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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.

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]

The Besicovitch pseudometric is a shift-invariant pseudometric over the set of infinite sequences, that enjoys interesting properties and is suitable for studying the dynamics of cellular automata. It corresponds to the asymptotic behavior of the Hamming distance on longer and longer prefixes. Though dynamics of cellular automata were already studied in the literature, we propose the first study of the dynamics of substitutions. We characterize those that yield a well-defined dynamical system as essentially the uniform ones. We also explore a variant of this pseudometric, the Feldman–Katok pseudometric, where the Hamming distance is replaced by the Levenshtein distance. Like in the Besicovitch space, cellular automata are Lipschitz in this space, but here also all substitutions are Lipschitz. In both spaces, we discuss equicontinuity of these systems, give a number of examples, and generalize our results to the class of dill maps, that embeds both cellular automata and substitutions. [ABSTRACT FROM AUTHOR]

Let X and Y be two pointed metric spaces. In this article, we give a generalization of the Cheng–Dong–Zhang theorem for coarse Lipschitz embeddings as follows: If f:X → Y is a standard coarse Lipschitz embedding, then for each x* ∈ Lip0(X) there exist α, γ > 0 depending only on f and Qx* ∈ Lip0(Y) with ‖ Q x * ‖ L i p ≤ α ‖ x * ‖ L i p such that | Q x * f (x) − x * (x) | ≤ γ ‖ x * ‖ L i p , f o r a l l x ∈ X. Coarse stability for a pair of metric spaces is studied. This can be considered as a coarse version of Qian Problem. As an application, we give candidate negative answers to a 58-year old problem by Lindenstrauss asking whether every Banach space is a Lipschitz retract of its bidual. Indeed, we show that X is not a Lipschitz retract of its bidual if X is a universally left-coarsely stable space but not an absolute cardinality-Lipschitz retract. If there exists a universally right-coarsely stable Banach space with the RNP but not isomorphic to any Hilbert space, then the problem also has a negative answer for a separable space. [ABSTRACT FROM AUTHOR]

We investigate dualities and inequalities related to noncommutative martingale Hardy-Orlicz spaces. More precisely, for a concave Orlicz function Φ, we characterize the dual spaces of noncommutative martingale Hardy-Orlicz spaces H Φ c (ℛ) and h Φ c (ℳ) , where ℛ denotes a hyperfinite finite von Neumann algebra and ℳ is a finite von Neumann algebra. The first duality is new even for classical martingales, which partially answers the problem raised by Conde-Alonso and Parcet (2016). We establish as well asymmetric martingale inequalities associated with Orlicz functions that are p-convex and q-concave for 0 < p ≼ q < 2. [ABSTRACT FROM AUTHOR]

We show that given a compact group G acting continuously on a metric space M$\mathcal {M}$ by bi‐Lipschitz bijections with uniformly bounded norms, the Lipschitz‐free space over the space of orbits M/G$\mathcal {M}/G$ (endowed with Hausdorff distance) is complemented in the Lipschitz‐free space over M$\mathcal {M}$. We also investigate the more general case when G is amenable, locally compact or SIN and its action has bounded orbits. Then, we get that the space of Lipschitz functions Lip0(M/G)$\mathrm{Lip}_0(\mathcal {M}/G)$ is complemented in Lip0(M)$\mathrm{Lip}_0(\mathcal {M})$. Moreover, if the Lipschitz‐free space over M$\mathcal {M}$, F(M)$\mathcal {F}(\mathcal {M})$, is complemented in its bidual, several sufficient conditions on when F(M/G)$\mathcal {F}(\mathcal {M}/G)$ is complemented in F(M)$\mathcal {F}(\mathcal {M})$ are given. Some applications are discussed. The paper contains preliminaries on projections induced by actions of amenable groups on general Banach spaces. [ABSTRACT FROM AUTHOR]

In this paper we investigate an iterated function system that defines a fractal interpolation function, where ordinate scaling, that is Lipschitz constant in Banach contraction principle is substituted by real-valued control function. In such a manner, fractal interpolation functions associated with Matkowski contractions are obtained and provide a new framework of approximating experimental data. Furthermore, given a data generating function f , we study a new class of fractal interpolation functions which converge to f. [ABSTRACT FROM AUTHOR]

