Lipschitz Continuous Research Articles

A fractal-fractional sex structured syphilis model with three stages of infection and loss of immunity with analysis and modeling

Treponema pallidum, a spiral-shaped bacterium, is responsible for the sexually transmitted disease syphilis. Millions of people in less developed countries are getting the disease despite the accessibility of effective preventative methods like condom use and effective and affordable treatment choices. The disease can be fatal if the patient does not have access to adequate treatment. Prevalence has hovered between endemic levels in industrialized countries for decades and is currently rising. Using the Mittag-Leffler kernel, we develop a fractal-fractional model for the syphilis disease. Qualitative as well as quantitative analysis of the fractional order system are performed. Also, fixed point theory and the Lipschitz condition are used to fulfill the criteria for the existence and uniqueness of the exact solution. We illustrate the system’s Ulam–Hyers stability for disease-free and endemic equilibrium. The analytical solution is supported by numerical simulations that show how the dynamics of the spread of syphilis within the population are influenced by fractional-order derivatives. The outcomes show that the suggested methods are effective in delivering better results. Overall, this research helps to develop more precise and comprehensive approaches to understanding and regulating syphilis disease transmission and progression.

Secant-inexact projection algorithms for solving a new class of constrained mixed generalized equations problems

In this paper, a new version of a secant-type method for solving constrained mixed generalized equations is addressed. The method is a combination of the secant method applied to generalized equations with the conditional gradient method. We use the contraction mapping principle to establish the convergence results. Moreover, by assuming the Lipschitz condition on the gradient and the metric regularity property, we show that the sequence generated by the proposed algorithm is well-defined and locally convergent for a solution with linear or superlinear rate.

Theoretical analysis of divide-and-conquer ERM: From the perspective of multi-view

Theoretical analysis of the divide-and-conquer based distributed learning with least square loss in the reproducing kernel Hilbert space (RKHS) have recently been explored within the framework of learning theory. However, the studies on learning theory for general loss functions and hypothesis spaces remain limited. In order to comprehend the properties and behaviors of distributed ERM, we introduce multi-view learning to explore it at a more profound level. In this work, we adopt an innovative perspective by examining the distributed ERM from multi-view, and study the risk performance of distributed ERM for general loss functions and hypothesis spaces. The main theoretical results are two-fold. First, we derive two tight risk bounds under certain basic assumptions on the hypothesis space, as well as the smoothness, Lipschitz continuity, strong convexity of the loss function. Second, we further develop two more general risk bounds for distributed ERM without the restriction of strong convexity. The present work not only bridges the knowledge deficit in learning theory of distributed ERM for general loss functions and hypothesis spaces, from multi-view sight, it also shows that under certain conditions the number of views can guarantee the performance.

Effects of Deterministic Measurement Noise on Barrier Function-Based Adaptive Sliding-Mode Control

The effect of deterministic measurement noise on barrier function-based adaptive sliding mode controllers (BFASMC) is analyzed. For general bounded deterministic noise signals, it is impossible to select the controller parameters to track some perturbation with unknown bound. Nonetheless, if the assumption of continuity of the noise is added, the tracking of such a perturbation is possible whenever the barrier function width depends on the bound of the noise. If Lipschitz continuity of the noise is assumed, then the barrier function width can be chosen arbitrarily.

An embedding observer for nonlinear dynamical systems with global convergence

AbstractState observers for nonlinear systems are often designed for a canonical form of this system. However, this form may possess singular points, where the vector field is not defined or a Lipschitz condition is not fulfilled. This unpleasant behavior can possibly be avoided using an embedding into a higher dimensional space. A construction of such an embedding and the corresponding inverse map is discussed for polynomial systems using methods from algebraic geometry.

Notes on pseudodifferential operators commutators and Lipschitz functions

Abstract This article focuses on the boundedness of the commutators generated by pseudodifferential operators with Lipschitz functions and obtains a sufficient condition such that these operators are bounded from L p ( R n ) {L}^{p}\left({{\bf{R}}}^{n}) into the homogeneous Triebel-Lizorkin space F ˙ p γ , ∞ ( R n ) {\dot{F}}_{p}^{\gamma ,\infty }\left({{\bf{R}}}^{n}) and from L p ( R n ) {L}^{p}\left({{\bf{R}}}^{n}) into L q ( R n ) {L}^{q}\left({{\bf{R}}}^{n}) .

