Lipschitz Continuous Research Articles

Applying fixed point techniques to solve fractional differential inclusions under new boundary conditions

<abstract><p>Many scholars have lately explored fractional-order boundary value issues with a variety of conditions, including classical, nonlocal, multipoint, periodic/anti-periodic, fractional-order, and integral boundary conditions. In this manuscript, the existence and uniqueness of solutions to sequential fractional differential inclusions via a novel set of nonlocal boundary conditions were investigated. The existence results were presented under a new class of nonlocal boundary conditions, Carathéodory functions, and Lipschitz mappings. Further, fixed-point techniques have been applied to study the existence of results under convex and non-convex multi-valued mappings. Ultimately, to support our findings, we analyzed an illustrative example.</p></abstract>

Weighted Milne-type inequalities through Riemann-Liouville fractional integrals and diverse function classes

<abstract><p>This research paper investigated weighted Milne-type inequalities utilizing Riemann-Liouville fractional integrals across diverse function classes. A key contribution lies in the establishment of a fundamental integral equality, facilitated by the use of a nonnegative weighted function, which is pivotal for deriving the main results. The paper systematically proved weighted Milne-type inequalities for various function classes, including differentiable convex functions, bounded functions, Lipschitzian functions, and functions of bounded variation. The obtained results not only contribute to the understanding of Milne-type inequalities but also offer insights that pave the way for potential future research in the considered topics. Furthermore, it is evident that the results obtained encompass numerous findings that were previously presented in various studies as special cases.</p></abstract>

On approximating properties of blended variant of Kantorovich-Stancu type Lupaş operators

We introduce blended variant of Kantorovich-Stancu type Lupa? operators and study convergence and q-statistical convergence properties using Korovkin theorem. We investigate rate of convergence interms of modulus of continuity, Peetre?s K-functional and Lipschitz functions. Some Direct results and Voronovskaja type theorem are established. Moreover, graphical analysis and error estimations are presented using MATLAB.

Lipschitz p-Approximate Schauder Frames

With the aim of representing subsets of Banach spaces as an infinite series using Lipschitz functions, we study a variant of metric frames which we call Lipschitz p-approximate Schauder frames (Lipschitz p-ASFs). We characterize Lipschitz p-ASFs and their duals completely using the canonical Schauder basis for classical sequence spaces. Similarity of Lipschitz p-ASF is introduced and characterized.

RESEARCH ON FRACTAL DIMENSIONS AND THE HÖLDER CONTINUITY OF FRACTAL FUNCTIONS UNDER OPERATIONS

Based on the previous studies, we make further research on how fractal dimensions of graphs of fractal continuous functions under operations change and obtain a series of new results in this paper. Initially, it has been proven that a positive continuous function under unary operations of any nonzero real power and the logarithm taking any positive real number that is not equal to one as the base number can keep the fractal dimension invariable. Then, a general method to calculate the Box dimension of two continuous functions under binary operations has been proposed. Using this method, the lower and upper Box dimensions of the product and the quotient of continuous functions without zero points have been investigated. On this basis, these conclusions will be generalized to the ring of rational functions. Furthermore, we discuss the Hölder continuity of continuous functions under operations and then prove that a Lipschitz function can be absorbed by any other continuous functions under certain binary operations in the sense of fractal dimensions. Some elementary results for vector-valued continuous functions have also been given.

Approximation by the modified $ \lambda $-Bernstein-polynomial in terms of basis function

<abstract><p>In this article by means of shifted knots properties, we introduce a new type of coupled Bernstein operators for Bézier basis functions. First, we construct the operators based on shifted knots properties of Bézier basis functions then investigate the Korovkin's theorem, establish a local approximation theorem, and provide a convergence theorem for Lipschitz continuous functions and Peetre's $ K $-functional. In addition, we also obtain an asymptotic formula of the type Voronovskaja.</p></abstract>

Finite-in-time flocking of the thermodynamic Cucker–Smale model

<abstract><p>We illustrate finite-in-time flocking in the thermodynamic Cucker–Smale (TCS) model. First, we extend the original TCS model to allow for a continuous vector field with a locally Lipschitz continuity. Then, within this system, we derive appropriate dissipative inequalities concerning the position-velocity-temperature using several preparatory estimates. Subsequently, based on initial data and system parameters, we formulate sufficient conditions to guarantee the desired finite-time flocking in each case where the communication weight conditions are divided into two scenarios: one with a positive lower bound and another with nonnegativity and monotonicity. Finally, we provide several numerical simulations and compare them with the analytical results.</p></abstract>

