Local Lipschitz Property Research Articles

Metric Approximation of Set-Valued Functions of Bounded Variation by Integral Operators

AbstractWe introduce an adaptation of integral approximation operators to set-valued functions (SVFs, multifunctions), mapping a compact interval [a, b] into the space of compact non-empty subsets of $${\mathbb {R}}^d$$ R d . All operators are adapted by replacing the Riemann integral for real-valued functions by the weighted metric integral for SVFs of bounded variation with compact graphs. For such a SVF F, we obtain pointwise error estimates for sequences of integral operators at points of continuity, leading to convergence at such points to F. At points of discontinuity of F, we derive estimates, which yield the convergence to a certain set described in terms of the metric selections of F. To obtain these estimates we refine and extend known results on approximation of real-valued functions by integral operators. Our analysis uses recently defined one-sided local quasi-moduli at points of discontinuity and several notions of local Lipschitz property at points of continuity. We also provide a global approach for error bounds. A multifunction F is represented by the set of all its metric selections, while its approximation (its image under the operator) is represented by the set of images of these metric selections under the operator. A bound on the Hausdorff distance between these two sets of single-valued functions in $$L^1$$ L 1 provides our global estimates. The theory is applied to concrete operators: the Bernstein–Durrmeyer operator and the Kantorovich operator.

Short Communication: Weak Sparse Superresolution is Well-Conditioned

This paper considers superresolution (SR) as the mapping of Fourier coefficients of a discrete measure on to its support and weights. We focus on weak SR assuming a condition on the separation of the involved measures similar to the Rayleigh criterion and prove that the reconstruction map satisfies a local Lipschitz property giving explicit estimates for the Lipschitz constant depending on the dimension and the sampling effort while improving a bound on the assumed separation from [6]. With the Wasserstein distance as the metric on the space of measures we even conclude that weak SR is globally Lipschitz continuous and hence well-conditioned.

Geometry of planar curves intersecting many lines at a few points

The local Lipschitz property is shown for the graphs avoiding multiple point intersection with lines directed in a given cone. The assumption is much stronger than those of Marstrand’s well-known theorem, but the conclusion is much stronger too. Additionally, a continuous curve with a similar property is σ \sigma -finite with respect to Hausdorff length and an estimate on the Hausdorff measure of each “piece” is found.

Characterizing Sobolev spaces of vector-valued functions

We are concerned here with Sobolev-type spaces of vector-valued functions. For an open subset Ω⊂RN and a Banach space V, we characterize the functions in the Sobolev-Reshetnyak space R1,p(Ω,V), where 1≤p≤∞, in terms of the existence of partial metric derivatives or partial w⁎-derivatives with suitable integrability properties. In the case p=∞ the Sobolev-Reshetnyak space R1,∞(Ω,V) is characterized in terms of a uniform local Lipschitz property. We also consider the special case of the space V=ℓ∞.

Application of Methods of Ordinary Differential Equations to Global Inverse Function Theorems

We obtain a global inverse function theorem guaranteeing that if a smooth mapping of finite-dimensional spaces is uniformly nonsingular, then it has a smooth right inverse. Global implicit function theorems are obtained guaranteeing the existence and continuity of a global implicit function under the condition that the mappings in question are uniformly nonsingular. The local Lipschitz property and the smoothness of the global implicit function are studied. The results are generalized to the case of mappings of Hilbert spaces.

Characterizations of local upper Lipschitz property of perturbed solutions to nonlinear second-order cone programs

We characterize the local upper Lipschitz property of the stationary point mapping and the Karush–Kuhn–Tucker (KKT) mapping for a nonlinear second-order cone programming problem using the graphical derivative criterion. We demonstrate that the second-order sufficient condition and the strict constraint qualification are sufficient for the local upper Lipschitz property of the stationary point mapping and are both sufficient and necessary for the local upper Lipschitz property of the KKT mapping.

Metric Properties of Orlicz–Sobolev Classes

The homeomorphisms of the Orlicz–Sobolev class Wloc1,φ under a condition of the Calderon type on φ in ℝn, n ≥ 3 are considered. For these classes of mappings, a number of theorems on the local behavior are established, and, in particular, an analog of the famous Gehring theorem on a local Lipschitz property, as well as various theorems on estimates of a distortion of the Euclidean distance are proved. In particular, the results hold for the homeomorphisms of the Sobolev classes Wloc1,p with p > n − 1.

The lower Q-homeomorphisms relative to a p-modulus

The paper is devoted to the development of the theory of lower Q-homeomorphisms relative to a p-modulus in ℝ n , n ≥ 2. For these classes of mappings, a number of theorems on the local behavior are established, and, in particular, an analog of the famous Gehring theorem on a local Lipschitz property is proved, various theorems on estimates of distortion of the Euclidean distance are given, an estimate of the ball image measure is established, and, as a consequence, an analog of the Ikoma–Schwartz lemma is proved.

Properties of the solutions of delocalised coagulation and inception problems with outflow boundaries

Well posedness is established for a family of equations modelling particle populations undergoing delocalised coagulation, advection, inflow and outflow in a externally specified velocity field. Very general particle types are allowed while the spatial domain is a bounded region of $d$-dimensional space for which every point lies on exactly one streamline associated with the velocity field. The problem is formulated as a semi-linear ODE in the Banach space of bounded measures on particle position and type space. A local Lipschitz property is established in total variation norm for the propagators (generalised semi-groups) associated with the problem and used to construct a Picard iteration that establishes local existence and global uniqueness for any initial condition. The unique weak solution is shown further to be a differentiable or at least bounded variation strong solution under smoothness assumptions on the parameters of the coagulation interaction. In the case of one spatial dimension strong differentiability is established even for coagulation parameters with a particular bounded variation structure in space. This one dimensional extension establishes the convergence of the simulation processes studied in [Patterson, Stoch. Anal. Appl. 31, 2013] to a unique and differentiable limit.

