Continuous Approximations of Multivalued Mappings and Fixed Points

In the present paper, we prove a fixed-point theorem for completely continuous multivalued mappings defined on a bounded convex closed subset X of the Hilbert space H which satisfies the tangential condition , where T _X (x) is the cone tangent to the set X at a point x. The proof of this theorem is based on the method of single-valued approximations to multivalued mappings. In this paper, we consider a simple approach for constructing single-valued approximations to multivalued mappings. This approach allows us not only to simplify the proofs of already-known theorems, but also to obtain new statements needed to prove the main theorem in this paper.__________Translated from Matematicheskie Zametki, vol. 78, no. 2, 2005, pp. 212–222.Original Russian Text Copyright © 2005 by B. D. Gel’man.     

On Some Almost-Periodicity Problems in Various Metrics

The Bohr-type and the Bochner-type definitions for almost periodic functions are examined in various metrics (Stepanov, Weyl and Besicovitch). The correct definitions of Besicovitch-like multifunctions are given. Weak almost-periodic solutions are proved for differential equations and inclusions. This problem is also discussed as a fixed-point problem in function spaces.     

非线性分数阶脉冲微分方程边值问题的解

通过Schauder不动点定理和Banach压缩映射原理得到了一类非线性分数阶脉冲微分方程边值问题解的存在性和唯一性结果.     

具有无限多个奇性点的一维p-Laplacian方程的正解

主要讨论奇异边值问题{(фp(x′))′ a(t)f(x(t))=0,t∈(0,1),αx(0)-βx′(0)=0,γx(1) δx′(1)=0.在奇性条件下无穷多个解的存在性问题,其中:фp(s)=│s│p-2s,p>1;a(t)在0,1/2上有可数个奇性点.     

Multiple positive solutions for a class of nonlinear four-point boundary value problem with <Emphasis Type="Italic">p</Emphasis>-Laplacian

This paper deals with the existence of multiple positive solutions for a class of nonlinear singular four-point boundary value problem with p-Laplacian：
{（φ（u′））′＋a（t）f（u（t））=0, 0〈t〈1,
αφ（u（0））-βφ（u′（ξ））=0,γφ（u（1））＋δφ（u′（η））0,
where φ（x） = ｜x｜^p-2x,p 〉 1, a（t） may be singular at t = 0 and/or t = 1. By applying Leggett-Williams fixed point theorem and Schauder fixed point theorem, the sufficient conditions for the existence of multiple （at least three） positive solutions to the above four-point boundary value problem are provided. An example to illustrate the importance of the results obtained is also given.     

Disintegration of projective measures

In this paper, we study a class of quasi-invariant measures on paths generated by discrete dynamical systems. Our main result characterizes the subfamily of these measures which admit a certain disintegration. This is a disintegration with respect to a random walk Markov process which is indexed by the starting point of the paths. Our applications include wavelet constructions on Julia sets of rational maps on the Riemann sphere.

     

Iterative Method for Set-Valued Mixed Quasi-variational Inequalities in a Banach Space

This paper introduces an iterative method for finding approximate solutions of a set-valued mixed quasivariational inequality in the setting of a Banach space. Existence of a solution of this rather general problem and the convergence of the proposed iterative method to a solution are established.The first two authors were partially supported by the National Science Council of the Republic of China. The third author was partially supported by the Teaching and Research Award Fund for Outstanding Young Teachers in Higher Education Institutions of MOE, China and by the Dawn Program Foundation in ShanghaiCommunicated by     

Controllability of Nonlinear Systems in Banach Spaces: A Survey

This paper presents a survey on research using fixed-point theorems and semigroup theory to study the controllability of nonlinear systems and functional integrodifferential systems in Banach spaces. Also discussed is the use of this technique in K-controllability and boundary controllability problems for nonlinear systems and integrodifferential systems in abstract spaces.     

A short and constructive proof of Tarski’s fixed-point theorem

I give short and constructive proofs of Tarski’s fixed-point theorem, and of Zhou’s extension of Tarski’s fixed-point theorem to set-valued maps.I thank Charles Blair, William Thomson, an associated editor and a referee for their helpful suggestions.I have taught Tarski’s Theorem with F continuous to Caltech undergraduates.     

On the existence of Nash strategies and solutions to coupled riccati equations in linear-quadratic games

The existence of linear Nash strategies for the linear-quadratic game is considered. The solvability of the coupled Riccati matrix equations and the stability of the closed-loop matrix are investigated by using Brower's fixed-point theorem. The conditions derived state that the linear closed-loop Nash strategies exist, if the open loop matrixA has a sufficient degree of stability which is determined in terms of the norms of the weighting matrices. WhenA is not necessarily stable, sufficient conditions for existence are given in terms of the solutions of auxiliary problems using the same procedure.This work was supported in part by the Joint Services Electronics Program (US Army, US Navy, and US Air Force) under Contract No. DAAG-29-78-C-0016, in part by the National Science Foundation under Grant No. ENG-74-20091, and in part by the Department of Energy, Electric Energy Systems Division, under Contract No. US-ERDA-EX-76-C-01-2088.