Implicit Function Theorems for Nondifferentiable Mappings

Sufficient conditions are given for a mapping to be γ-G inverse differentiable. Constrained implicit function theorems for γ-G inverse differentiable mappings are obtained, where the constraint is taken to be either a closed convex cone or a closed subset. A theorem without assuming the γ-G inverse differentiability in a finite-dimensional space is also presented. Communicated by F. A. Potra The author thanks the referees for valuable suggestions concerning the presentation of this paper. He also thanks Professor J. R. L. Webb and Dr. M. French for help.     

An Implicit Function Theorem for One-sided Lipschitz Mappings

Implicit function theorems are derived for nonlinear set valued equations that satisfy a relaxed one-sided Lipschitz condition. We discuss a local and a global version and study in detail the continuity properties of the implicit set-valued function. Applications are provided to the Crank–Nicolson scheme for differential inclusions and to the analysis of differential algebraic inclusions.     

Sard theorems for Lipschitz functions and applications in optimization

We establish a “preparatory Sard theorem” for smooth functions with a partially affine structure. By means of this result, we improve a previous result of Rifford [17, 19] concerning the generalized (Clarke) critical values of Lipschitz functions defined as minima of smooth functions. We also establish a nonsmooth Sard theorem for the class of Lipschitz functions from R^d to R^p that can be expressed as finite selections of C^k functions (more generally, continuous selections over a compact countable set). This recovers readily the classical Sard theorem and extends a previous result of Barbet–Daniilidis–Dambrine [1] to the case p > 1. Applications in semi-infinite and Pareto optimization are given.     

A General Fredholm Theory II: Implicit Function Theorems

This is the second paper in a series introducing a generalized Fredholm theory in a new class of smooth spaces called polyfolds. In general, these spaces are not locally homeomorphic to open sets in Banach spaces. The current paper develops the Fredholm theory in M-polyfold bundles. It consists of a transversality and a perturbation theory. In upcoming papers the generalized Fredholm theory will be applied to the Floer Theory, the Gromov–Witten Theory and the Symplectic Field Theory H.H.’s research partially supported by NSF grant DMS-0603957. K.W.’s research partially supported by NSF grant DMS-0606588.     

F-调和映照的不存在性定理

本文主要讨论一类F-调和映照的不存在性问题,从而得到相应的Liouville型定理．     

On Duality Theorems for Nonsmooth Lipschitz Optimization Problems

A nonsmooth Lipschitz vector optimization problem (VP) is considered. Using the Fritz John type necessary optimality conditions for (VP), we formulate the Mond–Weir dual problem (VD) and establish duality theorems for (VP) and (VD) under (strict) pseudoinvexity assumptions on the functions. Our duality theorems do not require a constraint qualification.     

Liouville Theorems for F-Harmonic Maps and Their Applications

We prove several Liouville theorems for F-harmonic maps from some complete Riemannian manifolds by assuming some conditions on the Hessian of the distance function, the degrees of F(t) and the asymptotic behavior of the maps at infinity. In particular, the results can be applied to F-harmonic maps from some pinched manifolds, and can deduce a Bernstein type result for an entire minimal graph.     

Extension Theorems for Spaces Arising from Approximation by Translates of a Basic Function

We establish extension theorems for functions in spaces which arise naturally in studying interpolation by radial basic functions. These spaces are akin in some way to the non-integer-valued Sobolev spaces, although they are considerably more general. Such extensions allow us to establish local error estimates in a way which we make precise in the introductory section of our paper. There are many other applications of these fundamental results, including improved L_p error estimates for interpolation by shifts of a single basic function, but these applications have been left to a later paper.     

Holomorphic Extension Theorems in Lipschitz Domains of {\mathbb{C}}^{2}

The holomorphic functions of several complex variables are closely related to the continuously differentiable solutions $f : {\mathbb{R}}^{2n} \mapsto {\mathbb{C}}_{n}$f : {\mathbb{R}}^{2n} \mapsto {\mathbb{C}}_{n} of the so called isotonic system

调和函数的Lipschitz型空间和Landau-Bloch型定理

主要研究调和函数和Poisson方程的解的性质.讨论了调和函数的Lipschitz型空间,建立了调和函数的Schwarz-Pick型引理,并利用所得结果证明了与调和Hardy空间有关的一个Landau-Bloch型定理.最后,还利用正规族理论讨论了与Poisson方程的解有关的Landau-Bloch型定理的存在性.     

Essential Equivariant Maps and Borsuk-Ulam Theorems

A full characterization is given of those compact Lie groupsG with the property that every G-map XX on a finite-dimensionalG-complex X of finite orbit type, X^G = Ø, is (non-equivariantly)essential. For arbitrary G, conditions are given on the G-spaceX which guarantee this property. Finally, conditions are givenfor the non-existence of a G-map XY inducing a homotopy equivalenceX^GY^G on the fixed point sets. These results have applicationsto critical point theory of almost G-invariant functionals.     

Some Fixed Point Theorems for Multivalued Maps and Applications

In order to obtain some existence results for certain generalized eigenvalue problems nd to study certain parametrized boundary value problems for second order differential nclusions, we generalize some fixed point theorems for completely continuous (single-valued) aps to the multivalued case. We illustrate our results with some examples.     

Moreau–Rockafellar Theorems for Nonconvex Set-Valued Maps

In this paper, we introduce the notion of (Benson) proper subgradient of a set-valued map and prove that, for some class of nonconvex set-valued maps, a proper subgradient of the sum of two set-valued maps can be expressed as the sum of two proper subgradients of these maps. This property is also established for weak subgradients. A result in Ref. [Lin, L.J.: J. Math. Anal. Appl. 186, 30–51 (1994)], obtained under some convexity assumption, is included as a special case of the corresponding result of this paper. The author thanks the anonymous referees for their valuable remarks.     

序压缩映射的随机不动点定理

在Banach空间中利用一个随机Mann迭代序列组,讨论了随机映射的随机不动点的存在性问题,得出了几个随机不动点定理,改进了相关文献中的相应结果.     

Extension Dimension and Refinable Maps

Extension dimension is characterized in terms of -maps. We apply this result to prove that extension dimension is preserved by refinable maps between metrizable spaces. It is also shown that refinable maps preserve some infinite-dimensional properties.     

An Improvement of Existence Theorems of Additive Selection Maps for Sub-additive Set-valued Maps

In this paper, we will construct a new class of subadditive set-valued maps and use Cantor theorem to prove that the set-valued map has an unique additive selection map when the set-valued map satisfies some certain conditions, and then compare the obtained result with the well-known results.     

Implicit Gamma Theorems (I): Pseudoroots and Pseudospectra

   Abstract. Let g : E → F be an analytic function between two Hilbert spaces E and F. We study the set g(B(x, ε)) ⊂ E, the image under g of the closed ball about x∈ E with radius ε . When g(x) expresses the solution of an equation depending on x , then the elements of g(B(x,ε )) are ε -pseudosolutions. Our aim is to investigate the size of the set g(B(x,ε )) . We derive upper and lower bounds of the following form: g(x) + Dg (x) ( B(0, c ₁ ε N))