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Fixed point theorems for Lipschitzian pseudo-contractive mappings


Authors: Juan A. Gatica and W. A. Kirk
Journal: Proc. Amer. Math. Soc. 36 (1972), 111-115
MSC: Primary 47H10
DOI: https://doi.org/10.1090/S0002-9939-1972-0306993-8
MathSciNet review: 0306993
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Abstract: Let X be a Banach space and $ D \subset X$. A mapping $ U:D \to X$ is said to be pseudo-contractive if, for all $ u,v \in D$ and all $ r > 0,\left\Vert {u - v} \right\Vert \leqq \left\Vert {(1 + r)(u - v) - r(U(u) - U(v))} \right\Vert$. A recent fixed point theorem of W. V. Petryshyn is used to prove: If G is an open bounded subset of X with $ 0 \in G$ and $ U:\bar G \to X$ is a lipschitzian pseudo-contractive mapping satisfying (i) $ U(x) \ne \lambda x$ for $ x \in \partial G,\lambda > 1$ , and (ii) $ (I - U)(\bar G)$ is closed, then U has a fixed point in $ \bar G$. This result yields fixed point theorems for pseudo-contractive mappings in uniformly convex spaces and for ``strongly'' pseudo-contractive mappings in reflexive spaces.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0306993-8
Keywords: Fixed point theorem, pseudo-contractive mapping, nonexpansive mapping, accretive mapping
Article copyright: © Copyright 1972 American Mathematical Society

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