Picard-Lindelof Research Articles

Mathematical modelling of COVID-19 outbreak using caputo fractional derivative: stability analysis

The novel coronavirus SARS-Cov-2 is a pandemic condition and poses a massive menace to health. The governments of different countries and their various prohibitory steps to restrict the virus's expanse have changed individuals' communication processes. Due to physical and financial factors, the population's density is more likely to interact and spread the virus. We establish a mathematical model to present the spread of the COVID-19 in worldwide. In this article, we propose a novel mathematical model (‘ S L I I q I h R P ’) to assess the impact of using hospitalization, quarantine measures, and pathogen quantity in controlling the COVID-19 pandemic. We analyse the boundedness of the model's solution by employing the Laplace transform approach to solve the fractional Gronwall's inequality. To ensure the uniqueness and existence of the solution, we rely on the Picard-Lindelof theorem. The model's basic reproduction number, a crucial indicator of epidemic potential, is determined based on the greatest eigenvalue of the next-generation matrix. We then employ stability theory of fractional differential equations to qualitatively examine the model. Our findings reveal that both locally and globally, the endemic equilibrium and disease-free solutions demonstrate symptomatic stability. These results shed light on the effectiveness of the proposed interventions in managing and containing the COVID-19 outbreak.

Transmission dynamics of breast cancer through Caputo Fabrizio fractional derivative operator with real data

<abstract><p>In this paper, we studied the dynamical behavior of various phases of breast cancer using the Caputo Fabrizio (CF) fractional order derivative operator. The Picard-Lindelof (PL) method was used to investigate the existence and uniqueness of the proposed system. Moreover, we investigated the stability of the system in the sense of Ulam Hyers (UH) criteria. In addition, the two-step Adams-Bashforth (AB) technique was employed to simulate our methodology. The fractional model was then simulated using real data, which includes reported breast cancer incidences among females of Saudi Arabia from 2004 to 2016. The real data was used to determine the values of the parameters that were fitted using the least squares method. Also, residuals were computed for the integer as well as fractional-order models. Based on the results obtained, the CF model's efficacy rates were greater than those of the existing classical model. Graphical representations were used to illustrate numerical results by examining different choices of fractional order parameters, then the dynamical behavior of several phases of breast cancer was quantified to show how fractional order affects breast cancer behavior and how chemotherapy rate affects breast cancer behavior. We provided graphical results for a breast cancer model with effective parameters, resulting in fewer future incidences in the population of phases Ⅲ and Ⅳ as well as the disease-free state. Chemotherapy often raises the risk of cardiotoxicity, and our proposed model output reflected this. The goal of this study was to reduce the incidence of cardiotoxicity in chemotherapy patients while also increasing the pace of patient recovery. This research has the potential to significantly improve outcomes of patients and provide information of treatment strategies for breast cancer patients.</p></abstract>

Monotone Inclusions, Acceleration, and Closed-Loop Control

We propose and analyze a new dynamical system with a closed-loop control law in a Hilbert space [Formula: see text], aiming to shed light on the acceleration phenomenon for monotone inclusion problems, which unifies a broad class of optimization, saddle point, and variational inequality (VI) problems under a single framework. Given an operator [Formula: see text] that is maximal monotone, we propose a closed-loop control system that is governed by the operator [Formula: see text], where a feedback law [Formula: see text] is tuned by the resolution of the algebraic equation [Formula: see text] for some [Formula: see text]. Our first contribution is to prove the existence and uniqueness of a global solution via the Cauchy–Lipschitz theorem. We present a simple Lyapunov function for establishing the weak convergence of trajectories via the Opial lemma and strong convergence results under additional conditions. We then prove a global ergodic convergence rate of [Formula: see text] in terms of a gap function and a global pointwise convergence rate of [Formula: see text] in terms of a residue function. Local linear convergence is established in terms of a distance function under an error bound condition. Further, we provide an algorithmic framework based on the implicit discretization of our system in a Euclidean setting, generalizing the large-step hybrid proximal extragradient framework. Even though the discrete-time analysis is a simplification and generalization of existing analyses for a bounded domain, it is largely motivated by the aforementioned continuous-time analysis, illustrating the fundamental role that the closed-loop control plays in acceleration in monotone inclusion. A highlight of our analysis is a new result concerning [Formula: see text]-order tensor algorithms for monotone inclusion problems, complementing the recent analysis for saddle point and VI problems. Funding: This work was supported in part by the Mathematical Data Science Program of the Office of Naval Research [Grant N00014-18-1-2764] and by the Vannevar Bush Faculty Fellowship Program [Grant N00014-21-1-2941].

