Arzela-Ascoli Research Articles

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

Theoretical and Numerical Studies of Fractional Volterra-Fredholm Integro-Differential Equations in Banach Space

This paper examines the theoretical, analytical, and approximate solutions of the Caputo fractional Volterra-Fredholm integro-differential equations (FVFIDEs). Utilizing Schaefer's fixed-point theorem, the Banach contraction theorem and the Arzel\`{a}-Ascoli theorem, we establish some conditions that guarantee the existence and uniqueness of the solution. Furthermore, the stability of the solution is proved using the Hyers-Ulam stability and Gronwall-Bellman's inequality. Additionally, the Laplace Adomian decomposition method (LADM) is employed to obtain the approximate solutions for both linear and non-linear FVFIDEs. The method's efficiency is demonstrated through some numerical examples.

Numerical analysis on fuzzy fractional human liver model using a novel double parametric approach

Abstract This paper introduces a fractal-fractional order model of the human liver (FFOHLM) incorporating a new fractional derivative operator with a generalized exponential kernel, specifically addressing uncertainties. The study delves into verifying the uniqueness and existence of this fuzzy FOHLM using Schauder’s Banach fixed point theorem and the Arzela-Ascoli theorem. It also investigates the fuzzy FOHLM using fixed-point theory and the Picard-Lindelof approach. Moreover, the research analyzes the stability and equilibrium points of the proposed model. To conduct this analysis, the study employs an innovative approach based on a double parametric generalized Adams-Bashforth technique within Newton’s polynomial framework. The numerical results of the proposed fuzzy FOHLM are validated by comparing them with real-world clinical data and other published results, and it shows that the fractal-fractional technique can yield greater efficacy and stimulation compared to the fractional operator when applied to epidemic simulations. Finally, the results of fractional fractal orders are illustrated graphically in a fuzzy environment.

Controllability of switched integro‐neutral fractional delay systems over arbitrary time domain

In this paper, we examine various qualitative properties for some class of fractional impulsive switched integro‐neutral dynamic systems over arbitrary time domain. We establish the existence, stability, and controllability of the scrutinized models. This article is divided into three parts: The first part is concerned with the existence and uniqueness, the second part is dedicated to the Hyers–Ulam stability, and in the third part, we examine the controllability analysis. We establish our results via the Banach and Leray–Schauder fixed‐point approaches along with the Arzela–Ascoli theorem. To show the implications and effectiveness of established results, we present a simulated numerical example at the end.

Inverse a time-dependent potential problem of a generalized time-fractional super-diffusion equation with a nonlinear source from a nonlocal integral observation

This paper focus on the problem of reconstructing the time-dependent potential in a class of generalized (including three special cases: the classical/multi-term/distributed order) time-fractional super-diffusion equations with nonlinear sources from a nonlocal integral observation. For such nonlinear equation, we investigate it for both the direct and inverse time-dependent potential problems. For the direct problem, given the time-dependent potential function p(t), we obtain the well-posedness of the corresponding initial–boundary value problem. For the inverse potential problem, by utilizing additional nonlocal measurement data and using the Arzelà–Ascoli theorem and Grönwall’s inequality, we prove the existence and uniqueness of the solution for such nonlinear problem. Meanwhile, we also demonstrate the ill-posedness of the inverse problem. Moreover, to validate the theoretical results, we numerically reconstruct the potential term from Bayesian perspective. Several numerical examples are presented to show the efficiency of the proposed method.

Delay volterra integrodifferential models of fractional orders and exponential kernels: Well-posedness theoretical results and Legendre–Galerkin shifted approximations

The utilized research work studies the well-posedness theoretical results and Galerkin-shifted approximations of delay Volterra integrodifferential models of fractional orders and exponential kernels in linear and nonlinear types in the Caputo–Fabrizio sense. Schauder’s and Arzela–Ascoli’s theorems are employed to prove a local existence theorem. After that, the Laplace continuous transform is employed to establish and confirm the global well-posedness theoretical results for the presented delay model. For numerical solutions, the Legendre–Galerkin shifted approximations’ algorithm has been used by the dependence on the orthogonal Legendre shifted polynomials to find out the required approximations. Utilizing this scheme, the considered delay model is reduced to an algebraic system with initial constraints. The algorithm precision, error behavior, and convergence are studied. Several numerical experiments together with several tables and figures are included to present the performance and validation of the algorithm presented. At last, some consequences and notes according to the given consequences were offered in the final section with several suggestions to guide future action.

