Extensions of Schauder’s Fixed Point Theorem for Lipschitz Operators under Weakened Compactness Conditions
DOI:
https://doi.org/10.32938/jpm.v7i1.9696Keywords:
Fixed Point, Schauder, Lipschitz, Compact, convexAbstract
This is a literature review focusing on the existence of fixed-point in Banach Space. This research aims to investigate the existence of fixed-point in Banach Space by examining the Schauder Fixed-Point Theorem and its extensions. This research began with a review of fundamental concepts such as metric space, compactness, convexity, and operator in Banach Space, the research proceeds to analyse the proof of the Schauder Fixed Point Theorem, which asserts that every continuous mapping on a compact and convex subset of Banach Space has a fixed point. This research further extends this result by examining cases where it is not the operator itself but the image of the operator that is compact yet still guarantees the existence of a fixed point. These findings broaden the applicability of Schauder Fixed-Point Theorem and suggest potential for its use in mathematical problems involving nonlinear-operators in infinite-dimensional spaces.
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