Modeling and analysis of the dynamics of HIV/AIDS with non-singular fractional and fractal-fractional operators

In the present study, we analyze the dynamics of HIV/AIDS epidemic model with fractional and the new definition of a nonlocal fractal-fractional derivative in the Atangana-Baleanu-Caputo (ABC) sense. The proposed model is initially formulated with the help of classical integer-order differential equations and then the ABC fractional derivative is used to obtain the fractional HIV/AIDS model with arbitrary order p 1. Then we present some basic properties including equilibria and the basic reproduction number of the model. The local and global stability of the fractional model at disease free equilibrium is shown and it is found that the model is stable when the threshold quantity is less than 1. Further, an efficient numerical approach is applied to present an iterative scheme for the fractional model and present graphical results for various values of p 1. After this, we make use of the new fractal-fractional operator to formulate the HIV/AIDS model in the ABC sense. The uniqueness and existence of the solutions are shown through the Picard-Lindelof theorem. Furthermore, we derive an iterative scheme of the fractal-fractional ABC model using a novel numerical method based on Adams-Bashforth fractal-fractional approachand depicts detailed graphical results for many values of fractional and fractal orders p 1 and p 2 respectively. The simulation results show that the HIV/AIDS model with the novel fractal-fractional operator provides biologically more feasible analysis than that of the classical fractional and integer-order models.

On the Existence and Uniqueness of the ODE Solution and Its Approximation Using the Means Averaging Approach for the Class of Power Electronic Converters

Power electronic converters are mathematically represented by a system of ordinary differential equations discontinuous right-hand side that does not verify the conditions of the Cauchy-Lipschitz Theorem. More generally, for the properties that characterize their discontinuous behavior, they represent a particular class of systems on which little has been investigated over the years. The purpose of the paper is to prove the existence of at least one global solution in Filippov’s sense to the Cauchy problem related to the mathematical model of a power converter and also to calculate the error in norm between this solution and the integral of its averaged approximation. The main results are the proof of this theorem and the analytical formulation that provides to calculate the cited error. The demonstration starts by a proof of local existence provided by Filippov himself and already present in the literature for a particular class of systems and this demonstration is generalized to the class of electronic power converters, exploiting the non-chattering property of this class of systems. The obtained results are extremely useful for estimating the accuracy of the averaged model used for analysis or control of the effective system. In the paper, the goodness of the analytical proof is supported by experimental tests carried out on a converter prototype representing the class of power electronics converter.

A Study of Banach Fixed Point Theorem and It’s Applications

This paper aims at treating a study of Banach fixed point theorem for mapping results that introduced in the setting of normed space. The classical Banach fixed point theorem is a generalization of this work. A fixed point theory is a beautiful mixture of Mathematical analysis to explain some conditions in which maps give excellent solutions. Here later many mathematicians used this fixed point theory to establish their results, see for instance, Picard-Lindel of Theorem, The Picard theorem, Implicit function theorem etc. Also, we developed ideas that many of known fixed point theorems can easily be derived from the Banach theorem. It extends some recent works on the extension of Banach contraction principle to metric space with norm spaces.

Stability of a Regularized Newton Method with Two Potentials

In a Hilbert space setting, we introduce dynamical systems, which are linked to Newton and Levenberg-Marquardt methods. They are intended to solve, by splitting methods, inclusions governed by structured monotone operators M = A + B, where A is a general maximal monotone operator, and B is monotone and locally Lipschitz continuous. Based on the Minty representation of A as a Lipschitz manifold, we show that these dynamics can be formulated as differential systems, which are relevant to the Cauchy-Lipschitz theorem, and involve separately B and the resolvents of A. In the convex subdifferential case, by using Lyapunov asymptotic analysis, we prove a descent minimizing property, and weak convergence to equilibria of the trajectories. Time discretization of these dynamics gives algorithms combining Newton’s method and forward-backward methods for solving structured monotone inclusions. The Levenberg-Marquardt regularization term acts in an open loop way. As a byproduct of our study, we can take the regularization coefficient of bounded variation. These stability results are directly related to the study of numerical algorithms that combine forward-backward and Newton’s methods.

