Contraction Mapping Principle Research Articles

Convergence results for controlled contractions and applications to a system of Volterra integral equations

This work presents the concept of controlled dislocated metric spaces, and the Banach contraction principle in these spaces is examined. Next, we provide extensive examples of numerical convergence of fixed points to demonstrate the practical implications of such spaces. Along with specific instances, it presents the idea of a rational contraction of the ϝ − type to determine common fixed points in such spaces. Moreover, numerical analysis and graphical comparisons are used to further assess the existence of common fixed points based on the provided notions. Lastly, using these ideas to solve a Volterra integral equation shows how useful they are for finding common solutions, improving our comprehension of fixed point theory in these spaces, and creating new avenues for applications.

Almost Periodic Solutions of Differential Equations with Generalized Piecewise Constant Delay

In this paper, we investigate differential equations with generalized piecewise constant delay, DEGPCD in short, and establish the existence and stability of a unique almost periodic solution that is exponentially stable. Our results are derived by utilizing the properties of the (μ1,μ2)-exponential dichotomy, Cauchy and Green matrices, a Gronwall-type inequality for DEGPCD, and the Banach fixed point theorem. We apply these findings to derive new criteria for the existence, uniqueness, and convergence dynamics of almost periodic solutions in both the linear inhomogeneous and quasilinear DEGPCD systems through the (μ1,μ2)-exponential dichotomy for difference equations. These results are novel and serve to recover, extend, and improve upon recent research.

Controllability of Impulsive Damped Fractional Order Systems Involving State Dependent Delay

In this article, the concept of controllability on fractional order impulsive systems involving state dependent delay and damping behavior is analysed by utilizing Caputo fractional derivative. The main motivation is to derive the sufficient conditions for the controllability of the considered systems. Based on the Laplace transform and inverse Laplace transform, the solution of fractional-order dynamical systems are obtained. The results are established by utilizing basic ideas of fractional calculus, Mittag-Leffler function and Banach fixed point theorem. Finally, an application is provided to illustrate the derived result.

Quasi‐stabilization/synchronization of quaternion‐valued memristive neural networks: Norm approach

This paper examines the quasi‐stabilization and synchronization of memristive neural networks in high dimensional forms. First, new criteria for the existence of the equilibrium point (EP) are delivered by contraction mapping principle. Then, a new Lyapunov function described with one‐norm form is imported for quasi‐stabilization/synchronization of the system. Finally, two numerical examples containing simulations are given to demonstrate the effectiveness.

Solving fractional integro-differential equations with delay and relaxation impulsive terms by fixed point techniques

This paper presents a systematic approach to investigating the existence of solutions for fractional integro-differential equation systems incorporating delay and relaxation impulsive terms. By employing suitable definitions of fractional derivatives, we establish physically interpretable boundary conditions. To account for abrupt state changes, impulsive conditions are integrated into the model. The system is transformed into an equivalent integral equation, facilitating the application of Banach and Schaefer fixed-point theorems to prove the existence and uniqueness of solutions. The practical applicability of our findings is demonstrated through an illustrative example.

On a Hybrid Class of $p$-Laplacian Initial Value Problems with Modified Mittag-Leffler Kernel

In this work, we establish key results on the existence theory for a category of initial value problems (IVPs) involving hybrid fractional integro-differential equations (HFIDEs) with a $p$-Laplacian operator, utilizing the modified Mittag-Leffler kernel. By employing Krasnoselskii and Banach fixed point theorems (FPTs), we determine the conditions required for the existence of solutions. Additionally, we examine the Hyers-Ulam (H-U) stability of the problem. Lastly, we present an example to confirm our theoretical results.

On the Family of Theorems on Metric Completeness

There have appeared a large family of theorems related the metric completeness. They are mainly concerned with generalizations of the Banach contraction on quasi-metric spaces and their extended artificial spaces. In this survey article, we classify the family according to our 2023 Metatheorem. Many known metric fixed point theorems belong to the family including the Rus-Hicks-Rhoades (RHR) theorem. Such results on metric spaces are consequences of our generalized forms of the Banach contraction principle for weak contractions or the RHR maps on quasi-metric spaces. We list a large number of examples of metric fixed point theorems which follow from our principles. Moreover, we add some comments on related papers in order to improve them.