The aim of this paper is threefold. Firstly, we prove the existence and uniqueness of a global strong (in both the probabilistic and the PDE senses) H¹2-valued solution to the 2D stochastic Navier--Stokes equations (SNSEs) driven by a multiplicative Lévy noise under the natural Lipschitz condition on balls and linear growth assumptions on the jump coefficient. Secondly, we prove a Girsanov-type theorem for Poisson random measures and apply this result to a study of the wellposedness of the corresponding stochastic controlled problem for these SNSEs. Thirdly, we apply these results to establish a Freidlin--Wentzell-type large deviation principle for the solutions of these SNSEs by employing the weak convergence method introduced by Budhiraja et al. (2011, 2013). [ABSTRACT FROM AUTHOR]

In this paper we study the transport equation in generalized Campanato spaces Ls The critical case is particularly interesting, and is applied to the local well-posedness problem for the incompressible Euler equations in a space close to the Lipschitz space in our companion paper [Ann. Inst. H. Poincaré Anal. Non Linéaire 38 (2021), no. 2, 201-241]. In the critical case s Dq DN D1, we have the embeddings B1 1;1.Rn/,!L1 1.p;1/.Rn/,!C0;1.Rn/, where B11; 1.Rn/and C0;1.Rn/are the Besov and Lipschitz spaces, respectively. For f0 2L1 1.p;1/.Rn/, v 2L1.0;T IL1 1.p;1/.Rn///and g 2L1.0;T IL1 1.p;1/.Rn///, we prove the existence and uniqueness of solutions to the transport equation in L1.0; T IL1 1.p;1/.Rn//such that Similar results for the other cases are also proved. [ABSTRACT FROM AUTHOR]

This paper presents four uniqueness criteria for the initial value problem of a differential equation which depends on conformable fractional derivative. Among them is the generalization of Nagumo-type uniqueness theory and Lipschitz conditional theory, and advances its development in proving fractional differential equations. Finally, we verify the main conclusions of this paper by providing four concrete examples. [ABSTRACT FROM AUTHOR]

In the present work, the degrees of convergence of a function of conjugate Fourier series in Lip φ and weighted Lipschitz classes using product Hausdorff operator are obtained. Some interesting applications of our results are also discussed in order to observe the nature of convergence of the function in both the classes. From our applications, we find that the rate of convergence obtained from our results are much better than those of earlier results and rate of convergence will be faster as α increases. [ABSTRACT FROM AUTHOR]

Let M be the Hardy-Littlewood maximal operator and b be a locally integrable function. Denote by Mb and [b,M] the maximal commutator and the nonlinear commutator of M with b. In this paper, we give necessary and sufficient conditions for the boundedness of Mb and [b,M] on slice spaces when the function b belongs to Lipschitz spaces, by which a new characterization of non-negative Lipschitz functions is obtained. [ABSTRACT FROM AUTHOR]

The celebrated theorem of Weierstrass, dating from 1885, states that continuous functions can be uniformly approximated by polynomials on any bounded, closed interval. But just how well can we approximate by polynomials of a certain degree? Let us introduce some notation to facilitate the discussion. For an interval I (which will usually be), denote by C (I) the space of continuous functions on I , and write for (the notation is often used). Uniform convergence of fn to f (on I) equates to the statement that. Denote by the space of polynomials of degree not more than n. This is a linear subspace of C (I) of dimension n + 1. We write [ABSTRACT FROM AUTHOR]

In this paper, we obtain the characterizations of Morrey Triebel-Lizorkin spaces by two families of operators. Applying the characterizations of Morrey Triebel-Lizorkin spaces, it is proved that b is a Lipschitz function if and only if the commutator [b, T] is bounded from Morrey spaces to Morrey Triebel-Lizorkin spaces, where T is singular integral operator or Riesz potential operator. [ABSTRACT FROM AUTHOR]