Regularity of Lagrangian flows over RCD*(K, N) spaces

Abstract The aim of this note is to provide regularity results for Regular Lagrangian flows of Sobolev vector fields over compact metric measure spaces verifying the Riemannian curvature dimension condition. We first prove, borrowing some ideas already present in the literature, that flows generated by vector fields with bounded symmetric derivative are Lipschitz, providing the natural extension of the standard Cauchy–Lipschitz theorem to this setting. Then we prove a Lusin-type regularity result in the Sobolev case (under the additional assumption that the m.m.s. is Ahlfors regular) therefore extending the already known Euclidean result.

A Formalization of Metric Spaces in HOL Light

We present a computer formalization of metric spaces in the HOL Light theorem prover. Basic results of the theory of complete metric spaces are provided, including the Banach Fixed-Point Theorem, the Baire Category Theorem and the completeness of the space of continuous bounded functions. A decision procedure for a fragment of the elementary theory of metric spaces is also implemented. As an application, the Picard–Lindelof theorem on the existence of the solutions of ordinary differential equations is proved by using the well-known argument which appeals to the Banach theorem.

Differential Equations for Closed Sets in a Banach Space

Nonempty subsets of a vector space suggest themselves for describing shapes (without restrictions of regularity) or states (together with their deterministic uncertainty). Hence, there have been developed several suggestions how to extend differential and integral equations respectively to time-dependent subsets although subsets do not form a linear space in an obvious way. In this article, we summarize three central approaches which can handle possibly non-convex closed subsets: integral funnel solution (a.k.a. R-solutions), morphological primitives, and reachable sets (generalizing Aumann integrals). They are extended to a separable Banach space and characterized by means of semilinear evolution inclusions.We formulate conditions sufficient for the equivalence of these generalized concepts and, then this joint basis is used for specifying differential equations for closed (not necessarily convex or compact) subsets. Several further approaches in the literature prove to be special cases. In this purely metric setting, the counterpart of the Picard–Lindelof theorem (a.k.a. Cauchy–Lipschitz theorem) ensures the existence and uniqueness of set-valued solutions to initial value problems.

A nonlinear transport problem of hyperbolic-elliptic type

A nonlinear transport problem of hyperbolic-elliptic type is studied. Using estimates of potentials over varying domains and the method of characteristics, the initial value problem for Holder continuous data can be treated as an abstract evolution equation via Picard-Lindelof theorem. In addition, a global existence result is proved. Modifications of these techniques yield shock front solutionswith smooth interfaces and an extension to the case of only bounded initial values. Moreover, long time behavior of solutions and the relation to diffeomorphisms with prescribed Jacobian determinants are analyzed.

Analysis and approximation of frictionless contact problems between two piezoelectric bodies with adhesion

Abstract We consider a mathematical frictionless contact problem between two electro-elastic bodies. The contact is modelled with normal compliance and adhesion. We provide a variational formulation for the problem and prove the existence of a unique weak solution. The proofs are based on arguments of time-dependent variational inequalities, the Cauchy–Lipschitz Theorem and the Banach Fixed-Point Theorem. Then, a discrete scheme is introduced based on the finite element method to approximate the spatial variable. Furthermore, we provide optimal a priori error estimates for the displacements, the electric potential and the bonding at the contact interface.

Similarity solutions for non-Newtonian power-law fluid flow

The problem of the boundary layer flow of power law non-Newtonian fluids with a novel boundary condition is studied. The existence and uniqueness of the solutions are examined, which are found to depend on the curvature of the solutions for different values of the power law index n. It is established with the aid of the Picard-Lindelof theorem that the nonlinear boundary value problem has a unique solution in the global domain for all values of the power law index n but with certain conditions on the curvature of the solutions. This is done after a suitable transformation of the dependent and independent variables. For 0 1, the solution has a negative or zero curvature on some part of the global domain. Some solutions are presented graphically to illustrate the results and the behaviors of the solutions.

