Fixed points of locally nonexpansive mappings in geodesic spaces

Kirk, W. A.; Shahzad, Naseer

doi:10.1016/j.jmaa.2016.09.059

Cited by 2 publications

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“…(We refer the reader to the recent survey of Ciesielski and Jasinski [18] for a comprehensive review of many local results in metric fixed point theory, and especially for numerous intricate examples. For recent results in geodesic spaces, see also [1,63]. )…”

Section: Preliminariesmentioning

confidence: 99%

Hyperbolic spaces and directional contractions

Kirk

Shahzad

2019

Bull. Math. Sci.

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The axiomatic approach to metric convexity goes back to the pioneering work of Karl Menger in 1928. This is an overview of this concept and the role it plays in metric fixed point theory especially in conjunction with spaces possessing a “hyperbolic” type structures. These include the CAT(0) spaces, hyperconvex metric spaces, and [Formula: see text]-trees. Much of the discussion involves the existence of “approximate” fixed point sequences for mappings satisfying weak contractive conditions. Applications of a well-known fixed point theorem due to Caristi are also included. These involve fixed and approximate fixed points for mappings satisfying local “directional” contractive and non-expansive conditions. Convexity plays a role in this part of the discussion as well. While the paper is semi-expository in nature, some detailed proofs appear here for the first time. Also the concept of a weak [Formula: see text]-directional contraction introduced in Sec. 8 appears to be new. Several suggestions for further research are also discussed.

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Section: Preliminariesmentioning

confidence: 99%