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Brouwer Fixed Point Theorem in the General Case

Brouwer Fixed Point Theorem in the General Case FORMALIZED Vol. MATHEMATICS 19, No. 3, Pages 151­153, 2011 DOI: 10.2478/v10037-011-0024-3 Karol Pk Institute of Informatics University of Bialystok Poland Summary. In this article we prove the Brouwer fixed point theorem for an arbitrary convex compact subset of E n with a non empty interior. This article is based on [15]. MML identifier: BROUWER2, version: 7.11.07 4.160.1126 The notation and terminology used here have been introduced in the following papers: [17], [12], [1], [4], [7], [16], [6], [13], [10], [2], [3], [14], [9], [20], [18], [8], [19], [11], [21], and [5]. 1. Preliminaries For simplicity, we adopt the following convention: n is a natural number, p, n n n q, u, w are points of ET , S is a subset of ET , A, B are convex subsets of ET , and r is a real number. Next we state several propositions: (1) (1 - r) · p + r · q = p + r · (q - p). (2) If u, w halfline(p, q) and |u - p| = |w - p|, then u = w. (3) Let given S. Suppose p S and p = q and S halfline(p, q) is Bounded. Then there exists w such that (i) w Fr S halfline(p, q), (ii) for every u such that u S halfline(p, q) holds |p - u| |p - w|, and (iii) for every r such that r > 0 there exists u such that u S halfline(p, q) and |w - u| < r. 151 2011 University of Bialystok ISSN 1426­2630(p), 1898-9934(e) karol pk (4) For every A such that A is closed and p Int A and p = q and A halfline(p, q) is Bounded there exists u such that Fr Ahalfline(p, q) = {u}. (5) If r > 0, then Fr Ball(p, r) = Sphere(p, r). n Let n be an element of N, let A be a Bounded subset of ET , and let p be a n . One can verify that p + A is Bounded. point of ET 2. Main Theorems Next we state four propositions: n (6) Let n be an element of N and A be a convex subset of ET . Suppose A is compact and non boundary. Then there exists a function h from n n ET A into Tdisk(0ET , 1) such that h is homeomorphism and h Fr A = n ), 1). Sphere((0ET (7) Let given A, B. Suppose A is compact and non boundary and B is n compact and non boundary. Then there exists a function h from ET A n into ET B such that h is homeomorphism and h Fr A = Fr B. (8)1 For every A such that A is compact and non boundary holds every n n continuous function from ET A into ET A has a fixpoint. n (9) Let A be a non empty convex subset of ET . Suppose A is compact and n non boundary. Let F1 be a non empty subspace of ET A. If (F1 ) = Fr A, n A. then F1 is not a retract of ET http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Formalized Mathematics de Gruyter

Brouwer Fixed Point Theorem in the General Case

Formalized Mathematics , Volume 19 (3) – Jan 1, 2011

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References (26)

Publisher
de Gruyter
Copyright
Copyright © 2011 by the
ISSN
1426-2630
eISSN
1898-9934
DOI
10.2478/v10037-011-0024-3
Publisher site
See Article on Publisher Site

Abstract

FORMALIZED Vol. MATHEMATICS 19, No. 3, Pages 151­153, 2011 DOI: 10.2478/v10037-011-0024-3 Karol Pk Institute of Informatics University of Bialystok Poland Summary. In this article we prove the Brouwer fixed point theorem for an arbitrary convex compact subset of E n with a non empty interior. This article is based on [15]. MML identifier: BROUWER2, version: 7.11.07 4.160.1126 The notation and terminology used here have been introduced in the following papers: [17], [12], [1], [4], [7], [16], [6], [13], [10], [2], [3], [14], [9], [20], [18], [8], [19], [11], [21], and [5]. 1. Preliminaries For simplicity, we adopt the following convention: n is a natural number, p, n n n q, u, w are points of ET , S is a subset of ET , A, B are convex subsets of ET , and r is a real number. Next we state several propositions: (1) (1 - r) · p + r · q = p + r · (q - p). (2) If u, w halfline(p, q) and |u - p| = |w - p|, then u = w. (3) Let given S. Suppose p S and p = q and S halfline(p, q) is Bounded. Then there exists w such that (i) w Fr S halfline(p, q), (ii) for every u such that u S halfline(p, q) holds |p - u| |p - w|, and (iii) for every r such that r > 0 there exists u such that u S halfline(p, q) and |w - u| < r. 151 2011 University of Bialystok ISSN 1426­2630(p), 1898-9934(e) karol pk (4) For every A such that A is closed and p Int A and p = q and A halfline(p, q) is Bounded there exists u such that Fr Ahalfline(p, q) = {u}. (5) If r > 0, then Fr Ball(p, r) = Sphere(p, r). n Let n be an element of N, let A be a Bounded subset of ET , and let p be a n . One can verify that p + A is Bounded. point of ET 2. Main Theorems Next we state four propositions: n (6) Let n be an element of N and A be a convex subset of ET . Suppose A is compact and non boundary. Then there exists a function h from n n ET A into Tdisk(0ET , 1) such that h is homeomorphism and h Fr A = n ), 1). Sphere((0ET (7) Let given A, B. Suppose A is compact and non boundary and B is n compact and non boundary. Then there exists a function h from ET A n into ET B such that h is homeomorphism and h Fr A = Fr B. (8)1 For every A such that A is compact and non boundary holds every n n continuous function from ET A into ET A has a fixpoint. n (9) Let A be a non empty convex subset of ET . Suppose A is compact and n non boundary. Let F1 be a non empty subspace of ET A. If (F1 ) = Fr A, n A. then F1 is not a retract of ET

Journal

Formalized Mathematicsde Gruyter

Published: Jan 1, 2011

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