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The paper aims to propose a two-step iterated method, which is derived from a regularized dynamical system of extragradient-type in terms of time discretizing, for solving an equilibrium problem. We prove that the iterative sequence generated by the method converges strongly to a solution of the equilibrium problem. Some numerical experiments are given to illustrate and compare the behavior of the new method with several other methods. [ABSTRACT FROM AUTHOR]

In this paper, we propose two self-adaptive extragradient-like algorithms for solving pseudomonotone variational inequalities. We consider two cases: the mapping is Lipschitz continuous (with unknown modulus) and is not Lipschitz continuous. The step size in our algorithms can either increase or decrease at each step. This feature makes our algorithms more efficient. We also give some numerical examples to compare the performance of our algorithm with the existing one. [ABSTRACT FROM AUTHOR]

This paper presents two new algorithms for solving multivalued variational inequality problems in a real Hilbert space. By combining the nonexpansiveness of proximal operators associated with the proper lower semicontinuous convex function of the problems and inertial techniques, we demonstrate the weak convergence of the iteration sequences generated by our first algorithm under monotone and Lipschitz continuous assumptions of the cost mappings. Next, we use Mann iteration technique to obtain the second algorithm and show its strong convergence. Finally, we give some numerical results for two proposed algorithms and comparison with some other known algorithms. [ABSTRACT FROM AUTHOR]

Let X$X$ and Y$Y$ be complex Banach spaces with BX$B_X$ denoting the open unit ball of X$X$. This paper studies various aspects of the holomorphic Lipschitz spaceHL0(BX,Y)$\mathcal {H}L_0(B_X,Y)$, endowed with the Lipschitz norm. This space consists of the functions in the intersection of the sets Lip0(BX,Y)$\operatorname{Lip}_0(B_X,Y)$ of Lipschitz mappings and H∞(BX,Y)$\mathcal {H}^\infty (B_X,Y)$ of bounded holomorphic mappings, from BX$B_X$ to Y$Y$. Thanks to the Dixmier–Ng theorem, HL0(BX,C)$\mathcal {H}L_0(B_X, \mathbb {C})$ is indeed a dual space, whose predual G0(BX)$\mathcal {G}_0(B_X)$ shares linearization properties with both the Lipschitz‐free space and Dineen–Mujica predual of H∞(BX)$\mathcal {H}^\infty (B_X)$. We explore the similarities and differences between these spaces, and combine techniques to study the properties of the space of holomorphic Lipschitz functions. In particular, we get that G0(BX)$\mathcal {G}_0(B_X)$ contains a 1‐complemented subspace isometric to X$X$ and that G0(X)$\mathcal {G}_0(X)$ has the (metric) approximation property whenever X$X$ has it. We also analyze when G0(BX)$\mathcal {G}_0(B_X)$ is a subspace of G0(BY)$\mathcal {G}_0(B_Y)$, and we obtain an analog of Godefroy's characterization of functionals with a unique norm preserving extension in the holomorphic Lipschitz context. [ABSTRACT FROM AUTHOR]

We investigate some Bernstein–Gelfand–Gelfand complexes consisting of Sobolev spaces on bounded Lipschitz domains in R n . In particular, we compute the cohomology of the conformal deformation complex and the conformal Hessian complex in the Sobolev setting. The machinery does not require algebraic injectivity/surjectivity conditions between the input spaces, and allows multiple input complexes. As applications, we establish a conformal Korn inequality in two space dimensions with the Cauchy–Riemann operator and an additional third-order operator with a background in Möbius geometry. We show that the linear Cosserat elasticity model is a Hodge–Laplacian problem of a twisted de Rham complex. From this cohomological perspective, we propose potential generalizations of continuum models with microstructures. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate the optimality conditions for a class of nonsmooth multiobjective fractional optimization problems subject to vanishing constraints. In particular, necessary and sufficient conditions for (weak) Pareto solution are presented in terms of the Clark subdifferential. Furthermore, we construct Wolfe and Mond–Weir-type dual models and derive some duality theorems by using generalized quasiconvexity assumptions. Some examples to show the validity of our conclusions are provided. [ABSTRACT FROM AUTHOR]