Robust machine learning modeling for predictive control using Lipschitz-Constrained Neural Networks

Neural networks (NNs) have emerged as a state-of-the-art method for modeling nonlinear systems in model predictive control (MPC). However, the robustness of NNs, in terms of sensitivity to small input perturbations, remains a critical challenge for practical applications. To address this, we develop Lipschitz-Constrained Neural Networks (LCNNs) for modeling nonlinear systems and derive rigorous theoretical results to demonstrate their effectiveness in approximating Lipschitz functions, reducing input sensitivity, and preventing over-fitting. Specifically, we first prove a universal approximation theorem to show that LCNNs using SpectralDense layers can approximate any 1-Lipschitz target function. Then, we prove a probabilistic generalization error bound for LCNNs using SpectralDense layers by using their empirical Rademacher complexity. Finally, the LCNNs are incorporated into the MPC scheme, and a chemical process example is utilized to show that LCNN-based MPC outperforms MPC using conventional feedforward NNs in the presence of training data noise.

Generalized Lagrange Theorem

The present paper is devoted to possible generalizations of the classic Lagrange Mean Value Theorem. We consider a real-valued function of several variables that is only assumed to be continuous. The main concept is to replace the notion of the derivative by the so called bisequential tangent cone. We first prove Rolle and Lagrange type results and then we turn to comparing this cone with the Clarke subdifferential in the case of a Lipschitz function. We also investigate an approach using normal cones.

Lipschitz Continuity for Harmonic Functions and Solutions of the α¯-Poisson Equation

In this paper we investigate the solutions of the so-called α¯-Poisson equation in the complex plane. In particular, we will give sufficient conditions for Lipschitz continuity of such solutions. We also review some recently obtained results. As a corollary, we can restate results for harmonic and (p,q)-harmonic functions.

Normal families and quasiregular mappings

AbstractBeardon and Minda gave a characterization of normal families of holomorphic and meromorphic functions in terms of a locally uniform Lipschitz condition. Here, we generalize this viewpoint to families of mappings in higher dimensions that are locally uniformly continuous with respect to a given modulus of continuity. Our main application is to the normality of families of quasiregular mappings through a locally uniform Hölder condition. This provides a unified framework in which to consider families of quasiregular mappings, both recovering known results of Miniowitz, Vuorinen and others and yielding new results. In particular, normal quasimeromorphic mappings, Yosida quasiregular mappings and Bloch quasiregular mappings can be viewed as classes of quasiregular mappings which arise through consideration of various metric spaces for the domain and range. We give several characterizations of these classes and obtain upper bounds on the rate of growth in each class.

Fractional Riesz–Feller type derivative for the one dimensional Dunkl operator and Lipschitz condition

In this paper, utilizing the Dunkl transform and a generalized translation operator, we derive an analogue of the Riesz-Feller fractional derivative for a class of functions satisfying Lipschitz conditions.

Bifurcation and Stability Analysis of a New Fractional-Order Prey–Predator Model with Fear Effects in Toxic Injections

This paper proposes a prey–predator model affected by fear effects and toxic substances. We used the Lipschitz condition to prove the uniqueness of the model solution and Laplace transform to prove the boundedness of the model solution. We used the fractional-order stability theorem to provide sufficient conditions for the local stability of equilibrium points, and selected fractional-order derivatives as parameters to perform Hopf bifurcation analysis on the system. Finally, the theoretical results are verified via numerical simulation. The results show that a value of α will affect the stability of the system and that the population size and the effect of toxic substances have a huge impact on the stability of the system.

Triebel–Lizorkin regularity and bi-Lipschitz maps: Composition operator and inverse function regularity

We study the stability of Triebel–Lizorkin regularity of bounded functions and Lipschitz functions under bi-Lipschitz changes of variables and the regularity of the inverse function of a Triebel–Lizorkin bi-Lipschitz map in Lipschitz domains. To obtain the results we provide an equivalent norm for the Triebel–Lizorkin spaces with fractional smoothness in uniform domains in terms of the first-order difference of the last weak derivative available averaged on balls.

Duality arguments in the analysis of a viscoelastic contact problem

We consider a mathematical model which describes the quasistatic frictionless contact of a viscoelastic body with a rigid-plastic foundation. We describe the mechanical assumptions, list the hypotheses on the data and provide three different variational formulations of the model in which the unknowns are the displacement field, the stress field and the strain field, respectively. These formulations have a different structure. Nevertheless, we prove that they are pairwise dual of each other. Then, we deduce the unique weak solvability of the contact problem as well as the Lipschitz continuity of its weak solution with respect to the data. The proofs are based on recent results on history-dependent variational inequalities and inclusions. Finally, we present numerical simulations in the study of the contact problem, together with the corresponding mechanical interpretations.

