مبرهنة شاودر للنقطة الثابتة

Authors

  • محمود باكير

Abstract

تــعد مبرهنــة شاودر للنقطــة الثابتـة تعميمـاً لمبرهنـة برووَرْ للنقطــة الثابتــة Brouwer’s Fixed Point Theorem التي تنص على أنّ: كل مجموعة جزئية غير خالية متراصة compact ومحدبة convex من En تتمتع بخاصة النقطة الثابتة fixed point property، (حيث  أو، R مجموعة الأعداد الحقيقية، ومجموعة الأعداد العقدية وn عدد صحيح موجب). وإثبات صحة مبرهنة شاودر للنقطة الثابتة يعتمد على مبرهنة برووَرْ آنفة الذكر، انظر مثلاً الصفحة 61 من [5]. كما أنَّ هناك إثباتاً آخر لها يعتمد على توطئة (تمهيدية) سبيرنر Sperner’s Lemma، انظر[2]. سنثبت صحتها دون استخدام مبرهنة بروور للنقطة الثابتة، وذلك اعتماداً على توطئة متعلقة بـ  - النقطة الثابتة fixed point -  .

Schauder’s Fixed Point Theorem is considered to be one of the most prominent and well-known theorem in Fixed Point Theory, since it is used in Economy, Game Theory, and Deferential Equations. It was proved by the Polish mathematician Juliusz Schauder in 1930. It is a generalization of Brouwer’s Fixed Point Theorem; and its proof depends on Brouwer’s Fixed Point Theorem; see, for example, [5]. It has also another proof using Sperner’s Lemma; see [2]. We will prove it without using Brouwer’s Fixed Point Theorem; our proof will depend on a lemma about - fixed point.

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Published

2018-12-05

How to Cite

1.
باكير م. مبرهنة شاودر للنقطة الثابتة. TUJ-BA [Internet]. 2018Dec.5 [cited 2024May17];30(4). Available from: https://journal.tishreen.edu.sy/index.php/bassnc/article/view/5083

Issue

Vol. 30 No. 4 (2008): العلوم الأساسية

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