General Cauchy–Lipschitz theory for Δ-Cauchy problems with Carathéodory dynamics on time scales

The aim of this paper was to complete some aspects of the classical Cauchy–Lipschitz (or Picard–Lindelöf) theory for general nonlinear systems posed on time scales. Despite a rich literature on Cauchy–Lipschitz type results on time scales, most of the existing results are concerned with rd-continuous dynamics (and -solutions) and do not cover the framework of general Carathéodory dynamics encountered for instance in control theory with measurable controls (which are not rd-continuous in general). In this paper, our main objective was to study -Cauchy problems with general Carathéodory dynamics. We introduce the notion of absolutely continuous solution (weaker regularity than ) and then the notion of maximal solution. We state and prove a Cauchy–Lipschitz theorem, providing existence and uniqueness of the maximal solution of a given -Cauchy problem under suitable assumptions such as regressivity and local Lipschitz continuity. Three new related issues are also discussed in this paper: the boundary value is not necessarily an initial or a final condition, the solutions are constrained to take their values in a non-empty open subset and the behaviour of maximal solutions at terminal points is studied.

Generating and Adding Flows on Locally Complete Metric Spaces

As a generalization of a vector field on a manifold, the notion of an arc field on a locally complete metric space was introduced in \cite{BC}. In that paper, the authors proved an analogue of the Cauchy-Lipschitz Theorem i.e they showed the existence and uniqueness of solution curves for a time independent arc field. In this paper, we extend the result to the time dependent case, namely we show the existence and uniqueness of solution curves for a time dependent arc field. We also introduce the notion of the sum of two time dependent arc fields and show existence and uniqueness of solution curves for this sum.

A Continuous Dynamical Newton-Like Approach to Solving Monotone Inclusions

We introduce nonautonomous continuous dynamical systems which are linked to the Newton and Levenberg–Marquardt methods. They aim at solving inclusions governed by maximal monotone operators in Hilbert spaces. Relying on the Minty representation of maximal monotone operators as lipschitzian manifolds, we show that these dynamics can be formulated as first-order in time differential systems, which are relevant to the Cauchy–Lipschitz theorem. By using Lyapunov methods, we prove that their trajectories converge weakly to equilibria. Time discretization of these dynamics gives algorithms providing new insight into Newton's method for solving monotone inclusions.

Nonlocal stochastic differential equations: Existence and uniqueness of solutions

The focus of interest is the existence of strong solutions to stochastic functional differential equations which are not restricted to pathwise dependencies. Indeed, the evolution of the wanted random process may be prescribed by its current features being nonlocal with respect to the probability space — like the expected value and second moments. This result is concluded from Cauchy-Lipschitz Theorem for mutational equations (a form of generalized ODEs beyond vector spaces) and a new aspect of weakening their a priori requirements.

Viscoelastic sliding contact problems with wear

We consider a mathematical model which describes the sliding contact with wear between a viscoelastic body and a rigid moving foundation. We consider both the dynamic and quasi-static cases and we model the wear with a version of Archard's law. We derive the variational formulation of the model and prove existence and uniqueness results. The proofs are based on arguments of evolution equations with monotone operators and Banach's fixed-point theorem, in the case of the dynamical model, and on Cauchy-Lipschitz theorem in the case of the quasi-static model. We also establish the continuous dependence of the solution with respect to parameters related to the velocity of the moving foundation.

Generating Flows on Metric Spaces

The notion of an arc field on a locally complete (but not necessarily locally compact) metric space is introduced as a generalization of a vector field on a manifold. Generalizing the Cauchy–Lipschitz Theorem, sufficient conditions on arc fields are given under which the existence and uniqueness of solution curves and flows are proven. A continuous analog of an iterated function system is given as an example.

Mutational equations in metric spaces

This paper summarizes an extension of differential calculus to a mutational calculus for maps from one metric space to another. The simple idea is to replace half-lines allowing to define difference quotients of maps and their various limits in the case of vector space by ‘transitions’ with which we can also define differential quotients of a map. Their various limits are called ‘mutations’ of a map. Many results of differential calculus and set-valued analysis, including the Inverse Function Theorem, do not really rely on the linear structure and can be adapted to the nonlinear case of metric spaces and exploited. Furthermore, the concept of differential equation can be extended tomutational equation governing the evolution in metric spaces. Basic Theorems as the Nagumo Theorem, the Cauchy-Lipschitz Theorem, the Center Manifold Theorem and the second Lyapunov Method hold true for mutational equations.