مبرهنة شاودر للنقطة الثابتة
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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