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On closed graph and implicit function theorems for multifunctions

Part of:Calculus on manifolds; nonlinear operatorsTopological linear spaces and related structures

Published online by Cambridge University Press:  09 April 2009

C. C. Chou,
L. R. Huang and
K. F. Ng
C. C. Chou
Affiliation:
Université de Perpignan, France
L. R. Huang
Affiliation:
South China Normal UniversityGuangzhou, China
K. F. Ng
Affiliation:
The Chinese University of Hong KongHong Kong e-mail: ngkf@cuhk.hk
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Abstract

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We give several general implicit function and closed graph theorems for ser-valued functions. Let Z be a normed space, X, Y metric spaces with X complete. Let f: X ⇉ Z, F: X × Y ⇉ Z be multifunctions with z0 ∈ f(x0) ∩ F(x0, y0) such that f is open at (x0, y0) and f ‘approximates’ F in an appropriate sense. Suppose that f−1(z) is closed, F(x, y) is compact for each x, y and z and suppose that F(x0, ·) is lower semi-continuous at y0. Then F(·, y) is of closed graph ‘locally’, is open at x0, and there exists a function x(·) with x(y) → x0 for y → y0 such that z0 ∈ F(x(y), (y)) for all y near y0. A more general form dealing with the non-linear rate situation is also established.

Keywords

Multifunctionimplicit functionopen mapping theoremimplicit function theorem.

MSC classification

Secondary:46A30: Open mapping and closed graph theorems; completeness (including $B$-, $B_r$-completeness)58C15: Implicit function theorems; global Newton methods

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 61 , Issue 1 , August 1996, pp. 129 - 142
Copyright
Copyright © Australian Mathematical Society 1996

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