Existence of Solutions for Vector Optimization on Hadamard Manifolds

In this paper, a relationship between a vector variational inequality and a vector optimization problem is given on a Hadamard manifold. An existence of a weak minimum for a constrained vector optimization problem is established by an analogous to KKM lemma on a Hadamard manifold.     

Resolvents of Set-Valued Monotone Vector Fields in Hadamard Manifolds

Firmly nonexpansive mappings are introduced in Hadamard manifolds, a particular class of Riemannian manifolds with nonpositive sectional curvature. The resolvent of a set-valued vector field is defined in this setting and by means of this concept, a strong relationship between monotone vector fields and firmly nonexpansive mappings is established. This fact is then used to prove that the resolvent of a maximal monotone vector field has full domain. The Yosida approximation of a set-valued vector field is also introduced, analyzing its properties from which the asymptotic behavior of the resolvent is studied. Regarding the singularities of a set-valued monotone vector field, existence results are proved under certain boundary condition. As a consequence, the existence of fixed points for continuous pseudo-contractive mappings is obtained.     

广义指数二分性与微分方程的不变流形

用 Hadamar和 Bogoliubov的方法,在常微分方程、泛函微分方程以及半线性抛物型方程所能满足的条件下对Banach或Hilbert空间上的非自治抽象微分方程建立了不变流形理论。首先,对相应的线性方程提出了“广义指数二分性”概念并讨论了它和线性方程谱的关系,然后,我们给出了不变流形存在性结论以及强稳定(不稳定)流形、弱稳定(不稳定)流形和弱双曲流形的分类。进而,我们对这些不变流形给出了C~k光滑性、周期性、概周期性和吸斥性的结论。     

广义Kuramoto—Sivashinsky型方程的惯性流形

本文及续文对一类广义Kuramoto-Sivashinsky型方程证明了惯性流形的存在性，并对其维数作了估计同时研究该方程的不变锥性质（ICP）和轨线的强挤压性（SSP）。     

A General Iterative Procedure for Solving Nonsmooth Generalized Equations

We present a general iterative procedure for solving generalized equations in the nonsmooth framework. To this end, we consider a class of functions admitting a certain type of approximation and establish a local convergence theorem that one can apply to a wide range of particular problems.Mathematics Subject Classification (2000): 47H04, 65K10     

广义紧向量场

本文是[1]的继续,主要研究固定的有界连续映象经紧连续映象摄动以后得到的广义紧向量场。我们沿用[1]的符号和术语,特别,采用如下基本假设: 设X、Z为线性赋范空间,Ω为Χ中有界开集。设I:→Z为确定的连续映象,且 设F:→Z紧连续,表f(x)=I(x)-F(x)。 首先建立广义紧向量场f(x)的平凡性和本质性判别法,特别是其中某些平凡性判别法,即使对Leray—Schauder映象而言,也是新结果。     

Generalized Jacobi Identities and Ball-Box Theorem for Horizontally Regular Vector Fields

Consider a family $\mathcal{H}:= \{X_{j} =: f_{j}\cdot\nabla: j=1,\ldots , m\}$ of C ¹ vector fields in ? ⁿ and let s∈?. We assume that for all p∈{1,…,s} and j ₁,…,j _p ∈{1,…,m} the horizontal derivatives $X_{j_{1}}X_{j_{2}}\cdots X_{j_{p-1}}f_{j_{p}}$ exist and are Lipschitz continuous with respect to the control distance defined by  $\mathcal{H}$ . Then we show that different notions of commutator agree. This involves an accurate analysis of some algebraic identities involving nested commutators which seem to have an independent interest. Our principal applications are a ball-box theorem, the doubling property, and the Poincaré inequality for Hörmander vector fields under an intrinsic “horizontal regularity” assumption on their coefficients.     

广义经典力学中的广义WHITTAKER方程和场方法

本文首先利用能量积分降阶广义经典力学的正则方程并得到广义Whittaker方程.其次,将场方法应用于求广义经典力学方程的积分.最后,举例说明新方程和新方法的应用.     

