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Proceedings of the American Mathematical Society

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On a fixed point theorem of Kirk


Authors: Claudio H. Morales and Simba A. Mutangadura
Journal: Proc. Amer. Math. Soc. 123 (1995), 3397-3401
MSC: Primary 47H10
DOI: https://doi.org/10.1090/S0002-9939-1995-1301520-6
MathSciNet review: 1301520
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Abstract: Let X be a reflexive Banach space, D an open and bounded subset of X, and $ T:\bar D \to X$ a continuous mapping which is locally pseudocontractive on D. Suppose there exists an element $ z \in D$ such that $ \left\Vert {z - Tz} \right\Vert < \left\Vert {x - Tx} \right\Vert$ for all x on the boundary of D. Then under the so-called condition (S), T has a fixed point in D. Although this result was proved earlier by Kirk, we show here a much easier approach.


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DOI: https://doi.org/10.1090/S0002-9939-1995-1301520-6
Keywords: Locally pseudo-contractive, convex hull, reflexivity
Article copyright: © Copyright 1995 American Mathematical Society