Remarks on metric transforms and fixed-point theorems
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RESEARCH
Open Access
Remarks on metric transforms and fixed-point theorems William A Kirk1 and Naseer Shahzad2* *
Correspondence: [email protected] 2 Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah, Saudi Arabia Full list of author information is available at the end of the article
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. Keywords: metric transform; local radial contraction; rectifiably pathwise connected space; Hausdorff metric; H+ -contraction; uniform local multivalued contraction; metric space
1 Introduction In this paper, we study fixed points of mappings satisfying local contractive conditions, with a special emphasis on the following concept due to L. M. Blumenthal. Definition . A strictly increasing concave function φ : [, ∞) → R for which φ() = is called a metric transform. Blumenthal has observed (see Exercise on p. of []) that if (X, d) is a metric space and if ρ(x, y) = φ(d(x, y)) for each x, y ∈ X, where φ is a metric transform, then (X, ρ) is also a metric space. He had introduced this concept earlier in [] to show that the metric transform φ(M) of any metric space M by φ(t) = t α , < α ≤ , has the Euclidean four point property, i.e., each four points of φ(M) are isometric to a quadruple of points in -dimensional Euclidean space. A mapping g defined on a metric space X is said to be a local radial contraction [] if there exists k ∈ (, ) such that for each x ∈ X there exists εx > such that d(x, u) < εx ⇒ d(g(x), g(u)) ≤ kd(x, u) for all u ∈ X. We begin by showing that if a mapping is a local radial contraction defined on a metric space (X, d) and taking values in a metric transform of (X, d), then in many instances 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. In Section , we turn our attention to set-valued contractions and prove, among other things, a set-valued analog to the main result of Section . Finally, because it is also based on the idea of a metric transform, we revisit a counter-example given in [] in somewhat more detail. © 2013 Kirk and Shahzad; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Kirk and Shahzad Fixed Point Theory and Appli
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