Modeling is an effective way of using mathematical concepts and tools to represent natural systems and phenomena. Fractional calculus is an essential part of modeling a biological system. Recently, many researchers have been interested in modeling real-time problems mathematically and analyzing them. In this paper, the tumor system under fractional order is considered, and it comprises normal cells, tumor cells and effector-immune cells. By taking chemotherapy drugs into account, the toxicity of the drug and concentration of the drug is also studied in the model. The main objective of this work is to establish the solution for the model using Laplace transform and analyze the stability of the model. Laplace transform, a simple and efficient method, is used in solving the system that proves the existence and uniqueness of the solution. The boundedness of the system is also verified using the Lipschitz condition. Further, the system is solved for numerical values, and the population dynamics of cells are provided for different values of a as a graphical representation. Also, after analyzing the effect of chemotherapy drugs on tumor cells for different a's, which signifies that a = 0.9 provides a sufficient decrease in the dynamics of tumor cells. The main and significant part of this work is presenting that the usage of chemotherapy drugs reduces the number of tumor cells. The importance of the work is that apart from the immune system, chemotherapy drugs play a significant role in destroying tumor cells. The Hyers Ulam stability has a significant application that one need not find the exact solution to when analyzing a Hyers Ulam stable system. Thus, the stability of this tumor model under Caputo fractional order is presented using Hyers-Ulam stability and Hyers-Ulam-Rassias stability. [ABSTRACT FROM AUTHOR]

The main goal of this work is to develop and analyze an accurate truncated stochastic Runge-Kutta (TSRK2) method to obtain strong numerical solutions of nonlinear one-dimensional stochastic differential equations (SDEs) with continuous Hölder diffusion coefficients. We will establish the strong L1-convergence theory to the TSRK2 method under the local Lipschitz condition plus the one-sided Lipschitz condition for the drift coefficient and the continuous Hölder condition for the diffusion coefficient at a time T and over a finite time interval [0, T], respectively. We show that the new method can achieve the optimal convergence order at a finite time T compared to the classical Euler-Maruyama method. Finally, numerical examples are given to support the theoretical results and illustrate the validity of the method. [ABSTRACT FROM AUTHOR]

In this paper the control system with integral constraint on the control functions is studied where the behavior of the system by the Urysohn type integral equation is described. The admissible control functions are chosen from the closed ball of the space Lp ([a, b]; Rm) (p > 1) centered at the origin with radius r. Dependence of the set of trajectories on r and p is investigated. It is proved that the set of trajectories is Lipschitz continuous with respect to r and is continuous with respect to p. The robustness of the trajectory with respect to the fast consumption of the remaining control resource is established. [ABSTRACT FROM AUTHOR]

To approximate a function, some simpler functions are used: simpler as definitory structure (formula) and also as regularity, continuity and/or smoothness. Any Lipschitz function is a uniformly continuous function, but conversely it not always true. We add to the usual examples some other, especially of the class of differentiable functions. [ABSTRACT FROM AUTHOR]

We prove sufficient conditions for the weighted integrability of the Fourier–Jacobi transform. These results generalize a Titchmarsh type result due to Daher and Tyr and also use a generalized Jacobi translation defined by Flensted-Jensen and Koornwinder. Also some results connected with the integrability of Fourier–Jacobi transforms of Jacobi convolutions are given. [ABSTRACT FROM AUTHOR]

By refining the Bismut formula for SDEs and analysing the distribution variable, the derivative formula is established for singular McKean-Vlasov SDEs, where the noise coefficient belongs to a local Sobolev space, and the drift contains a locally integrable time-space term as well as a time-space-distribution term Lipschitz continuous in the space and distribution variables. The results characterize the regularity in initial distributions for distribution dependent stochastic systems, and are new also for classical SDEs. [ABSTRACT FROM AUTHOR]

In this paper, we propose an explicit diffuse the split-step truncated Euler-Maruyama (DSSTEM) method for stochastic differential equations with non-global Lipschitz coefficients. We investigate the strong convergence of the new method under local Lipschitz and Khasiminskii-type conditions. We show that the newly proposed method achieves a strong convergence rate arbitrarily close to half under some additional conditions. Finally, we illustrate the efficiency and performance of the proposed method with numerical results. [ABSTRACT FROM AUTHOR]

This paper is concerned with the pathwise dynamics of the stochastic evolution equation: du + Audt = F(u)dt + G(u)dW(t) on a separable Hilbert space H with the Lipschitz continuous drift term F(u) as well as the Lipschitz continuous diffusion term G(u). We first introduce the notion of generalized random dynamical systems (GRDSs) and show that the equation can generate a GRDS. We then construct a pathwise unstable manifold for the GRDS provided that the Lipschitz constants of the drift term and the diffusion term satisfy a spectral gap condition. At last, we present a pathwise unstable manifold reduction for the GRDS. [ABSTRACT FROM AUTHOR]

