Lipschitz Continuous Research Articles

Double inertial extrapolations method for solving split generalized equilibrium, fixed point and variational inequity problems

<abstract><p>This article proposes an iteration algorithm with double inertial extrapolation steps for approximating a common solution of split equilibrium problem, fixed point problem and variational inequity problem in the framework of Hilbert spaces. Unlike several existing methods, our algorithm is designed such that its implementation does not require the knowledge of the norm of the bounded linear operator and the value of the Lipschitz constant. The proposed algorithm does not depend on any line search rule. The method uses a self-adaptive step size which is allowed to increase from iteration to iteration. Furthermore, using some mild assumptions, we establish a strong convergence theorem for the proposed algorithm. Lastly, we present a numerical experiment to show the efficiency and the applicability of our proposed iterative method in comparison with some well-known methods in the literature. Our results unify, extend and generalize so many results in the literature from the setting of the solution set of one problem to the more general setting common solution set of three problems.</p></abstract>

MODIFIED SHRINKING TARGET PROBLEM FOR MATRIX TRANSFORMATIONS OF TORI

In this paper, we calculate the Hausdorff dimension of the fractal set [Formula: see text] where [Formula: see text] is the standard [Formula: see text]-transformation with [Formula: see text], [Formula: see text] is a positive function on [Formula: see text] and [Formula: see text] is the usual metric on the torus [Formula: see text]. Moreover, we investigate a modified version of the shrinking target problem, which unifies the shrinking target problems and quantitative recurrence properties for matrix transformations of tori. Let [Formula: see text] be a [Formula: see text] non-singular matrix with real coefficients. Then, [Formula: see text] determines a self-map of the [Formula: see text]-dimensional torus [Formula: see text]. For any [Formula: see text], let [Formula: see text] be a positive function on [Formula: see text] and [Formula: see text] with [Formula: see text]. We obtain the Hausdorff dimension of the fractal set [Formula: see text] where [Formula: see text] is a hyperrectangle and [Formula: see text] is a sequence of Lipschitz vector-valued functions on [Formula: see text] with a uniform Lipschitz constant.

Re-Thinking the Effectiveness of Batch Normalization and Beyond.

Batch normalization (BN) is used by default in many modern deep neural networks due to its effectiveness in accelerating training convergence and boosting inference performance. Recent studies suggest that the effectiveness of BN is due to the Lipschitzness of the loss and gradient, rather than the reduction of internal covariate shift. However, questions remain about whether Lipschitzness is sufficient to explain the effectiveness of BN and whether there is room for vanilla BN to be further improved. To answer these questions, we first prove that when stochastic gradient descent (SGD) is applied to optimize a general non-convex problem, three effects will help convergence to be faster and better: (i) reduction of the gradient Lipschitz constant, (ii) reduction of the expectation of the square of the stochastic gradient, and (iii) reduction of the variance of the stochastic gradient. We demonstrate that vanilla BN only with ReLU can induce the three effects above, rather than Lipschitzness, but vanilla BN with other nonlinearities like Sigmoid, Tanh, and SELU will result in degraded convergence performance. To improve vanilla BN, we propose a new normalization approach, dubbed complete batch normalization (CBN), which changes the placement position of normalization and modifies the structure of vanilla BN based on the theory. It is proven that CBN can elicit all the three effects above, regardless of the nonlinear activation used. Extensive experiments on benchmark datasets CIFAR10, CIFAR100, and ILSVRC2012 validate that CBN makes the training convergence faster, and the training loss converges to a smaller local minimum than vanilla BN. Moreover, CBN helps networks with multiple nonlinear activations (Sigmoid, Tanh, ReLU, SELU, and Swish) achieve higher test accuracy steadily. Specifically, benefitting from CBN, the classification accuracies for networks with Sigmoid, Tanh, and SELU are boosted by more than 15.0%, 4.5%, and 4.0% on average, respectively, which is even comparable to the performance for ReLU.

RESEARCH ON THE K-DIMENSION OF THE SUM OF TWO CONTINUOUS FUNCTIONS AND ITS APPLICATION

In this paper, we have done some research studies on the fractal dimension of the sum of two continuous functions with different [Formula: see text]-dimensions and approximation of [Formula: see text]-dimensional fractal functions. We first investigate the [Formula: see text]-dimension of the linear combination of fractal function whose [Formula: see text]-dimension is [Formula: see text] and the function satisfying Lipschitz condition is still [Formula: see text]-dimensional. Then, based on the research of fractal term and the Weierstrass approximation theorem, an approximation of the [Formula: see text]-dimensional continuous function is given, which is composed of finite triangular series and partial Weierstrass function. In addition, some preliminary results on the approximation of one-dimensional and two-dimensional fractal continuous functions have been given.