Multiregional SIR Model with Infection during Transportation

We present a general epidemic model to describe the spread of an infectious disease in several regions connected by transportation. We take into account that infected individuals not only carry the disease to a new place while traveling from one region to another, but transmit the disease during travel as well. We obtain that a model structured by travel time is equivalent to a large system of differential equations with multiple delays. By showing the local Lipschitz property of the dynamically defined delayed feedback function, we obtain existence and uniqueness of solutions of the system.

On set-valued mappings with starlike graphs

The paper studies some topological properties of starlike bodies. It is proved that the boundary of a starlike body is a Lipschitz surface. A separability theorem for starlike bodies is proved. It is shown that under some additional assumptions the starlike property of the graph provides the local Lipschitz property of the set-valued mapping itself. It is shown that F. Clark’s contingent and tangential cones are Boltyansky tents. On the base of these results, some lower and upper differentials for set-valued mappings with starlike graphs are constructed. Some theorems on fixed points of set-valued mappings with starlike values are proved.

Cone-valued maps in optimization

Cone-valued maps are special set-valued maps where the image sets are cones. Such maps play an important role in optimization, for instance in optimality conditions or in the context of Bishop–Phelps cones. In vector optimization with variable ordering structures, they have recently attracted even more interest. We show that classical concepts for set-valued maps as cone-convexity or monotonicity are not appropriate for characterizing cone-valued maps. For instance, every convex or monotone cone-valued map is a constant map. Similar results hold for cone-convexity, sublinearity, upper semicontinuity or the local Lipschitz property. Therefore, we also propose new concepts for cone-valued maps.

Remarks on Lipschitz properties of matrix groups actions

It is proved that a large class of matrix group actions, including joint similarity and congruence-like actions, as well as actions of the type of matrix equivalence, have local Lipschitz property. Under additional hypotheses, global Lipschitz property is proved. These results are specialized and applied to obtain local Lipschitz property of canonical bases of matrices that are selfadjoint in an indefinite inner product. Real, complex, and quaternionic matrices are considered.

Lipschitz property of harmonic function on graphs

This paper proves the local Lipschitz property for harmonic (or positive subharmonic) functions on graphs. An example is also obtained to show that the global Lipschitz property of harmonic function on graphs does not hold.

First order optimality conditions in vector optimization involving stable functions

We study a nonsmooth vector optimization problem with an arbitrary feasible set or a feasible set defined by a generalized inequality constraint and an equality constraint. We assume that the involved functions are nondifferentiable. First, we provide some calculus rules for the contingent derivative in which the stability (a local Lipschitz property at a point) of the functions plays a crucial role. Second, another calculus rules are established for steady functions. Third, necessary optimality conditions are stated using tangent cones to the feasible set and the contingent derivative of the objective function. Finally, some necessary and sufficient conditions are presented through Lagrange multiplier rules.

On the Lipschitz property of strict triangular norms

This paper deals with Lipschitz triangular norms (t-norms). A partial answer to an open problem of Alsina, Frank and Schweizer is given with regard to strict t-norms with smooth additive generators. A new notion of local Lipschitz property for arbitrary t-norms is introduced. Some remarkable examples of non-Lipschitz continuous ones are provided.

N-bit stabilization of n-dimensional nonlinear systems in feedforward form

A methodology is presented which allows to design encoder, decoder and controller for stabilizing a nonlinear system in feedforward form using saturated encoded state feedback basically under standard assumption, namely local Lipschitz property of the vector field defining the system. n (respectively, n + 1) bits are used to encode the state information needed to the purpose of semiglobally (globally) stabilizing an n-dimensional system. Minimality of the data rate is discussed.

N-Bit stabilization of n-Dimensional nonlinear systems in feedforward form

A methodology is presented which allows to design encoder, decoder and controller for stabilizing a nonlinear system in feedforward form using saturated encoded state feedback basically under standard assumption, namely local Lipschitz property of the vector field defining the system. n bits are used to encode the state information needed to the purpose of semi-globally stabilizing an n-dimensional system.

THE BOUNDEDNESS OF SOLUTIONS FOR STOCHASTIC DIFFERENTIAL INCLUSIONS

We consider the stochastic differential inclusion of the form <TEX>$dX_t\;\in\;\sigma(t,\;X_t)db_t+b(t,\;X_t)dt$</TEX>, where <TEX>$\sigma$</TEX>, b are set-valued maps, B is a standard Brownian motion. We prove the boundedness of solutions under the assumption that <TEX>$\sigma$</TEX> and b satisfy the local Lipschitz property and linear growth.

Strong Convergence of Euler-Type Methods for Nonlinear Stochastic Differential Equations

Traditional finite-time convergence theory for numerical methods applied to stochastic differential equations (SDEs) requires a global Lipschitz assumption on the drift and diffusion coefficients. In practice, many important SDE models satisfy only a local Lipschitz property and, since Brownian paths can make arbitrarily large excursions, the global Lipschitz-based theory is not directly relevant. In this work we prove strong convergence results under less restrictive conditions. First, we give a convergence result for Euler--Maruyama requiring only that the SDE is locally Lipschitz and that the pth moments of the exact and numerical solution are bounded for some p >2. As an application of this general theory we show that an implicit variant of Euler--Maruyama converges if the diffusion coefficient is globally Lipschitz, but the drift coefficient satisfies only a one-sided Lipschitz condition; this is achieved by showing that the implicit method has bounded moments and may be viewed as an Euler--Maruyama approximation to a perturbed SDE of the same form. Second, we show that the optimal rate of convergence can be recovered if the drift coefficient is also assumed to behave like a polynomial.