In this paper, we considered the boundedness of Burkholder's martingale transforms for martingale Hardy spaces with variable exponents. In addition, through martingale transforms, some characterizations of predictable variable exponent martingale Hardy spaces were also provided. [ABSTRACT FROM AUTHOR]

We give necessary and sufficient conditions for the boundedness of the maximal commutators Mb$$ {M}_b $$ and the commutators of the maximal operator [b,M]$$ \left[b,M\right] $$ in total Morrey spaces Lp,λ,μ(ℝn)$$ {L}^{p,\lambda, \mu}\left({\mathrm{\mathbb{R}}}^n\right) $$ when b$$ b $$ belongs to Lipschitz spaces Λ˙β(ℝn)$$ {\dot{\Lambda}}_{\beta}\left({\mathrm{\mathbb{R}}}^n\right) $$, whereby some new characterizations for certain subclasses of Lipschitz spaces Λ˙β(ℝn)$$ {\dot{\Lambda}}_{\beta}\left({\mathrm{\mathbb{R}}}^n\right) $$ are obtained. [ABSTRACT FROM AUTHOR]

Operators of the form f ⟼ xβf(xα) are treated. Among other things, it is proved that such an operator acts on the class of operator Lipschitz functions on (0,+∞) tending to zero at zero if and only if α + β = 1. [ABSTRACT FROM AUTHOR]

We consider the long-term dynamics of the vanishing stepsize subgradient method in the case when the objective function is neither smooth nor convex. We assume that this function is locally Lipschitz and path differentiable, i.e., admits a chain rule. Our study departs from other works in the sense that we focus on the behavior of the oscillations, and to do this we use closed measures, a concept that complements the technique of asymptotic pseudotrajectories developed in this setting by Benaïm--Hofbauer--Sorin. We recover known convergence results, establish new ones, and show a local principle of oscillation compensation for the velocities. Roughly speaking, the time average of gradients around one limit point vanishes. Various cases are discussed, providing new insight into the oscillation and the stabilization phenomena. [ABSTRACT FROM AUTHOR]

Recently, a serious problem in communications is security. Hiding data is one of the most important security techniques. Steganography is the art and science of hiding information in a cover media. Texts are the most usual method of communication and so they are very suitable for cover objects. In this paper, we give a new technique for text steganography. There exist different models, such as RGB, HSL, HSV to determine a color. The main goal of the proposed algorithm is the fact that some different but very similar colors in RGB have the same code in HSL. We use this fact to hide data. For this purpose, first we find a color B′ in RGB, which is very similar to black in such a way that they have the same code in HSL. Then, by changing the color of each character of the text to black or B′, we conceal the information. We will show that the capacity of this method is better than some other methods of text steganography, and then we show that the invisibility of this algorithm is very high, which is the most prominent feature of the proposed technique. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a notion of Connes biprojectivity for a dual Banach algebra A with respect to its w∗-closed ideal I, say I-Connes biprojectivity. Some Lipschitz algebras Lipα(X) and some matrix algebras are studied under this new notion. Also, with some mild assumptions, the relation between IConnes biprojectivity and left ϕ-contractibility is given, where ϕ is a ω∗-continuous multiplicative linear functional on A. As an application, we characterize Connes biprojectivity of some Lipschitz algebras. [ABSTRACT FROM AUTHOR]

In this note, we investigate some new characterizations of the p-adic version of Lipschitz spaces via the boundedness of commutators of the p-adic maximal-type functions, including padic sharp maximal functions, p-adic fractional maximal functions, and p-adic fractional maximal commutators on p-adic Morrey spaces, when a symbol function b belongs to the Lipschitz spaces. [ABSTRACT FROM AUTHOR]

The aim of present article is to introduce the Szász-integral type sequences of operators in terms of Hermite polynomials and gamma function. Further, we calculate some estimates at test functions and central moments. Moreover, we discuss uniform convergence theorem and order of approximation via Korovkin theorem and first order modulus of smoothness respectively. Next, we study pointwise approximation results in view of Peetre's K-functional, second order modulus of smoothness and Lipschitz type space. Lastly, bivariate version of these sequences of operators are introduced. Moreover, their rate of convergence and order of approximation are investigated. [ABSTRACT FROM AUTHOR]

