全局性隐函数定理及其应用

A GLOBAL IMPLICIT FUNCTION THEOREM

得到两个全局性隐函数定理:定理1设D_1是第一可数的拓扑空间E_1的开子集.D_2是Banach空间E_2的开子集.映象f:(?)_1×(?)_2→Y(?)E关于第一变元连续且满足条件:1°|f(x,y_1)-f(x,y_2)|≤L(x)|y_2-y_1|.Ax∈(?)_1.y_1.y_2∈D_2.其中Y=D_2或D_2=Y=E_2,L(x)<1.L:(?)_1→R^+连续.则方程f(x.y)=y有连续解y:(?)_1→Y,即f(x.y(x))=y(x).(?)x∈(?)_1.定理2 设f:(?)_1×(?)_2→C((?)_2)满足条件:1°d(f(x,y_1).f(x,y_2))≤k|y_2-y_1|.(?)x∈(?)_1.y_1.y_2∈(?)_2.其中k<1是常数.d(·,·)表示:对有界闭子集A_1,A_2(?)(?)_2d(A_l,A_2)=sup{|y_1-y_2||y_1∈A_1,y_2∈A_2}2°(?)y∈(?)_2,多值映象,f(·,y)弱下半连续.C((?)_2)为(?)_2的有界闭凸子集类.则包含方程y∈f(x,y)有连续单值解y;(?)_1→(?)_2即y(x)∈f(x,y(x)) (?)x∈(?)_1还给出了对随机映象不动点存在性的一个应用.

In this paper, he proved a global Implicit function theoremTheorem Let Dt be an open set of a first countable toplogical space, D2 an open set of a Banach space E2. f:Di X D2 - F CI E is continuous on first variable and satisfiesR+ continuous. L(z) < 1 then the equations/(z, y) = y have a solution y. DI -,D2continuous, i. e.He also gave a example and a application to the existence of random operator fixed point.