The Bałaban variational problem in the non-linear sigma model

The minimization of the action of a QFT with a constraint dictated by the block averaging procedure is an important part of Bałaban’s approach to renormalization. It is particularly interesting for QFTs with non-trivial target spaces, such as gauge theories or non-linear sigma models on a lattice. We analyze this step for the [Formula: see text] non-linear sigma model in two dimensions and demonstrate, in this case, how various ingredients of Bałaban’s approach play together. First, using variational calculus on Lie groups, the equation for the critical point is derived. Then, this non-linear equation is solved by the Banach contraction mapping theorem. This step requires detailed control of lattice Green functions and their integral kernels via random walk expansions.

Analysis of Hyers–Ulam stability and controllability of non-linear switched impulsive systems with delays on time scales

In this article, we have used the concepts of time scales theory to discuss nonlinear switched impulsive systems with delay. Our main objective is to determine the Hyers–Ulam stability and controllability of nonlinear switched impulsive systems with delay on non-uniform time domains. To obtain the necessary and sufficient conditions for existence, Hyers–Ulam stability, and controllability, we utilize the Banach fixed-point theorem and Krasnoselskii’s fixed-point theorem. In order to demonstrate our conclusions, we have discussed some simulation-based examples along with the three tank liquid control problem and a potential practical situation related to the infectious disease with switching rules. The results of this manuscript provide all the necessary and sufficient conditions for Hyers–Ulam stability and controllability that are true for discrete, continuous as well as unified time domains simultaneously.

Existence and uniqueness of the solution to initial and inverse problems for integro-differential heat equations with fractional load

This work is devoted to the unique solvability of the direct and inverse problems for a multidimensional heat equation with a fractional load in Holder spaces. In the problem under consideration, the loaded term is in the form of a fractional integral operator for the time variable. We prove the existence and uniqueness of the solution to these problems by the contraction mapping theorem and the theory of integral equations. For more information see https://ejde.math.txstate.edu/Volumes/2024/64/abstr.html

On type I blowup of some nonlinear heat equations with a potential

In this paper, we are concerned with the following initial-boundary value problem{ut=Δu+Q(|x|)|u|p−1u,x∈BR,t>0u(x,t)=0,x∈∂BR,t>0u(x,0)=u0(x),x∈BR, where p≥ps:=N+2N−2, u0∈L∞(BR), and Q(r)∈C1([0,R]), 0<C_≤Q(r)≤C‾<∞,Q′(r)≤0. We extend the asymptotic behavior results, which is well-known when Q is constant according to Matano-Merle (cf. [25]), for the blow-up solutions. More precisely, we show that when ps≤p<p⁎, the blowup of radial solution to this problem is always of Type I. This result partially generalizes the conclusions in [25] for Q≡1. This extension is nontrivial due to the appearance of Q. The quasi-monotonicity formula established by the third author and Cheng in [8] allows us to use an energy method to get a priori estimates on the rescaled solutions. The contraction mapping principle shows the existence of singular stationary solutions to an associated elliptic equation with a potential. In the end, the properties of zero number for solutions lead to the nonexistence of type II singularity for the problem.

On the Generalized (p,q)-ϕ-Calculus with Respect to Another Function

In the present paper, we generalized some of the operators defined in (p,q)-calculus with respect to another function. More precisely, the generalized (p,q)-ϕ-derivatives and (p,q)-ϕ-integrals were introduced with respect to the strictly increasing function ϕ with the help of different orders of the q-shifting, p-shifting, and (q/p)-shifting operators. Then, after proving some related properties, and as an application, we considered a generalized (p,q)-ϕ-difference problem and studied the existence property for its unique solutions with the help of the Banach contraction mapping principle.

Granular fuzzy calculus on time scales and its applications to fuzzy dynamic equations

This paper introduces the foundational theory of fuzzy calculus on time scales, utilizing granular arithmetic operations between fuzzy intervals. These operations are developed based on the concept of the horizontal membership function (HMF), which is applied in multidimensional fuzzy arithmetic (MFA). Furthermore, the paper explores the existence of a unique solution and the continuous dependence of the solution to fuzzy dynamic equations on initial data, employing the Banach fixed-point theorem under a new metric for fuzzy functions in time scales involving the generalized exponential function. Finally, to highlight the practical significance of these results and their potential applications, the paper presents mathematical models relevant to nuclear physics and biology.