Unsteady MHD Free Convection Flow of Fluid in Micro Channel in the Presence of Viscous Dissipation Between Vertical Parallel Two Walls

Convection is a process by which heat is transferred by movement of a heated water. The phenomenon where heat transfer process is driven by natural fluid motion due to temperature and density gradients is called free convection. The effect of slip velocity on thermal behaviours of MHD free convection flow over a vertical parallel two walls is the focus of this research. The mathematical equations which described the phenomenom includes momentum, magnetic field and energy equations. Condition for the existence and uniqueness of solution of the model equations were established using the approach of Lipschitz continuity. The model equations in dimensionless form were solved using Olayiwola’s generalized polynomial approximation method (OGPAM). The results obtained which contains the dimensionless parameters were presented graphically and discussed. It was observed that increase in Reynolds number and Knudsen number both enhanced the velocity distribution at the two walls. Also, magnetic field and temperature field were enhanced by magnetic prandtl number and Eckert number respectively.

Kernel Masked Image Modeling Through the Lens of Theoretical Understanding.

Masked image modeling (MIM) has been considered as the state-of-the-art (SOTA) self-supervised learning (SSL) technique in terms of visual pretraining. The impressive generalization ability of MIM also paves the way for the remarkable success of large-scale vision foundation models. In this article, we further discuss the validity and advantages of implementing MIM techniques in the reproducing kernel Hilbert spaces (RKHSs) and we associate the analysis with a novel MIM method named R-MIM (short for RKHS-MIM). Through the careful construction of an augmentation graph and by using spectral decomposition techniques, we establish a systematic theoretical understanding between the proposed R-MIM's generalization ability and the choice of kernel function used during training. Specifically, we reach a conclusion that both of the local Lipschitz constant of the resultant R-MIM model and the corresponding expected pretraining error can have a strong composite effect on bounding downstream task error, depending on the kernel options. We demonstrate that under mild mathematical assumptions, R-MIM method is guaranteed to return a lower bound on downstream tasks in comparison to vanilla MIM techniques, such as masked autoencoder (MAE) and SimMIM. Empirical justification well corroborates our theoretical hypothesis and analysis in showing the superior generalization of the proposed R-MIM and the theoretical link to kernel choices. The code is available at: https://github.com/yurui-q/R-MIM.

Distributed Nash Equilibrium Seeking With Dynamics Subject to Disturbance of Unknown Frequencies Over Jointly Strongly Connected Switching Networks

In this paper, we study the problem of distributed Nash equilibrium seeking of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> -player games with single integrator dynamics subject to disturbances generated by an uncertain exosystem. Similar problems have been studied for games with single integrator dynamics subject to disturbances generated by a known exosystem. In addition, this paper offers some other new features. First, the existing results only apply to static, connected, and undirected graphs while the result of this paper applies to jointly strongly connected switching graphs, which can be disconnected at every time instant. Second, the existing results are asymptotic or ultimately uniformly bounded, while our result guarantees the exponential convergence when the exosystem is known. Third, the validity of existing results relies on an inequality involving some Lipschitz constant and the smallest nonzero eigenvalue of the Laplacian of the network graph. In contrast, the result of this paper is unconditional other than satisfying the same assumptions on the cost functions as those in the literature. Our design is illustrated by an application to velocity-actuated robots in sensor networks.

A new double inertial subgradient extragradient method for solving a non-monotone variational inequality problem in Hilbert space

In this paper, we introduced a new double inertial subgradient extragradient method for solving a variational inequality problem in Hilbert space. In our method, the mapping needed not to satisfy any assumption of monotonicity and two different self-adaptive step sizes were used for avoiding the need of Lipschitz constant of the mapping. The strong convergence of the proposed method was proved under some new conditions. Finally, some numerical examples were presented to illustrate the convergence of our method and compare with some related methods in the literature.

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

&lt;abstract&gt;&lt;p&gt;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.&lt;/p&gt;&lt;/abstract&gt;

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.

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

&lt;p&gt;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.&lt;/p&gt;

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.

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.