In this study, we take into account the of modified Szász–Mirakyan–Kantorovich operators to obtain their rate of convergence using the modulus of continuity and for the functions in Lipschitz space. Then, we obtain the statistical convergence of this form. In addition, we determine the weighted statistical convergence and compare it with the statistical one for the same operator. Medical applications and traditional mathematics; one way to get a close approximation of the Riemann integrable functions is through the use of the Kantorovich modification of positive linear operators. The use of Kantorovich operators is tremendously helpful from a medical point of view. Their application is shown as an approximation of the rate of convergence in respect of modulus of continuity. [ABSTRACT FROM AUTHOR]

We prove thet a Banach space X has the λ -bounded compact approximation property (respectively, weak λ -bounded approximation property) if and only if X has the λ -Lipschitz bounded compact approximation property (respectively, weak λ -Lipschitz bounded approximation property). Also, it is shown that the dual space of X has the approximation property if and only if for every separable reflexive Banach space Y, the space of finite rank Lipschitz maps from X to Y is dense in the Lipschitz norm in the space of compact operators from X to Y. [ABSTRACT FROM AUTHOR]

In this paper, we propose a parameterized three-operator splitting algorithm to solve nonconvex minimization problems with the sum of three non-convex functions, where two of them have Lipschitz continuous gradients. We establish the convergence of the proposed algorithm under the KurdykaŁojasiewicz assumption by constructing a suitable energy function with a non-increasing property. As applications, we employ the proposed algorithm to solve low-rank matrix recovery and image inpainting problems. Numerical results demonstrate the efficiency and effectiveness of the proposed algorithm compared to other algorithms. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate a Bregman projection algorithm for solving the split variational inequality problem governed by pseudomonotone and not necessarily Lipschitz continuous operators in real Hilbert spaces. The proposed algorithm is motivated by the ideas of the Halpern method, the CQ method, and Tseng's extragradient method. The step size sequences are determined by employing Armijo line search techniques. The strong convergence theorem is established without the prior knowledge of the operator norm and the Lipschitz continuous assumption on the operators involved. Some numerical experiments with graphical illustrations are presented to demonstrate the effectiveness and the performance of our proposed algorithm in comparison with some existing ones. [ABSTRACT FROM AUTHOR]

We deal with the boundedness properties of higher order commutators related to some generalizations of the multilinear fractional integral operator of order m, Iαm, from a product of weighted Lebesgue spaces into adequate weighted Lipschitz spaces, extending some previous estimates for the linear case. Our study includes two different types of commutators and suffi- cient conditions on the weights in order to guarantee the continuity properties described above. We also exhibit the optimal range of the parameters involved. The optimality is understood in the sense that the parameters defining the corresponding spaces belong to a certain region, being the weights trivial outside of it. We further show examples of weights for the class which cover the mentioned area. [ABSTRACT FROM AUTHOR]

In this paper, we consider the commutators associated with the fractional maximal function and sharp maximal function when symbol function b in Lipschitz spaces. We give some characterizations of the Lipschitz spaces via the boundedness of these commutators on total Morrey spaces. [ABSTRACT FROM AUTHOR]

The square root normal field (SRNF) framework is a method in the area of shape analysis that defines a (pseudo)distance between unparametrized surfaces. For piecewise linear surfaces it was recently proved that the SRNF distance between unparametrized surfaces is equivalent to the Wasserstein Fisher Rao (WFR) metric on the space of finitely supported measures on S2. In the present article we extend this point of view to a much larger set of surfaces; we show that the SRNF distance on the space of Lipschitz surfaces is equivalent to the WFR distance between Borel measures on S2. For the space of spherical surfaces this result directly allows us to characterize the noninjectivity and the (closure of the) image of the SRNF transform. In the last part of the paper we further generalize this result by showing that the WFR metric for general measure spaces can be interpreted as an optimization problem over the diffeomorphism group of an independent background space. [ABSTRACT FROM AUTHOR]