Discontinuous Measure-Valued Branching Processes and Generalized Stochastic Equations

We study a class of integrable and discontinuous measure-valued branching processes. They are constructed as limits of renormalized spatial branching processes, the underlying branching distribution belonging to the domain of attraction of a stable law. These processes, computed on a test function f, are semimartingales whose martingale terms are identified with integrals of f with respect to a martingale measure. According to a representation theorem of continuous (respectively purely discontinuous) martingale measures as stochastic integrals with respect to a white noise (resp. to a POISSON process), we prove that the measure-valued processes that we consider are solutions of stochastic differential equations in the space of L² (Ω)-valued vector measures.     

Optimality Conditions for Metrically Consistent Approximate Solutions in Vector Optimization

In this paper, approximate solutions of vector optimization problems are analyzed via a metrically consistent ε-efficient concept. Several properties of the ε-efficient set are studied. By scalarization, necessary and sufficient conditions for approximate solutions of convex and nonconvex vector optimization problems are provided; a characterization is obtained via generalized Chebyshev norms, attaining the same precision in the vector problem as in the scalarization. This research was partially supported by the Ministerio de Educación y Ciencia (Spain), Project MTM2006-02629 and by the Consejería de Educación de la Junta de Castilla y León (Spain), Project VA027B06. The authors are grateful to the anonymous referees for helpful comments and suggestions.     

Generalized Solutions to Boundary-Value Problems for Quasilinear Elliptic Equations on Noncompact Riemannian Manifolds

In the present work we develop approximation approach to evaluation of solutions to boundary-value problems for quasilinear equations of the elliptic type on arbitrary noncompact Riemannian manifolds. Our technique essentially bases on an approach from the papers of E. A. Mazepa and S. A. Korol’kov connected with introduction of equivalency classes of functions and representations. On the other hand, it generalizes the method of building of generalized solution to the Dirichlet problem for linear elliptic Laplace–Beltrami and Schrödinger equations in bounded domains in ? ⁿ , which is described in details in the works of M. V. Keldysh and E. M. Landis.     

广义强向量拟平衡问题解的存在性和Hadamard适定性

首先,在映射－f(·,y,u)自然拟C-凸和映射f上半(－C)-连续的条件下,构造一个重要辅助函数,利用不同的证明方法,在不要求C*具有弱*紧基的情况下,建立了广义强向量拟平衡问题解的存在性定理.然后在适当条件下,给出问题序列收敛的定义,建立解集映射的上半连续性,并讨论广义强向量拟平衡问题的Hadamard适定性,得到广义强向量拟平衡问题的Hadamard适定性成立的充分条件.     

Generalized Solutions of Nonlinear Parabolic Equations and Diffusion Processes

We reduce the construction of a weak solution of the Cauchy problem for a quasilinear parabolic equation to the construction of a solution to a stochastic problem. Namely, we construct a diffusion process that allows us to obtain a probabilistic representation of a weak (in distributional sense) solution to the Cauchy problem for a nonlinear PDE.      

一类非线性方程Mann和Ishikawa迭代程序的稳定性

设x是实Banach空间，H：X→X是Lipschitz算子，T:x→x是一致连续的且值域有界，H T是强增生的，则Mann和Ishikawa迭代程序几乎稳定地强收敛到方程Hx Tx＝f的唯一解．     

Comparison Theorems on Convex Hypersurfaces in Hadamard Manifolds

In a Hadamard manifold with sectional curvaturebounded from below by –k ₂ ², we give sharp upper estimates for the difference circumradius minus inradiusof a compact k ₂-convex domain, and we getalso estimates for the quotient (Total d-mean curvature)/Area of a convex domain.     

Convergence Domains for Some Iterative Processes in Banach Spaces Using Outer or Generalized Inverses

We provide a semilocal Ptak–Kantorovich-type analysis for inexact Newton-like methods using outer and generalized inverses to approximate a locally unique solution of an equation in a Banach space containing a nondifferentiable term. We use Banach-type lemmas and perturbation bounds for outer as well as generalized inverses to achieve our goal. In particular we determine a domain such that starting from any point of our method converges to a solution of the equation. Our results can be used to solve undetermined systems, nonlinear least-squares problems, and ill-posed nonlinear operator equations in Banach spaces. Finally, we provide two examples to show that our results compare favorably with earlier ones.