Invariant measures of stochastic delay complex Ginzburg-Landau equations

This paper is concerned with the existence of invariant measures for stochastic delay complex Ginzburg-Landau equations defined on the entire integer set. When the nonlinear drift and diffusion terms are globally Lipschitz continous, the existence of invariant measures of the equation is proved by estiblishing the tightness of probability distributions of solutions in the space of continuous functions from a finite interval to an infinite-dimensional space, based on the idea of uniform tail-estimates, the technique of diadic division and the Arzela-Ascoli theorem.

Impulsive fractional dynamic equation with non-local initial condition on time scales

In this manuscript we investigate the existence and uniqueness of an im-pulsive fractional dynamic equation on time scales involving non-local initial condition with help of Caputo nabla derivative. The existency is based on the Scheafer’s fixed point theorem along with the Arzela-Ascoli theorem and Banach contraction theorem. The comparison of the Caputo nabla derivative and Riemann-Liouvile nabla derivative of fractional order are also discussed in the context of time scale.

On an m-dimensional system of quantum inclusions by a new computational approach and heatmap

Recent research indicates the need for improved models of physical phenomena with multiple shocks. One of the newest methods is to use differential inclusions instead of differential equations. In this work, we intend to investigate the existence of solutions for an m-dimensional system of quantum differential inclusions. To ensure the existence of the solution of inclusions, researchers typically rely on the Arzela–Ascoli and Nadler’s fixed point theorems. However, we have taken a different approach and utilized the endpoint technique of the fixed point theory to guarantee the solution’s existence. This sets us apart from other researchers who have used different methods. For a better understanding of the issue and validation of the results, we presented numerical algorithms, tables, and some figures. The paper ends with an example.

Qualitative dynamical study of hybrid system of Pantograph equations with nonlinear p-Laplacian operator in Banach’s space

This study provides a comprehensive exploration of the qualitative analysis of a hybrid system of pantograph equations with fractional order and a p-Laplacian operator. The existence of the solution of the system is explicitly established within the context of Riemann–Liouville’s fractional order operator, employing the Arzelà–Ascoli theorem for validation. The establishment of uniqueness criteria is accomplished by the utilization of the Banach contractive technique. In addition, the examination of solution stability is conducted using the Hyers–Ulam (HU) stability technique. In order to enhance the credibility of our main conclusions, we have included a representative and illustrative example in the concluding section of the study. This work serve to offer a thorough and applicable comprehension of the mathematical framework that has been proposed.

Controllability of Hilfer Fractional Semilinear Integro-Differential Equation of Order α ∊ (0, 1), β ∊ [0, 1

In this paper, Hilfer fractional differential equation is studied. Firstly, we used Laplace transform and semigroup theory to find the mild solution of the system. Then exact controllability of proposed system is established using the Arzela-Ascoli theorem and Schauder fixed point theorem. To illustrate the developed theory we provide an example at the end.

Complex-valued neural networks with time delays in the [formula omitted] sense: Numerical simulations and finite time stability

A discrete fractional-order complex-valued neural network is taken into consideration in the present study. For the existence of the solution of the considered model to be stable in finite time, certain requirements are specified. Our strategy focuses on the use of the recently formulated discrete fractional calculus, mathematical inequalities, Krasnoselskii’s fixed point theorem, and the Arzelà–Ascoli theorem. We present afew numerical examples that demonstrate the theoretical results’ implementation.

Stability analysis for a fractional coupled Hybrid pantograph system with p-Laplacian operator

In this article, we study the qualitative analysis of a fractional coupled hybrid pantograph system with a p-Laplacian operator. We developed the criteria for the existence of solutions for the proposed system with boundary conditions in the context of Caputo’s operator. In addition, existence and uniqueness criteria are carried out with the help of Banach’s and Arzelá Ascoli theorems by contractive approach. Furthermore, the Hyers–Ulam stability method was applied for stability analysis. For the sake of the reliability of the main results, we provided some examples at the end of the article for illustrative purposes.

Analysis of the Solution of a Model of SARS-CoV-2 Variants and Its Approximation Using Two-Step Lagrange Polynomial and Euler Techniques

In this paper, a vaccination model for SARS-CoV-2 variants is proposed and is studied using fractional differential operators involving a non-singular kernel. It is worth mentioning that variability in transmission rates occurs because of the particular population that is vaccinated, and hence, the asymptomatic infected classes are classified on the basis of their vaccination history. Using the Banach contraction principle and the Arzela–Ascoli theorem, existence and uniqueness results for the proposed model are presented. Two different numerical approaches, the fractional Euler and Lagrange polynomial methods, are employed to approximate the model’s solution. The model is then fitted to data associated with COVID-19 deaths in Pakistan between 1 January 2022 and 10 April 2022. It is concluded that our model is much aligned with the data when the order of the fractional derivative ζ=0.96. The two different approaches are then compared with different step sizes. It is observed that they behave alike for small step sizes and exhibit different behaviour for larger step sizes. Based on the numerical assessment of the model presented herein, the impact of vaccination and the fractional order are highlighted. It is also noted that vaccination could remarkably decrease the spikes of different emerging variants of SARS-CoV-2 within the population.