The purpose of this erratum is to point out an error in Theorem 1.3 of [SIAM J. Math. Anal., 53 (2021), pp. 5948--5978] that results in the invalidity of its conclusion. In addition, we include an example showing that Theorem 1.3 cannot hold and propose alternative (although weaker) results. More specifically, a Lipschitz stability in an L¹ -norm is replaced by one involving an H-2 -norm. We also revise other results that were obtained from Theorem 1.3; in particular, we modify the stability theorem for the inverse problem in light sheet fluorescence microscopy, which is now a logarithmic one in the natural Lebesgue spaces or Lipschitz in the weaker H-2 -norm. [ABSTRACT FROM AUTHOR]

In the present article, we construct the Szász-Jakimovski-Leviatan operators in parametric form by including the sequences of continuous functions and then investigate the approximation properties. We have successfully estimated the convergence by use of modulus of continuity in the spaces of Lipschitz functions, Peetres $ K $-functional and weighted functions. [ABSTRACT FROM AUTHOR]

We show that the Zariski canonical stratification of complex hypersurfaces is locally bi-Lipschitz trivial along the strata of codimension two. More precisely, we study the Zariski equisingular families of surface, not necessarily isolated, singularities in C³. We show that a natural stratification of such a family, given by the singular set and the generic family of polar curves, provides a Lipschitz stratification in the sense of Mostowski. In particular such families are biLipschitz trivial, with trivializations obtained by integrating Lipschitz vector fields. [ABSTRACT FROM AUTHOR]

In this paper, by weak convergence method, large and moderate deviation principles are established for path-distribution-dependent stochastic differential equations. To prove large deviation principle, both of the drift and diffusion coefficients are required to be Lipschitz continuous in the space variable as well as the distribution term, uniformly with respect to the time parameter $ t $. In further, to establish the moderate deviation principle, the drift coefficient is assumed to be Frechet differentiable with Lipschitz continuous derivative in the space variable. [ABSTRACT FROM AUTHOR]

In this work, we consider the fractional Stefan-type problem in a Lipschitz bounded domain Ω ⊂ ℝd with time-dependent Dirichlet boundary condition for the temperature ϑ = ϑ(x, t), ϑ = g on Ωc×]0, T[, and initial condition η0 for the enthalpy η = η(x, t), given in Ω×]0, T[ by ∂η/∂t + LsAϑ = f with η ∈ β(ϑ), where LsA is an anisotropic fractional operator defined in the distributional sense by sAu, v> = ∫ℝd ADsu · Ds v dx, β is a maximal monotone graph, A(x) is a symmetric, strictly elliptic and uniformly bounded matrix, and Ds is the distributional Riesz fractional gradient for 0 < s < 1. We show the existence of a unique weak solution with its corresponding weak regularity. We also consider the convergence as s ↗ 1 towards the classical local problem, the asymptotic behaviour as t → ∞, and the convergence of the two-phase Stefan-type problem to the one-phase Stefan-type problem by varying the maximal monotone graph β. [ABSTRACT FROM AUTHOR]

Let Ω ⊂ R 2 be an open, bounded and Lipschitz set. We consider the torsion problem for the Laplace operator associated to Ω with Robin boundary conditions. In this setting, we study the equality case in the Talenti-type comparison, proved in Alvino et al. (Commun Pure Appl Math 76:585–603, 2023).. We prove that the equality is achieved only if Ω is a disk and the torsion function u is radial. [ABSTRACT FROM AUTHOR]

Given an RCD(K, N) space (X , d , m) , one can use its heat kernel ρ to map it into the L 2 space by a locally Lipschitz map Φ t (x) : = ρ (x , · , t) . The space (X , d , m) is said to be an isometrically heat kernel immersing space, if each Φ t is an isometric immersion after a normalization. A main result states that any compact isometrically heat kernel immersing RCD(K, N) space is isometric to an unweighted closed smooth Riemannian manifold. This is justified by a more general result: if a compact non-collapsed RCD(K, N) space has an isometrically immersing eigenmap, then the space is isometric to an unweighted closed Riemannian manifold, which greatly improves a regularity result in by Honda (Comment Math Helv 96(3):515–559, 2021). As an application of these results, we give a C ∞ -compactness theorem for a certain class of Riemannian manifolds with a curvature-dimension-diameter bound and an isometrically immersing eigenmap. [ABSTRACT FROM AUTHOR]