In this paper, we give necessary and sufficient conditions for a continuous on $ \mathbb R_+ $ R + function f to belong the symmetric generalized Lipschitz classes defined by the Mellin translation in terms of growth or decay of some integrals connected with the Mellin transform. The results are counterparts of ones for classical Lipschitz classes and the classical (complex) Fourier transform obtained earlier by the author. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to investigate characterizations of composition operators on Bloch and Hardy type spaces. Initially, we use general doubling weights to study the composition operators from harmonic Bloch type spaces on the unit disc D to pluriharmonic Hardy spaces on the Euclidean unit ball B n . Furthermore, we develop some new methods to study the composition operators from harmonic Bloch type spaces on D to pluriharmonic Bloch type spaces on D . Additionally, some application to new characterizations of the composition operators between pluriharmonic Lipschitz type spaces to be bounded or compact will be presented. The obtained results of this paper provide the improvements and extensions of the corresponding known results. [ABSTRACT FROM AUTHOR]

We consider weighted composition operators, that is operators of the type g ↦ w · g ∘ f , acting on spaces of Lipschitz functions. Bounded weighted composition operators, as well as some compact weighted composition operators, have been characterized quite recently. In this paper, we provide a different approach involving their pre-adjoint operators, namely the weighted Lipschitz operators acting on Lipschitz free spaces. This angle allows us to retrieve and sometimes improve some results from the literature. Notably, we obtain a distinct characterization of boundedness with a precise estimate of the norm. We also characterise injectivity, surjectivity, compactness and weak compactness in full generality. [ABSTRACT FROM AUTHOR]

In this article, we study the existence of mild solutions and approximate controllability for non-autonomous impulsive evolution equations with nonlocal conditions in Banach space. The existence of mild solutions and some conditions for approximate controllability of these non-autonomous impulsive evolution equations are given by using the Krasnoselskii's fixed point theorem, the theory of evolution family and the resolvent operator. In particular, the impulsive functions are supposed to be continuous and the nonlocal item is divided into Lipschitz continuous and completely bounded. An example is given as an application of the results. [ABSTRACT FROM AUTHOR]

This paper concerns the study singular integrals along twisted surfaces of the form We prove Lp bounds for the corresponding operators when the surfaces are defined by mappings more general than polynomials and convex functions, provided that the kernels are in L(logL)2(Sn-1x Sm-1) . [ABSTRACT FROM AUTHOR]

In this article, we examine a nonsmooth vector optimization problem with locally Lipschitz approximately convex mappings in terms of the convexificator and provide some ideas for approximate effective solutions. Additionally, we define the relationship between the convexificator-based solutions of Stampacchia type vector variational inequalities (V V I) and the approximate efficient approximation convex function of nonsmooth vector optimization problems using the locally Lipschitz function. Furthermore, we provide a numerical example to demonstrate the veracity of our findings. [ABSTRACT FROM AUTHOR]

In this paper we study the space of strongly Lipschitz (lp,lq)-factorable operators between metric spaces and a Banach spaces. In particular, a factorization of this class through lp and lq spaces is given. We show that this type of operators fits in the theory of composition a-Banach Lipschitz operator ideal. As a special case, we get a Lipschitz version of weakly p-nuclear operators. [ABSTRACT FROM AUTHOR]

The aim of this paper is to prove a generalization of Titchmarsh’s theorems for the generalized Fourier transform called ( κ , n {\kappa,n} )-Fourier transform, where n is a positive integer and κ is a constant coming from Dunkl theory. As an application, we derive a ( κ , n ) {(\kappa,n)} -Fourier multiplier theorem for L 2 {L^{2}} Lipschitz spaces. Moreover, we give necessary conditions to ensure that f belongs to either one of the generalized Lipschitz classes of order m. This allows us to establish the analogue of the Boas-type result for ℱ κ , n {\mathcal{F}_{\kappa,n}} . [ABSTRACT FROM AUTHOR]

In this article we show, in a concise manner, a result of uniform in time propagation of chaos for non exchangeable systems of particles interacting according to a random graph. Provided the interaction is Lipschitz continuous, the restoring force satisfies a general one-sided Lipschitz condition (thus allowing for non-convex confining potential) and the graph is dense enough, we use a coupling method suggested by Eberle (2016) known as reflection coupling to obtain uniform in time mean-field limit with bounds that depend explicitly on the graph structure. [ABSTRACT FROM AUTHOR]

