On a Fixed Point Theorem of Kirk
Claudio H. Morales, Simba A. Mutangadura · Proceedings of the American Mathematical Society · 1995
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.