This paper is concerned with the wave equation having a nonlinear distributed delay. First, we prove the local existence of the solutions by using the semigroup theory, where the source term is globally Lipschitz. Next, we establish the global existence of solutions and the energy decay result under the local Lipschitz source and suitable conditions on the initial data. [ABSTRACT FROM AUTHOR]

We obtain several variants of the classic von Neumann–Morgenstern expected utility theorem with and without the completeness axiom in which the derived Bernoulli utility functions are Lipschitz. The prize space in these results is an arbitrary separable metric space, and the utility functions are allowed to be unbounded. The main ingredient of our results is a novel (behavioral) axiom on the underlying preference relations, which is satisfied by virtually all stochastic orders. The proof of the main representation theorem is built on the fact that the dual of the Kantorovich–Rubinstein space is (isometrically isomorphic to) the Banach space of Lipschitz functions that vanish at a fixed point. An application to the theory of nonexpected utility is also provided. [ABSTRACT FROM AUTHOR]

Let n ∈ Z + = { 0 , 1 , ⋯ } , ν > - 1 / 2 . For a function f in the space L n , ν 1 (R +) of special kind, we consider the generalized Fourier-Bessel transform F n , ν (f) . We prove a Boas type result about necessary and sufficient conditions for f to belong to the generalized uniform Lipschitz classes in terms of F n , ν (f) . Also an analogue of classical Titchmarsh theorem describing Lipschits classes in L 2 space with power weight is established in a more general setting. [ABSTRACT FROM AUTHOR]

In this paper, the authors first discuss the characterization of Herz Triebel–Lizorkin spaces with variable exponent via two families of operators. By this characterization, the authors prove that the Lipschitz commutators of sublinear operators is bounded from Herz spaces with variable exponent to Herz Triebel–Lizorkin spaces with variable exponent. As applications, the corresponding boundedness estimates for the commutators of maximal operator, Riesz potential operator and Calderón–Zygmund operator are established. [ABSTRACT FROM AUTHOR]

We deal with the boundedness of the multilinear fractional integral operator I γ , m from a product of weighted Lebesgue spaces into adequate weighted Lipschitz spaces. Our results generalize some previous estimates not only for the linear case but also for the unweighted problem in the multilinear context. We characterize the classes of weights for which the problem described above holds and show 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. We further exhibit examples of weights for the class which cover the mentioned area. [ABSTRACT FROM AUTHOR]

In this paper, we consider the characterizations of the Lipschitz spaces and homogeneous Lipschitz spaces associated with the biharmonic operator Δ 2. With these characterizations, we prove the boundedness of the Bessel potentials, fractional integrals, fractional powers, Riesz transforms and multipliers of Laplace transforms type associated with Δ 2 on Lipschitz spaces and homogeneous Lipschitz spaces. The proofs of these results need the language of semigroups in an essential way. [ABSTRACT FROM AUTHOR]

Variants of the notion of (q, p)-summing operator are introduced in the setting of Lipschitz mappings acting between metric spaces. Some classes of these operators from C(K)-spaces to metric spaces are studied. An integral domination estimate is proved for a class of the mentioned Lipschitz (q, p)-summing maps. It is shown that under some conditions this domination is equivalent to (q, 1)-summability of these Lipschitz maps. As an application, we recover Pisier's result, which provides this equivalence in the setting of the linear operators. [ABSTRACT FROM AUTHOR]

We prove that the Lipschitz-free space over a Banach space X of density \kappa, denoted by \mathcal {F}(X), is linearly isomorphic to its \ell _1-sum \left (\bigoplus _{\kappa }\mathcal {F}(X)\right)_{\ell _1}. This provides an extension of a previous result from Kaufmann in the context of non-separable Banach spaces. Further, we obtain a complete classification of the spaces of real-valued Lipschitz functions that vanish at 0 over a \mathcal {L}_p-space. More precisely, we establish that, for every 1\leq p\leq \infty, if X is a \mathcal {L}_p-space of density \kappa, then \mathrm {Lip}_0(X) is either isomorphic to \mathrm {Lip}_0(\ell _p(\kappa)) if p<\infty, or \mathrm {Lip}_0(c_0(\kappa)) if p=\infty. [ABSTRACT FROM AUTHOR]