Asymptotic Behavior of Solution for Coupled Reaction Diffusion System by Order m

The aim of this paper is to prove that asymptotic behavior in the time of solutions for the weakly coupled reaction diffusion system:∂ui/∂t − di∆ui = fi (u1, u2, …, um) in Ω×R+,∂ui/∂η = 0 in ∂Ω×R+, (0.1)ui(., 0) = ui0(.) in Ω,where Ω is an open bounded domain of class C1 in Rn, ui(t, x), i=1, m, t≥0, x∈Ω are real valued functions. We treat the system (0.1) as a dynamical system in C(Ω) × C(Ω) × ... × C(Ω) and apply Lyapunov type stability techniques. A key ingredient in this analysis is a result which establishes that the orbits of the dynamical system are precompact in C(Ω) × C(Ω) × ... × C(Ω). As a consequence of Arzela-Ascoli theorem, this will be satisfied if the orbits are, for example, uniformly bounded in C1(Ω) × C1(Ω) × ... × C1(Ω) for t>0.

Problem on piecewise Caputo-Fabrizio fractional delay differential equation under anti-periodic boundary conditions

This manuscript considers a class of piecewise differential equations (DEs) modeled with the Caputo-Fabrizio differential operator. The proposed problem involves a proportional delay term and is equipped with anti-periodic boundary conditions. The piecewise derivative can be applied to model many complex nature real-world problems that show a multi-step behavior. The existence theory and Hyer-Ulam (HU) stability results are studied for the proposed problem via fixed point techniques such as Banach contraction theorem, Schauder’s fixed point theorem and Arzelá Ascoli theorem. A numerical problem is presented as an example to see the validity and effectiveness of the applied concept.

Existence of a weak solution for a nonlinear parabolic problem with fractional derivates

The primary objective of this study was to demonstrate the existence and uniqueness of a weak solution for a nonlinear parabolic problem with fractional derivatives for the spatial and temporal variables on a one-dimensional domain. Using the Nehari manifold method and its relationship with the Fibering maps, the existence of a weak solution for the stationary case was demonstrated. Finally, using the Arzela-Ascoli theorem and Banach's fixed point theorem, the existence and uniqueness of a weak solution for the nonlinear parabolic problem were shown.

Endograph Metric and a Version of the Arzelà–Ascoli Theorem for Fuzzy Sets

In this paper, we provide several Arzelà–Ascoli-type results on the space of all continuous functions from a Tychonoff space X into the fuzzy sets of Rn, (FUSCB(Rn),Hend), which are upper semi-continuous and have bounded support endowed with the endograph metric. Namely, we obtain positive results when X is considered to be a kr-space and C(X,(FUSCB(Rn),Hend)) is endowed with the compact open topology, as well as when we assume that X is pseudocompact and C(X,(FUSCB(Rn),Hend)) is equipped with the uniform topology.

Invariant measures for stochastic delay nonlocal lattice dynamical systems

This paper deals with the dynamics of the stochastic delay lattice systems with fractional discrete Laplacian driven by nonlinear noise. The existence of a invariant measure for the stochastic delay nonlocal lattice systems is established under certain conditions. The idea of the uniform estimates on the tails of the solutions, the technique of diadic division and the Arzela-Ascoli theorem are employed to show the tightness of a family of probability distributions of the solutions.

Controllability of a generalized multi-pantograph system of non-integer order with state delay

<abstract><p>This paper presents the dynamical aspects of a nonlinear multi-term pantograph-type system of fractional order. Pantograph equations are special differential equations with proportional delays that are employed in many scientific disciplines. The pantograph mechanism, for instance, has been applied in numerous scientific disciplines like electrodynamics, engineering, and control theory. Because of its key rule in diverse fields, the current study establishes some necessary criteria for its controllability. The main idea of the proof is based on converting the system into a fixed point problem and introducing a suitable controllability Gramian matrix $ \mathcal{G}_{c} $. The Gramian matrix $ \mathcal{G}_{c} $ is used to demonstrate the linear system's controllability. Controllability criteria for the associated nonlinear system have been established in the sections that follow using the Schaefer fixed-point theorem and the Arzela-Ascoli theorem, as well as the controllability of the linear system and a few key assumptions. Finally, a computational example is listed.</p></abstract>