in time scales. The main results are derived by the Gronwall inequality and the Banach fixed point theorem in time scales. Eventually, we offered an example to demonstrate the suitability of the conclusions on the fractional Cauchy issue on a time scale.

Averaging principle for Hifer–Katugampola fractional stochastic differential equations

In this paper, we mainly study the averaging principle for a class of Hifer–Katugampola fractional stochastic differential equations driven by standard Brownian motion. Firstly, we establish the existence and uniqueness of mild solution for the considered system using Banach contraction principle. Then, under suitable assumptions, we demonstrate that the solution to the original differential equations converges to that of the averaged differential equations in the sense of mean square and probability as the time scale goes to zero. Finally, an illustrative example is provided to verify our theoretical results.

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.

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 Weighted Cauchy-Type Problem of Riemann-Liouville Fractional Differential Equations in Lebesgue Spaces with Variable Exponent

This paper aims to investigate the existence, uniqueness, and stability properties for a class of fractional weighted Cauchy-type problem in the variable exponent Lebesgue space $L^{p(.)}$. The obtained results are set up by employing generalized intervals and piece-wise constant functions so that the $L^{p(.)}$ is transformed into the classical Lebesgue spaces. Moreover, the usual Banach Contraction Principle is utilized, and the Ulam-Hyers (UH) stability is studied. At the final stage, we provide an example to support the accuracy of the obtained results.

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.

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.

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.

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.