Existence and uniqueness of solutions for stochastic differential equations with locally one-sided Lipschitz condition

<p>This paper investigated stochastic differential equations (SDEs) with locally one-sided Lipschitz coefficients. Apart from the local one-sided Lipschitz condition, a more general condition was introduced to replace the monotone condition. Then, in terms of Euler's polygonal line method, the existence and uniqueness of solutions for SDEs was established. In the meanwhile, the $ p $th moment boundedness of solutions was also provided.</p>

The Strong Convexity Supporting Condition and the Lipschitz Condition for the Metric Projection

В работе доказана типичность опорного условия сильной выпуклости для произвольного выпуклого компакта из $\mathbb R^n$. Показано, что для почти всех в определенном смысле точек метрическая проекция на выпуклый компакт удовлетворяет по точке условию Липшица с константой Липшица строго меньше 1. Последнее условие характеризует опорное условие сильной выпуклости. Доказана линейная сходимость метода альтернативных проекций для выпуклого компакта с опорным условием сильной выпуклости и проксимально гладкого множества при определенном соотношении между константами опорного условия сильной выпуклости и проксимальной гладкости. Библиография: 18 названий.

Existence and asymptotic behavior of square-mean $ S $-asymptotically periodic solutions of fractional stochastic evolution equations

This paper investigates the abstract fractional stochastic evolution equations. A new existence result of the square-mean $ S $-asymptotically periodic mild solutions are obtained under the assumption that the nonlinear terms only satisfy some growth conditions. Moreover, the uniqueness and asymptotic stability results of the square-mean $ S $-asymptotically periodic solution are presented when the nonlinear functions satisfy the general Lipschitz condition. Finally, two examples are given to illustrate our main results.

Semilocal convergence analysis for a three-step scheme in Banach spaces with a new type majorant: using average Lipschitz conditions

Semilocal convergence analysis for a three-step scheme in Banach spaces with a new type majorant: using average Lipschitz conditions

Observer-Based Distributed Adaptive Consensus Tracking of Nonlinear Multi-agent Systems on Directed Graphs

This paper studies the leader-following consensus problem of nonlinear multi-agent systems (MAS) with one-sided Lipschitz conditions under the directed graph, and a novel observer-based distributed tracking protocol that only utilizes the relative output information of adjacent agents is proposed. To avoid the disadvantage of solving Laplace matrix eigenvalues, a fully distributed adaptive consensus scheme is constructed. The sufficient conditions for linear matrix inequalities (LMIs) for the consensus of MAS systems are derived using the Lyapunov function and the properties of one-sided Lipschitz nonlinear systems. Finally, the effectiveness of the proposed control method can be verified by two examples-based simulations with a leader and five followers.

Investigation of SEIR model with vaccinated effects using sustainable fractional approach for low immune individuals

<abstract><p>Mathematical formulations are crucial in understanding the dynamics of disease spread within a community. The objective of this research is to investigate the SEIR model of SARS-COVID-19 (C-19) with the inclusion of vaccinated effects for low immune individuals. A mathematical model is developed by incorporating vaccination individuals based on a proposed hypothesis. The fractal-fractional operator (FFO) is then used to convert this model into a fractional order. The newly developed SEVIR system is examined in both a qualitative and quantitative manner to determine its stable state. The boundedness and uniqueness of the model are examined to ensure reliable findings, which are essential properties of epidemic models. The global derivative is demonstrated to verify the positivity with linear growth and Lipschitz conditions for the rate of effects in each sub-compartment. The system is investigated for global stability using Lyapunov first derivative functions to assess the overall impact of vaccination. In fractal-fractional operators, fractal represents the dimensions of the spread of the disease, and fractional represents the fractional ordered derivative operator. We use combine operators to see real behavior of spread as well as control of COVID-19 with different dimensions and continuous monitoring. Simulations are conducted to observe the symptomatic and asymptomatic effects of the corona virus disease with vaccinated measures for low immune individuals, providing insights into the actual behavior of the disease control under vaccination effects. Such investigations are valuable for understanding the spread of the virus and developing effective control strategies based on justified outcomes.</p></abstract>

The Lipschitz condition of the metric projection and the convergence of gradient methods

Рассмотрены разные опорные условия для замкнутого множества из вещественного гильбертова пространства $\mathcal H$ в точке границы множества. Указанные условия обеспечивают некоторое локальное условие Липшица метрического проектора точки на множество по точке. Также имеет место локальная липшицевость проектора в метрике Хаусдорфа как функции множества. Полученное условие Липшица применено для доказательства линейной сходимости ряда градиентных методов (метода проекции градиента, метода условного градиента) без предположения сильной выпуклости или даже выпуклости функции и без выпуклости множества. Функция при этом предполагается дифференцируемой с непрерывным по Липшицу градиентом. Библиография: 29 названий.