In this paper we present and prove some new results concerning the rate of approximation of T means of functions in Lp(G) and in lip (α, p), for 1 ≤ p < ∞ and α > 0. As a corollary, we obtain some new as well as known approximation inequalities. [ABSTRACT FROM AUTHOR]

In this article, John-Nirenberg inequality for Lipschitz martingale spaces is established. We further prove that Lipschitz martingale spaces Λp(α) are equivalent for 0 < p < ∞, which generalizes an important result in classical martingale Hp theory. [ABSTRACT FROM AUTHOR]

We prove that if M is an infinite complete metric space, then the set of strongly norm-attaining Lipschitz functions SNA.M / contains a linear subspace isomorphic to c0. This solves an open question posed by V. Kadets and Ó. Roldán. [ABSTRACT FROM AUTHOR]

Let $ \mathcal{L} = -\Delta+V $ be the Schrödinger operators on $ \mathbb{R}^n $ with nonnegative potential $ V $ belonging to the reverse Hölder class $ RH_q $ for some $ q \geq \frac{n}{2} $. We prove the boundedness of fractional integral operator $ \mathcal{I}_\alpha $ related to the Schrödinger operators $ \mathcal{L} $ from strong and weak variable exponent Lebesgue spaces into suitable variable exponent Lipschitz type spaces.

We derive new representations for the generalised Jacobian of a locally Lipschitz map between finite dimensional real Euclidean spaces as the lower limit (i.e., limit inferior) of the classical derivative of the mapwhere it exists. The new representations lead to significantly shorter proofs for the basic properties of the subgradient and the generalised Jacobian including the chain rule. We establish that a sequence of locally Lipschitz maps between finite dimensional Euclidean spaces converges to a given locally Lipschitz map in the L-topology--that is, theweakest refinement of the sup norm topology on the space of locally Lipschitz maps that makes the generalised Jacobian a continuous functional--if and only if the limit superior of the sequence of directional derivatives of the maps in a given vector direction coincides with the generalised directional derivative of the given map in that direction, with the convergence to the limit superior being uniform for all unit vectors. We then prove our main result that the subspace of Lipschitz C8 maps between finite dimensional Euclidean spaces is dense in the space of Lipschitz maps equipped with the L-topology, and, for a given Lipschitz map, we explicitly construct a sequence of Lipschitz C8 maps converging to it in the L-topology, allowing global smooth approximation of a Lipschitz map and its differential properties. As an application, we obtain a short proof of the extension of Green's theorem to interval-valued vector fields. For infinite dimensions, we show that the subgradient of a Lipschitz map on a Banach space is upper continuous, and, for a given real-valued Lipschitz map on a separable Banach space, we construct a sequence of Gateaux differentiable functions that converges to the map in the sup norm topology such that the limit superior of the directional derivatives in any direction coincides with the generalised directional derivative of the Lipschitz map in that direction. [ABSTRACT FROM AUTHOR]

We study the well-posedness of the initial value problem and the long-time behavior of almost periodic solutions to stochastic scalar conservation laws in any space dimension, under the assumption of Lipschitz continuity of the flux functions and a non-degeneracy condition. We show the existence and uniqueness of an invariant measure in a separable subspace of the space of Besicovitch almost periodic functions. [ABSTRACT FROM AUTHOR]

In this paper, we study the boundedness of p-adic Riesz potential on two-weighted Morrey–Herz spaces with both power weight and Muckenhoupt weight. Moreover, the boundedness for commutators of p-adic Riesz potential with symbols in Lipschitz spaces and weighted central BMO spaces on such spaces are also established. [ABSTRACT FROM AUTHOR]

We establish a factorization theorem for the Hardy space in terms of a multilinear fractional integral operator. As an application, a characterization of Lipschitz spaces via the boundedness of commutators of the multilinear fractional integral operator is obtained. [ABSTRACT FROM AUTHOR]

