On a fixed point theorem of Kirk

Morales, Claudio H.; Mutangadura, Simba A.

doi:10.1090/S0002-9939-1995-1301520-6

On a fixed point theorem of Kirk
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by Claudio H. Morales and Simba A. Mutangadura PDF

Proc. Amer. Math. Soc. 123 (1995), 3397-3401 Request permission

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 $\left \| {z - Tz} \right \| < \left \| {x - Tx} \right \|$ 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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