Local well-posedness results for the nonlinear fractional diffusion equation involving a Erdélyi-Kober operator

<p>In this paper, we study an initial boundary value problem of a nonlinear fractional diffusion equation with the Caputo-type modification of the Erdélyi-Kober fractional derivative. The main tools are the Picard-iteration method, fixed point principle, Mittag-Leffler function, and the embedding theorem between Hilbert scales spaces and Lebesgue spaces. Through careful analysis and precise calculations, the priori estimates of the solution and the smooth effects of the Erdélyi-Kober operator are demonstrated, and then the local existence, uniqueness, and stability of the solution of the nonlinear fractional diffusion equation are established, where the nonlinear source function satisfies the Lipschitz condition or has a gradient nonlinearity.</p>

Well-posedness and invariant measures for 2D stochastic Oldroyd model of order one with pure jumps

In this paper, we consider the stochastic two-dimensional viscoelastic fluid flow equations driven by a multiplicative Lévy noise, arising from the Oldroyd model for the non-Newtonian fluid flows. The system carries the natural Lipschitz condition on balls and linear growth assumptions on the jump coefficient. We first prove the existence and uniqueness of a global strong solution by using stopping time technique and Banach fixed point theorem. Meanwhile, we overcome the difficulties posed by the jump and memory term, the latter makes some convergence findings more natural. By demonstrating the exponential stability of strong solutions, we establish the existence of a unique invariant measure for the stochastic system. Numerical results provide further support for these theoretical findings, the dissipative velocity of solution is more rapid and rigorous when compared with the stochastic Navier-Stokes equation.

Mild solutions for fractional non-instantaneous impulses integro-differential equations with nonlocal conditions

<abstract><p>In this paper, we investigated Caputo fractional integro-differential equations with non-instantaneous impulses and nonlocal conditions. By employing the solution operator, the Mönch fixed point theorem, and the stepwise estimation method, we eliminated the Lipschitz condition of the nonlinear term, while also dispensing with the requirement for the compressibility coefficient condition $ 0 < k < 1 $. The main results presented represented a generalization and enhancement of previous findings. Furthermore, an example was provided to verify the application of our main results.</p></abstract>

Analysis of stochastic delay differential equations in the framework of conformable fractional derivatives

<abstract><p>In numerous domains, fractional stochastic delay differential equations are used to model various physical phenomena, and the study of well-posedness ensures that the mathematical models accurately represent physical systems, allowing for meaningful predictions and analysis. A fractional stochastic differential equation is considered well-posed if its solution satisfies the existence, uniqueness, and continuous dependency properties. We established the well-posedness and regularity of solutions of conformable fractional stochastic delay differential equations (CFrSDDEs) of order $ \gamma\in(\frac{1}{2}, 1) $ in $ \mathbb{L}^{\mathrm{p}} $ spaces with $ \mathrm{p}\geq2 $, whose coefficients satisfied a standard Lipschitz condition. More specifically, we first demonstrated the existence and uniqueness of solutions; after that, we demonstrated the continuous dependency of solutions on both the initial values and fractional exponent $ \gamma $. The second section was devoted to examining the regularity of time. As a result, we found that, for each $ \Phi\in(0, \gamma-\frac{1}{2}) $, the solution to the considered problem has a $ \Phi- $H$ \ddot o $lder continuous version. Lastly, two examples that highlighted our findings were provided. The two main elements of the proof were the Burkholder-Davis-Gundy inequality and the weighted norm.</p></abstract>

Stability Analysis of Fractional Bidirectional Associative Memory Neural Networks With Multiple Proportional Delays and Distributed Delays.

This article investigates the finite-time stability of a class of fractional-order bidirectional associative memory neural networks (FOBAMNNs) with multiple proportional and distributed delays. Different from the existing Gronwall integral inequality with single proportional delay ( N = 1 ), we establish the Gronwall integral inequality with multiple proportional delays for the first time in the case of N ≥ 2 . Since the existing fractional-order single-constant delay Gronwall inequality with two different orders cannot be directly applied to the stability analysis of the aforementioned system, initially, we skillfully develop a novel one with generalized fractional multiproportional delays' Gronwall inequalities of different orders. Furthermore, combined with the newly constructed generalized inequality, the stability criteria of FOBAMNNs with fractional orders and under weaker conditions, i.e., at most linear growth and linear growth conditions rather than the global Lipschitz condition, are given respectively. Finally, numerical experiments verify the effectiveness of the proposed method.