This paper addressed the problem of observer-based memory state feedback control design for semi-Markovian jump systems subject to input delays and external disturbances, where the measurement output was vulnerable to randomly occurring cyber attacks. To facilitate analysis, the cyber attacks were described by a nonlinear function that meets Lipschitz continuity and the possible attack scenarios were represented by a stochastic parameter that follows the Bernoulli distribution. Based on the information from the considered system and state observer, an augmented closed loop system was constructed. Then, by using the Lyapunov stability theory, an extended Wirtinger's integral inequality and stochastic analysis, the required stability criterion was proposed in the form of linear matrix inequalities. As a result, the control and observer gain matrices were efficiently derived, ensuring the stochastic stability of closed-loop systems with H ∞ performance, regardless of cyber attacks. To demonstrate the effectiveness and theoretical value of the proposed robust memory state feedback control design, simulation results were presented. [ABSTRACT FROM AUTHOR]

Let b → = (b 1 , b 2 , ... , b m) be a collection of locally integrable functions and T Σ b → the commutator of multilinear singular integral operator T. Denote by L (δ) and L (δ (⋅)) the Lipschitz spaces and the variable Lipschitz spaces, respectively. The main purpose of this paper is to establish some new characterizations of the (variable) Lipschitz spaces in terms of the boundedness of multilinear commutator T Σ b → in the context of the variable exponent Lebesgue spaces, that is, the authors give the necessary and sufficient conditions for bj (j = 1, 2, ..., m) to be L (δ) or L (δ (⋅)) via the boundedness of multilinear commutator from products of variable exponent Lebesgue spaces to variable exponent Lebesgue spaces. The authors do so by applying the Fourier series technique and some pointwise estimate for the commutators. The key tool in obtaining such pointwise estimate is a certain generalization of the classical sharp maximal operator. [ABSTRACT FROM AUTHOR]

In this paper, we give definitions of p-energy forms on homogenous p.c.f. self-similar sets and point that the domains of non-localp-energy form, localp-energy form are Lipschitz spaces B p , p σ , B p , ∞ σ respectively. By constructing equivalent semi-norms of p-energy forms, we obtain the convergence of the B p , p σ -norms to the B p , ∞ σ ∗ -norm as σ↑σ∗, where the critical exponentσ∗ is the supremum of σ such that B p , ∞ σ ∩ C (K) is dense in C (K) . [ABSTRACT FROM AUTHOR]

In this paper, we propose an inertial shrinking Tseng's extragradient algorithm with a self-adaptive step size for solving pseudomonotone variational inequality problem with non-Lipschitz operators in the framework of 2-uniformly convex Banach spaces which are also uniformly smooth. Moreover, we prove a strong convergence result for the proposed algorithm under mild conditions on the control parameters. The main advantages of our algorithm are: our proposed algorithm solves the variational inequality problem with a larger class of mappings (pseudomonotone and non-Lipschitz operators); unlike the existing results in the literature, our algorithm does not require any linesearch technique even while the operator is non-Lipschitz; minimized number of projections per iteration compared to related results in the literature; and the inertial technique employed which speeds up the rate of convergence. Finally, we present some numerical examples to illustrate the efficiency of our algorithm in comparison with related methods in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of generalized split feasibility problem over the solution set of monotone variational inclusion problems, which includes the split common null point problem, the split variational problems, and other related split type problems. We propose a new self-adaptive stepsize method coupled with inertial extrapolation techniques for solving this problem in real Hilbert spaces. We prove that the sequence generated by the proposed method converges strongly to a solution of the problem under the assumption that the associated singlevalued operator in the monotone variational inclusion problem is not required to be inverse-strongly monotone. Our method uses the stepsizes that are generated at each iteration by some simple calculations, which allows it to be easily implemented without the prior knowledge of the operator norm or the Lipschitz constant of the singlevalued operator. Finally, we apply our results to solve optimal control problems, split linear inverse problems, and least absolute selection and shrinkage operator problems. [ABSTRACT FROM AUTHOR]