A necessary and sufficient condition for the global existence of solutions to nonlinear reaction‐diffusion equations on the half‐spaces in ℝN

In this paper, we study the existence and nonexistence of the global solutions to nonlinear reaction‐diffusion equations where is the half‐space , is a nonnegative continuous function, and is a locally Lipschitz function with some additional properties. The purpose of this paper is to give a necessary and sufficient condition for the existence of global solutions as follows: There is no global solution for any nonnegative and nontrivial initial data if and only if for every . In fact, we introduce a very special curve in to obtain the lower bound of decay of the heat semigroup, which is essential to prove the main result.

Random periodic solutions of SDEs: Existence, uniqueness and numerical issues

The aim of this paper is to discuss existence and uniqueness of random periodic solutions to stochastic differential equations (SDEs) with multiplicative noise under a one-sided Lipschitz condition, as well as on their numerical approximation via two classes of stochastic θ-methods, i.e., θ-Maruyama methods with θ∈[1/2,1] and θ-Milstein ones with θ∈[0,1]. The existence of the random periodic solutions as the limit of the pull-back flows of the discretized SDEs and the strong convergence rate of the aforementioned methods are also investigated. Selected numerical experiments confirming the theoretical analysis are also given.

Leader-following cluster-delay consensus control for first-order nonlinear multi-agent systems based on event-triggered mechanism

In this paper, the event-triggered-based cluster-delay consensus control problem is investigated for leader-following nonlinear multi-agent systems (MASs). In control design, both state triggering in the agents’ network and constant time-delay in the leaders’ communication network are considered. Under the framework of Lyapunov function stability theory, the Lipschitz condition is used to overcome the influence caused by time-delay. In order to further effectively utilize data transmission resources, and reduce the communication load of the topology network between each agent, an event-triggered mechanism is established. Subsequently, a robust cluster consensus control strategy is proposed based on event-triggered mechanism, which can ensure all signals of the controlled system are bounded, and the tracking errors converge to zero. In addition, it can also effectively avoid the Zeno behavior. Finally, the effectiveness of the presented control method and theory is verified by a simulation example.

On the localization of fractal discontinuity lines from noisy data

We consider the ill-posed problem of localizing (finding the position of) the discontinuity lines of a function of two variables: the function is smooth outside the discontinuity lines, and at each point on the line it has a discontinuity of the first kind. We construct averaging procedures and study global discrete regularizing algorithms for approximating discontinuity lines. Lipschitz conditions are imposed on the discontinuity lines. A parametric family of fractal lines is constructed, for which all conditions can be checked analytically. A fractal is indicated that has a large fractal dimension, for which the efficiency of the constructed methods can be guaranteed.

Formula omitted] error estimates of unsymmetric RBF collocation for second order quasilinear elliptic equations

The paper proves the convergence of unsymmetric radial basis functions collocation for second order quasilinear elliptic equations. L2 error is obtained based on the kernel-based trial spaces generated by the compactly supported radial basis functions. For the unsymmetric collocation case, we obtain the convergence result when the testing discretization is finer than the trial discretization. The convergence rates depend on the regularity of the solution, the Lipschitz continuity of the Frechét derivative of quasilinear operator, the smoothness of the computing domain, and the approximation of scaled kernel-based spaces.

Numerical Analysis of Split-Step Backward Euler Method with Truncated Wiener Process for a Stochastic Susceptible-Infected-Susceptible Model.

This article deals with the numerical positivity, boundedness, convergence, and dynamical behaviors for stochastic susceptible-infected-susceptible (SIS) model. To guarantee the biological significance of the split-step backward Euler method applied to the stochastic SIS model, the numerical positivity and boundedness are investigated by the truncated Wiener process. Motivated by the almost sure boundedness of exact and numerical solutions, the convergence is discussed by the fundamental convergence theorem with a local Lipschitz condition. Moreover, the numerical extinction and persistence are initially obtained by an exponential presentation of the stochastic stability function and strong law of the large number for martingales, which reproduces the existing theoretical results. Finally, numerical examples are given to validate our numerical results for the stochastic SIS model.