A study of strong convergence of differential equations based on Euler’s algorithm

Abstract Differential equations have important applications in many fields, such as chemistry, biology, epidemiology, and finance. Most analytic solutions of differential equations are difficult to obtain. Therefore numerical solutions of differential equations become an important tool. The truncated Euler method is proposed in this paper, and we investigate how the truncated EM solution of the derived SDDE converges strongly under the local Lipschitz condition and the one-sided linear growth condition after relaxation. The basic strong convergence theorem is set up and the new notation X(t,x;s) is introduced as an analytic solution of the stochastic differential equation. Establish the assumption that the coefficients of the drift term and the coefficients of the diffusion term of the stochastic differential equation satisfy a contractionary monotonicity condition in order to prove, by induction, that the exact solution of this stochastic differential equation is bounded for a long time. Different examples are given to compare the simulated deviations between the numerical and analytical solutions of Euler’s algorithm and the modified Euler’s algorithm for the initial value problem of fractional order differential equations and to analyze the convergence of the two numerical solutions. The truncated Euler method is applied to highly nonlinear time-transformed stochastic differential equations by means of the dyadic principle, and it is proved that the order of convergence of the strong convergence of the truncated Euler-Maruyama method for time-transformed stochastic differential equations is min ( α , γ , 1 2 − ɛ ) \min \left( {\alpha ,\gamma ,{1 \over 2} - \varepsilon } \right) .

Event-Triggered Finite-Time Stabilization of Fuzzy Neural Networks With Infinite Time Delays and Discontinuous Activations

This paper unifies the stability criteria of asymptotic, exponential and finite-time control within a single framework for fuzzy neural networks (FNNs) with infinite time delays. First, the boundedness and differentiability for time delays are removed. Then, Lipschitz condition for activation function is relaxed, which is allowed to have jumping discontinuous points. To stabilize FNNs, the analytical method is established by comparison principle, contradiction method and inequality techniques. Moreover, different from the traditional Lyapunov method and finite-time stability theorem, several sufficient conditions are deduced and the suppression functions are designed to guarantee asymptotic, exponential and finite-time stabilization for FNNs by adjusting the parameters of the same controller. There is not necessary to construct the complex integral-type Lyapunov functional to deal with infinite time delays and to design power function in controller for finite-time stabilization. In addition, the designed event-triggered mechanism (ETM) has the inherent advantages of saving communication resources and indirectly eliminates the chattering caused by signum function. Finally, simulations are presented to illustrate the feasibility and effectiveness of the theoretical results.

Local convergence study of tenth-order iterative method in Banach spaces with basin of attraction

<abstract><p>Many applications from computational mathematics can be identified for a system of non-linear equations in more generalized Banach spaces. Analytical methods do not exist for solving these type of equations, and so we solve these equations using iterative methods. We introduced a new numerical technique for finding the roots of non-linear equations in Banach space. The method is tenth-order and it is an extension of the fifth-order method which is developed by Arroyo et.al. <sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup>. We provided a convergence analysis to demonstrate that the method exhibits tenth-order convergence. Also, we discussed the local convergence properties of the suggested method which depends on the fundamental supposition that the first-order Fréchet derivative of the involved function $ \Upsilon $ satisfies the Lipschitz conditions. This new approach is not only an extension of prior research, but also establishes a theoretical concept of the radius of convergence. We validated the efficacy of our method through various numerical examples. Our method is comparable with the methods of Tao Y et al. <sup>[<xref ref-type="bibr" rid="b2">2</xref>]</sup>. We also compared it with higher-order iterative methods, and we observed that it either performs similarly or better for the numerical examples. We also gave the basin of attraction to demonstrate the behaviour in the complex plane.</p></abstract>

Milne-Type inequalities via expanded fractional operators: A comparative study with different types of functions

<abstract><p>This study focused on deriving Milne-type inequalities using expanded fractional integral operators. We began by establishing a key equality associated with these operators. Using this equality, we explored Milne-type inequalities for functions with convex derivatives, supported by an illustrative example for clarity. Additionally, we investigated Milne-type inequalities for bounded and Lipschitzian functions utilizing fractional expanded integrals. Finally, we extended our exploration to Milne-type inequalities involving functions of bounded variation.</p></abstract>