In this paper, we propose a Mann type self-adaptive Tseng's extragradient method for solving the classical variational inequality problem with a Lipschitz continuous and quasi-monotone mapping in a real Hilbert space. The strong convergence of the proposed algorithm is proven without the prior knowledge of the Lipschitz constant of the corresponding function. Finally, we give some numerical examples to illustrate the superiority of our proposed algorithm. [ABSTRACT FROM AUTHOR]

Let p ∈ [1, ∞] and F : Set → Set be a functor with finite supports in the category Set of sets. Given a non-empty metric space (X, dX), we introduce the distance on the functor-space FX as the largest distance such that for every n ∈ ℕ and a ∈ Fn the map Xn → FX, f → Ff(a), is non-expanding with respect to the ℓp-metric on Xn. We prove that the distance is a pseudometric if and only if the functor F preserves singletons; is a metric if F preserves singletons and one of the following conditions holds: (1) the metric space (X, dX) is Lipschitz disconnected, (2) p = 1, (3) the functor F has finite degree, (4) F preserves supports. We prove that for any Lipschitz map f : (X, dX) → (Y, dY) between metric spaces the map is Lipschitz with Lipschitz constant Lip(Ff) ≤ Lip(f). If the functor F is finitary, has finite degree (and preserves supports), then F preserves uniformly continuous function, coarse functions, coarse equivalences, asymptotically Lipschitz functions, quasi-isometries (and continuous functions). For many dimension functions we prove the formula dim FpX ≤ deg(F) dim X. Using injective envelopes, we introduce a modification of the distance and prove that the functor Dist → Dist, , in the category Dist of distance spaces preserves Lipschitz maps and isometries between metric spaces. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a stochastic approximation based algorithm for solving nonmonotone stochastic variational inequalities. Our algorithm combines the projection and contraction method with the inertial extrapolation technique. The self-adaptive step size sequence is generated by employing the Armijo's line search rule. We also investigate the almost sure convergence property without using the prior knowledge of the Lipschitz constant of the involved operator in our algorithm. Theoretical results related to the convergence rate and the oracle complexity are provided under mild assumptions. Primary numerical experiments are presented to demonstrate the efficiency of the algorithm. [ABSTRACT FROM AUTHOR]

We consider relativistic plasma particles subjected to an external gravitation force in a 3D half space whose boundary is a perfect conductor. When the mean free path is much bigger than the variation of electromagnetic fields, the collision effect is negligible. As an effective PDE, we study the relativistic Vlasov–Maxwell system and its local-in-time unique solvability in the space-time locally Lipschitz space, for several basic mesoscopic (kinetic) boundary conditions: the inflow, diffuse, and specular reflection boundary conditions. We construct weak solutions to these initial-boundary value problems and study their locally Lipschitz continuity with the aid of a weight function depending on the solutions themselves. Finally, we prove the uniqueness of a solution, by using regularity estimate and realizing the Gauss's law at the boundary within Lipschitz continuous space. [ABSTRACT FROM AUTHOR]

In this paper, the boundedness of the Hausdorff operator on weak central Morrey space is obtained. Furthermore, we investigate the weak bounds of the p-adic fractional Hausdorff operator on weighted p-adic weak Lebesgue spaces. We also obtain the sufficient condition of commutators of the p-adic fractional Hausdorff operator by taking symbol function from Lipschitz spaces. Moreover, strong type estimates for fractional Hausdorff operator and its commutator on weighted p-adic Lorentz spaces are also acquired. [ABSTRACT FROM AUTHOR]

In this paper, the main aim is to consider the boundedness of the nonlinear commutator [ b , M α ] and the maximal commutator M α , b on the Lebesgue spaces over some stratified Lie group G when the symbol b belongs to the Lipschitz space. As a result, some new characterizations of the Lipschitz spaces on Lie group via [ b , M α ] and M α , b are given. [ABSTRACT FROM AUTHOR]

We characterize Bergman-Orlicz spaces of eigenfunctions on hyperbolic space in terms of Lipschitz type conditions with respect to pseudo-hyperbolic, hyperbolic and Euclidean metrics. As an application, some properties of symmetric lifting operators defined by the difference quotients of eigenfunctions are obatined. Our results of this paper are extensions of the corresponding known ones in classical weighted Bergman spaces. [ABSTRACT FROM AUTHOR]