Abstract
It is proved that if D is a bounded open subset of a uniformly convex Banach space X and\(T:\bar D \to X\) is a continuous mapping which is a local pseudo-contraction (e.g., locally nonexpansive) on D, then T has a fixed point in D if there exists x∈D such that ‖z−Tz‖<‖x−Tx‖ for all x in the boundary of D.
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Kirk, W.A. A fixed point theorem for local pseudo-contractions in uniformly convex spaces. Manuscripta Math 30, 89–102 (1979). https://doi.org/10.1007/BF01305991
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DOI: https://doi.org/10.1007/BF01305991