On the relation of Kannan contraction and Banach contraction

Abstract Kannan contraction is broadly investigated topic in Metric Fixed Point Theory due to its importance in omitting the continuity presumption. Of a great significance is also its role in characterizing completeness of a metric space through existence and uniqueness of a fixed point of arbitrary Kannan contraction in the observed setting. The concept of Kannan contraction has been adapted, extended and transferred to various types of spaces including cone metric spaces, quasi metric spaces, b-metric spaces, partial metric spaces, among others. The main aim of this article is to prove that for any Kannan contraction T on a complete metric space (X, d) there exists another metric on the set X in relation to which T is a Banach contraction while the completeness is preserved. In that way, all results on Kannan contraction may be derived as corollaries of the Banach contraction principle. The converse also holds since, by altering the metric, Banach contraction becomes a Kannan contraction. The obtained theoretical results are substantiated with adequate examples.

Some Results for a Class of Pantograph Integro-Fractional Stochastic Differential Equations

Symmetrical fractional differential equations have been explored through a variety of methods in recent years. In this paper, we analyze the existence and uniqueness of a class of pantograph integro-fractional stochastic differential equations (PIFSDEs) using the Banach fixed-point theorem (BFPT). Also, Gronwall inequality is used to demonstrate the Ulam–Hyers stability (UHS) of PIFSDEs. The results are illustrated by two examples.

Fractional-order analysis of temperature- and rainfall-dependent mathematical model for malaria transmission dynamics

Malaria remains a substantial public health challenge and economic burden globally. Currently, malaria has been declared as endemic in 85 countries. In this study, we developed and analyzed a fractional-order mathematical model for malaria transmission dynamics that incorporates variability of temperature and rainfall using Caputo-type AB operators. The existence and uniqueness of the model's solutions were established using the Banach fixed-point theorem. The model system's equilibria (both disease-free and endemic) were identified, and lemmas and theorems were developed to prove their stability. Furthermore, we used different temperature ranges and rainfall data, validating them against existing literature. Numerical simulations using the Toufik-Atangana schemes with various fractional-order alpha values revealed that as the value of alpha approaches 1, the behavior of the fractional-order model converges to that of the classical model. The numerical results are promising and are expected to be valuable for future research related to fractional-order models.

Mathematical Analysis of a Navier–Stokes Model with a Mittag–Leffler Kernel

In this paper, we establish the existence and uniqueness results of the fractional Navier–Stokes (N-S) evolution equation using the Banach fixed-point theorem, where the fractional order β is in the form of the Atangana–Baleanu–Caputo fractional order. The iterative method combined with the Laplace transform and Sumudu transform is employed to find the exact and approximate solutions of the fractional Navier–Stokes equation of a one-dimensional problem of unsteady flow of a viscous fluid in a tube. In the domains of science and engineering, these methods work well for solving a wide range of linear and nonlinear fractional partial differential equations and provide numerical solutions in terms of power series, with terms that are simple to compute and that quickly converge to the exact solution. After obtaining the solutions using these methods, we use Mathematica software Version 13.0.1.0 to present them graphically. We create two- and three-dimensional plots of the obtained solutions at various values of β and manipulate other variables to visualize and model relationships between the variables. We observe that as the fractional order β becomes closer to the integer order 1, the solutions approach the exact solution. Lastly, we plot a 2D graph of the first-, second-, third-, and fourth-term approximations of the series solution and observe from the graph that as the number of iterations increases, the approximate solutions become close to the series solution of the fourth-term approximation.

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.

Model-free adaptive consensus design for a class of unknown heterogeneous nonlinear multi-agent systems with packet dropouts

This paper studies the consensus problem for a class of unknown heterogeneous nonlinear multi-agent systems via a network with random packet dropouts. Based on the dynamic linearization technique, novel model-free adaptive consensus protocols with the data compensation mechanism are designed for both leaderless and leader-following cases. The advantage of this approach is that only neighborhood input and output data of the agents are required in the protocol design. For the stability analysis, a new Squeeze Theorem based method is developed to derive the theoretic results instead of the traditional contraction mapping principle used in model-free adaptive control. It is shown that the consensus can be achieved for both leaderless and leader-following cases if the communication topology is strongly connected. Finally, numerical simulations verifying the correctness of the theoretical results are given.

RESULTS ON IMPULSIVE ψ-CAPUTO FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS WITH BOUNDARY CONDITIONS

In this research article, we investigate the existence and uniqueness of solutions for a type of boundary value problems involving fractional integro-differential equations with the ψ-Caputo fractional derivative. Our method utilizes several classical fixed point theorems, such as the Banach contraction principle and Krasnoselskii's fixed point theorem. We establish our main results through theoretical analysis and present a specific case study to illustrate the practical significance of our findings.