On a Fixed Point Theorem of Ky Fan

We generalize a theorem of Ky Fan about the nearest distance between a closed convex set D in a Banach space E and its image by a function ƒ:D→E, in several directions: (1) for noncompact sets D, when ƒ(D) precompact; (2) for compact D and upper semicontinuous multifunction ƒ and more generally, (3) for noncompact D and upper semicontinuous multifunction ƒ with ƒ(D) Hausdorff precompact. In particular, we prove a version of the fixed point theorem of Kakutani-Ky Fan for multifunctions whose values are convex closed bounded, thus not necessarily compact. Received May 23, 2000, Accepted September 4, 2001     

Nonlinear hemivariational inequalities of second order using the method of upper-lower solutions

In this paper we examine a nonlinear hemivariational inequality of second order. The differential operator is set-valued, nonlinear and depends on both and its gradient . The same is true for the zero order term , while the right-hand side nonlinearity satisfies a one-sided Lipschitz condition. We use the method of upper and lower solutions, coupled with truncation and penalization techniques and the fixed point theory for multifunctions in an ordered Banach space.

     

Five Kinds of Maximal Monotonicity

We discuss Gossez's type (D) maximal monotone multifunctions and the newer type (ED) subfamily (for which an analog of the Brøndsted-Rockafellar property holds). We then discuss the locally maximal monotone (= type (FP)) and maximal monotone locally (= type (FPV)) multifunctions of Fitzpatrick-Phelps and Fitzpatrick-Phelps-Verona-Verona. Finally, we discuss the strongly maximal monotone multifunctions. We prove that every maximal monotone multifunction of type (D) is locally maximal monotone, and every maximal monotone multifunction of type (ED) is both maximal monotone locally and strongly maximal monotone.     

On strong pseudomonotonicity and (semi)strict quasimonotonicity

New concepts of strong pseudomonotonicity, strict quasimonotonicity, and semistrict quasimonotonicity of a map are introduced and their properties are studied. In the case of a differentiable gradient map, we show that strong pseudomonotonicity of the gradient is equivalent to strong pseudoconvexity of the underlying function. This does not hold for a different concept of strong pseudomonotonicity in Ref. 1. Analogous results are shown for strict quasimonotonicity and semistrict quasimonotonicity.     

集值条件期望的一个Fatou型引理

本文讨论了Ｂａｎａｃｈ空间只订合序列弱收敛的一些性质，给出了集值条件期望的表示定量，证明了集值条件期望在弱收敛意义下的Ｆａｔｏｕ型引理和控制收敛定理，并由此得到了一个可积选择空间的收敛定理。     

Relations approximated by continuous functions

Let be a Tychonoff space, let be the space of all continuous real-valued functions defined on and let be the hyperspace of all nonempty closed subsets of . We prove the following result. Let be a locally connected, countably paracompact, normal -space without isolated points, and let . Then is in the closure of in with the locally finite topology if and only if is the graph of a cusco map. Some results concerning an approximation in the Vietoris topology are also given.

     

Dunford-Pettis composition operators on in several variables

A bounded composition operator on , where is the unit ball in , is Dunford-Pettis if and only if the radial limit function of takes values on the unit sphere only on a set of surface measure zero. A similar theorem holds on bounded strongly pseudoconvex domains with smooth boundary.

     

Lempert mappings and symplectic forms

We use Lempert's version of Riemann mapping to construct non-equivalent symplectic forms on an ellipsoid in .

     

Berezin Quantization and Reproducing Kernels on Complex Domains

Let be a non-compact complex manifold of dimension , a Kähler form on , and the reproducing kernel for the Bergman space of all analytic functions on square-integrable against the measure . Under the condition

F. A. Berezin [Math. USSR Izvestiya 8 (1974), 1109--1163] was able to establish a quantization procedure on which has recently attracted some interest. The only known instances when the above condition is satisfied, however, are just and a bounded symmetric domain (with the euclidean and the Bergman metric, respectively). In this paper, we extend the quantization procedure to the case when the above condition is satisfied only asymptotically, in an appropriate sense, as . This makes the procedure applicable to a wide class of complex Kähler manifolds, including all planar domains with the Poincaré metric (if the domain is of hyperbolic type) or the euclidean metric (in the remaining cases) and some pseudoconvex domains in . Along the way, we also fix two gaps in Berezin's original paper, and discuss, for a domain in , a variant of the quantization which uses weighted Bergman spaces with respect to the Lebesgue measure instead of the Kähler-Liouville measure .

     

Pseudoconvexity and Analytic Discs

The purpose of this paper is to investigate the geometry of pseudoconvex hypersurfaces and their interrelation with analytic discs. The method of analytic discs is a very powerful tool in the study of pseudoconvexity. One way to use this tool is via the so-called type function introduced by Dwilewicz and Hill for arbitrary Cauchy–Riemann manifolds. In the hypersurface case, and moreover, under the assumption of pseudoconvexity, it has additional properties which are given in this paper. As an application, boundary estimates of plurisubharmonic functions are given.