Hölder Continuity of Solutions to Parametric Vector Equilibrium Problems with Nonlinear Scalarization

In this article, using the nonlinear scalarization approach by virtue of the nonlinear scalarization function, commonly known as the Gerstewitz function in the theory of vector optimization, Hölder continuity of solution mappings for both set-valued and single-valued cases to parametric vector equilibrium problems is studied. The nonlinear scalarization function is a powerful tool that plays a key role in the proofs, and its main properties (such as sublinearity, continuity, convexity) are fully employed. Especially, its locally and globally Lipschitz properties are provided and the Lipschitz property is first exploited to investigate the Hölder continuity of solutions.     

Hölder continuity results for nonconvex parametric generalized vector quasiequilibrium problems via nonlinear scalarizing functions

In this paper, new results for Hölder continuity of the unique solution to a parametric generalized vector quasiequilibrium problem are established via nonlinear scalarization, with and without using the free-disposal condition. Especially, a new kind of monotonicity hypothesis is proposed. The globally Lipschitz property together with other useful properties of the well-known Gerstewitz nonlinear scalarization function are fully exploited for proving. Moreover, our approach does not impose any convexity condition on the considered model. The oriented distance function is also employed for studying Hölder continuity.     

Hölder Continuity for Solution Mappings of Parametric Non-convex Strong Generalized Ky Fan Inequalities

Abstract

This article focuses on a new approach to investigate the Hölder continuity for the solution mapping of a parametric non-convex strong generalized Ky Fan inequality. Based on a non-convex separation theorem, the union relation between the solution set of the parametric non-convex strong generalized Ky Fan inequality and the solution sets of a series of Ky Fan inequalities, is established. Without density results and any information on the solution mapping, a sufficient condition for the Hölder continuity of the solution mapping to the parametric non-convex strong generalized Ky Fan inequality is given by using the key union relation. Our method does not impose any convexity, monotonicity, and the single-valuedness of the solution mapping.     

Continuity and Convexity of a Nonlinear Scalarizing Function in Set Optimization Problems with Applications

Hern\(\acute{\mathrm{a}}\)ndez and Rodríguez-Marín (J Math Anal Appl 325:1–18, 2007) introduced a nonlinear scalarizing function for sets, which is a generalization of the Gerstewitz’s function. This paper aims at investigating some properties concerned with the nonlinear scalarizing function for sets. The continuity and convexity of the nonlinear scalarizing function for sets are showed under some suitable conditions. As applications, the upper semicontinuity and the lower semicontinuity of strongly approximate solution mappings to the parametric set optimization problems are also given.     

Convergence of Rothe's method for fully nonlinear parabolic equations

Convergence of Rothe's method for the fully nonlinear parabolic equation u_t+F(D²u, Du, u, x, t)=0 is considered under some continuity assumptions on F. We show that the Rothe solutions are Lipschitz in time, Hölder in space, and they solve the equation in the viscosity sense. As an immediate corollary we get Lipschitz behavior in time of the viscosity solutions of our equation.     

On the Local Convergence of Modified Homeier-Like Method in Banach Spaces

The aim of this article is to investigate the local convergence analysis of the multi-step Homeier-like approach in order to approximate the solution of nonlinear equations in Banach spaces, which fulfilled the Lipschitz as well as Hölder continuity condition. The Hölder condition is more relax than Lipschitz condition. Also, the existence and uniqueness theorem has been derived and found their error bounds. Numerical examples are available to appear the importance of theoretical discussions.     

A convergence analysis of an inexact Newton-Landweber iteration method for nonlinear problem

In this paper, we study the convergence and the convergence rates of an inexact Newton–Landweber iteration method for solving nonlinear inverse problems in Banach spaces. Opposed to the traditional methods, we analyze an inexact Newton–Landweber iteration depending on the Hölder continuity of the inverse mapping when the data are not contaminated by noise. With the namely Hölder-type stability and the Lipschitz continuity of DF, we prove convergence and monotonicity of the residuals defined by the sequence induced by the iteration. Finally, we discuss the convergence rates.     

Hölder continuity of the unique solution to parametric vector quasiequilibrium problems via nonlinear scalarization

In this paper, by virtue of the nonlinear scalarization function commonly known as the Gerstewitz function in the theory of vector optimization, Hölder continuity of the unique solution to a parametric vector quasiequilibrium problem is studied based on nonlinear scalarization approach, under three different kinds of monotonicity hypotheses. The globally Lipschitz property of the nonlinear scalarization function is fully employed. Our approach is totally different from the ones used in the literature, and our results not only generalize but also improve the corresponding ones in some related works.     

Semilocal convergence analysis for the modified Newton-HSS method under the Hölder condition

The present paper is concerned with theoretical properties of the modified Newton-HSS method for large sparse non-Hermitian positive definite systems of nonlinear equations. Assuming that the nonlinear operator satisfies the Hölder continuity condition, a new semilocal convergence theorem for the modified Newton-HSS method is established. The Hölder continuity condition is milder than the usual Lipschitz condition. The semilocal convergence theorem is established by using the majorizing principle, which is based on the concept of majorizing sequence given by Kantorovich. Two real valued functions and two real sequences are used to establish the convergence criterion. Furthermore, a numerical example is given to show application of our theorem.     

Global Continuity of the Integrated Density of States for Random Landau Hamiltonians

Abstract

We prove that the integrated density of states (IDS) for the randomly perturbed Landau Hamiltonian is Hölder continuous at all energies with any Hölder exponent 0 < q < 1. The random Anderson-type potential is constructed with a nonnegative, compactly supported single-site potential u. The distribution of the iid random variables is required to be absolutely continuous with a bounded, compactly supported density. This extends a previous result Combes et al. [Combes, J. M., Hislop, P. D., Klopp, F. (2003a). Hölder continuity of the integrated density of states for some random operators at all energies. Int. Math. Res. Notices 2003: 179--209] that was restricted to constant magnetic fields having rational flux through the unit square. We also prove that the IDS is Hölder continuous as a function of the nonzero magnetic field strength.     

