Remarks on metric transforms and fixed-point theorems

Abstract

It is shown that if a mapping is a local radial contraction defined on a metric space (X, d) which takes values in a metric transform of (X, d), then for many metric transforms it is also a local radial contraction (with possibly different contraction constant) relative to the original metric. Several specific examples are given. This in turn implies that the mapping has a fixed point if the space is rectifiably pathwise connected. Some results about set-valued contractions are also discussed. © 2013 Kirk and Shahzad; licensee Springer.