We consider a model for elastic dislocations in geophysics. We model a portion of the Earth's crust as a bounded, inhomogeneous elastic body with a buried fault surface, along which slip occurs. We prove well-posedness of the resulting mixed-boundary-value-transmission problem, assuming only bounded elastic moduli. We establish uniqueness in the inverse problem of determining the fault surface and the slip from a unique measurement of the displacement on an open patch at the surface, assuming in addition that the Earth's crust is an isotropic, layered medium with Lamé coefficients piecewise Lipschitz on a known partition and that the fault surface satisfies certain geometric conditions. These results substantially extend those of the authors in [Arch. Ration. Mech. Anal. 263, 71-111 (2020)]. [ABSTRACT FROM AUTHOR]

In the present paper, we introduce a modification of Szfisz operators involving generalized Brenke type polynomials, First, we investigate Korovkin's type approximation theorem, Next, we established an approximation result for the functions in Lipschitz type space, Furthermore, we discuss the rate of convergence by means of the Ditzian-Totik modulus of smoothness, Lastly, we established an approximation result for the functions having derivatives of bounded variation. [ABSTRACT FROM AUTHOR]

In this paper, we present an enriched Ciarlet-Raviart scheme for the biharmonic equation with variable coefficient on Lipschitz (maybe nonconvex)polyhedral domains. With the enriched finite element space for the Laplacian of the true solution, we manage to prove the discrete $ H^{2} $-stability of numerical solution. Error analysis is provided for solutions with low regularity. [ABSTRACT FROM AUTHOR]

We analyze the relationship between Borel measures and continuous linear functionals on the space |$\textrm{Lip}_0(M)$| of Lipschitz functions on a complete metric space |$M$|⁠. In particular, we describe continuous functionals arising from measures and vice versa. In the case of weak |$^\ast $| continuous functionals, that is, members of the Lipschitz-free space |$\mathcal{F}(M)$|⁠ , measures on |$M$| are considered. For the general case, we show that the appropriate setting is rather the uniform (or Samuel) compactification of |$M$| and that it is consistent with the treatment of |$\mathcal{F}(M)$|⁠. This setting also allows us to give a definition of support for all elements of |$\textrm{Lip}_0(M)^\ast $| with similar properties to those in |$\mathcal{F}(M)$|⁠ , and we show that it coincides with the support of the representing measure when such a measure exists. We deduce that the members of |$\textrm{Lip}_0(M)^\ast $| that can be expressed as the difference of two positive functionals admit a Jordan-like decomposition into a positive and a negative part. [ABSTRACT FROM AUTHOR]

By combining suitable estimates for Bessel kernel and an interpolation argument, we give an alternative proof of the lifting property (- Δ + 1) - α : Λ β → Λ β + 2 α is an isomorphism, where α ≥ 0 , β > 0 , and Λ β denotes the classical Lipschitz space. [ABSTRACT FROM AUTHOR]

This paper is devoted to generalizing Ξ-metric spaces and b-metric-like spaces to present the structure of generalized Ξ-metric-like spaces. The topological properties of this space and examples to support it are being investigated. Moreover, as demonstrated in the previous literature, the concept of Lipschitz mappings is presented more generally and some results of fixed points are derived in the aforementioned space. Finally, some theoretical results have been implicated in the discussion of the existence and uniqueness of the solution to the Fredholm integral equation. [ABSTRACT FROM AUTHOR]

Many problems arising from science and engineering are in the form of a system of nonlinear equations. In this work, a new derivative-free inertial-based spectral algorithm for solving the system is proposed. The search direction of the proposed algorithm is defined based on the convex combination of the modified long and short Barzilai and Borwein spectral parameters. Also, an inertial step is introduced into the search direction to enhance its efficiency. The global convergence of the proposed algorithm is described based on the assumption that the mapping under consideration is Lipschitz continuous and monotone. Numerical experiments are performed on some test problems to depict the efficiency of the proposed algorithm in comparison with some existing ones. Subsequently, the proposed algorithm is used on problems arising from robotic motion control. [ABSTRACT FROM AUTHOR]