Regularity for P-Harmonic Type Systems with the Gradients below the Controllable Growth

In this paper, we shall establish that each weak solution of p-harmonic type systems with the gradients below the controllable growth belongs to, Holder continuity spaces with any HSlder exponent α∈ [0, 1）. Furthermore, we can obtain that the gradients of the corresponding weak solutions also belong to locally Hoelder continuity spaces with some Hoelder exponent. Keywords controllable growth, p-harmonic systems, full regularity MR（2000） Subject Classification 35J60, 35B65     

On Hölder continuity of solution maps to parametric vector primal and dual equilibrium problems

In this paper, we first propose some kinds of the strong convexity (concavity) for vector functions. Then we apply these assumptions to establish sufficient conditions for the Hölder continuity of solution maps of the vector primal and dual equilibrium problems in metric linear spaces. As applications, we derive the Hölder continuity of solution maps of vector optimization problems and vector variational inequalities. Our results improve and generalize some recent existing ones in the literature.     

Hölder continuity of the solutions for a class of nonlinear SPDE’s arising from one dimensional superprocesses

The Hölder continuity of the solution X _t (x) to a nonlinear stochastic partial differential equation (see (1.2) below) arising from one dimensional superprocesses is obtained. It is proved that the Hölder exponent in time variable is arbitrarily close to 1/4, improving the result of 1/10 in Li et al. (to appear on Probab. Theory Relat. Fields.). The method is to use the Malliavin calculus. The Hölder continuity in spatial variable x of exponent 1/2 is also obtained by using this new approach. This Hölder continuity result is sharp since the corresponding linear heat equation has the same Hölder continuity.     

Continuity of the solution mappings to parametric generalized vector equilibrium problems

The aim of this paper is to establish the continuity of the efficient solution mappings to a parametric generalized strong vector equilibrium problem, by using the Hölder relation. Our result extends and improves some recent results in the references therein.     

On distributionally robust optimization problems with k-th order stochastic dominance constraints induced by full random quadratic recourse

In this paper, we consider the optimization problems with k-th order stochastic dominance constraint on the objective function of the two-stage stochastic programs with full random quadratic recourse. By establishing the Lipschitz continuity of the feasible set mapping under some pseudo-metric, we show the Lipschitz continuity of the optimal value function and the upper semicontinuity of the optimal solution mapping of the problem. Furthermore, by the Hölder continuity of parameterized ambiguity set under the pseudo-metric, we demonstrate the quantitative stability results of the feasible set mapping, the optimal value function and the optimal solution mapping of the corresponding distributionally robust problem.     

Hölder continuity for a class of X-elliptic equations with singular lower order term

The regularity for a class of X-elliptic equations with lower order term
Lu＋vu=-∑i,j=1 mXj^＊（aij（x）Xiu）＋vu=μ
is studied, where X = {X1,..., Xm} is a family of locally Lipschitz continuous vector fields, v is in certain Morrey type space and μ a nonnegative Radon measure. The HSlder continuity of the solution is proved when μ satisfies suitable growth condition, and a converse result on the estimate of μ is obtained when u is in certain HSlder class.     

Hölder continuity for nonlinear elliptic problem in Musielak–Orlicz–Sobolev space

Under appropriate assumptions on the $N(\mathrm{\Omega })$-function, the De Giorgi process is presented by the tools recently developed in Musielak–Orlicz–Sobolev space to prove the Hölder continuity of fully nonlinear elliptic problems. As the applications, the Hölder continuity of the minimizers for a class of the energy functionals in Musielak–Orlicz–Sobolev spaces is proved; and furthermore, the local Hölder continuity of the weak solutions for a class of fully nonlinear elliptic equations is provided.     

Hölder continuity of solutions to a basic problem in the calculus of variations

For the basic problem in the calculus of variations where the Lagrangian is convex and depends only on the gradient, we establish the continuity of the solutions when the Dirichlet boundary condition is defined by a continuous function ϕ. When ϕ is Lipschitz continuous, then the solutions are Hölder continuous. To cite this article: P. Bousquet et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Semilocal convergence for the Super-Halley’s method

The semilocal convergence of Super-Halley’s method for solving nonlinear equations in Banach spaces is established under the assumption that the second Frëchet derivative satisfies the ω-continuity condition. This condition is milder than the well-known Lipschitz and Hölder continuity conditions. The importance of our work lies in the fact that numerical examples can be given to show that our approach is successful even in cases where the Lipschitz and the Hölder continuity conditions fail. The difficult computation of second Frëchet derivative is also avoided by replacing it with the divided difference containing only the first Frëchet derivatives. A number of recurrence relations based on two parameters are derived. A convergence theorem is established to estimate a priori error bounds along with the domains of existence and uniqueness of the solutions. The R-order convergence of the method is shown to be at least three. Two numerical examples are worked out to demonstrate the efficacy of our method. It is observed that in both examples the existence and uniqueness regions of solution are improved when compared with those obtained in [7].     

Partial regularity for nonlinear elliptic systems with continuous growth exponent

For weak solutions of nonlinear elliptic systems of the type ${- {\rm div}a(x, u(x), Du(x)) = 0,}$ with nonstandard p(x) growth, we show interior partial Hölder continuity for any Hölder exponent ${\alpha \in (0,1)}$ , provided that the exponent function is ‘logarithmic Hölder continuous’. The result also covers the up to now open partial regularity for systems with constant growth with exponent p less than two in the case of merely continuous dependence on the spacial variable x.