Measure of Nonhyperconvexity and Fixed-point Theorems

In this paper, we work with the notion of the measure of nonhyperconvexity introduced by Cianciaruso and De Pascale [6] in order to obtain new fixed-point theorems in hyperconvex metric spaces. This class of metric spaces was introduced by Aronszajn and Panitchpakdi [1] in 1956 to study problems on extension of uniformly continuous mappings. Several and interesting properties of these spaces were shown in Aronszajn and Panitchpakdi’s original paper, some of these properties turned out to be crucial in the successful searching for fixedpoint theorems in hyperconvex metric spaces. More precisely, this research began when Sine [13] and Soardi [14] independently proved in 1979 that bounded hyperconvexmetric spaces have the fixed-point property for nonexpansive mappings. Since then, authors like J. B. Baillon, M. A. Khamsi, W. A. Kirk, S. Park, G. Yuan, and many others, including both authors of this paper, have been attracted by this subject and have contributed to a wide development of it (for a recent survey see [11, Chapter 13]). Hyperconvex metric spaces can be defined as follows: given twometric spaces (Y,d) and (X,ρ), we say that a mapping T : Y → X is nonexpansive if ρ(Tx,T y)≤ d(x, y) for any x and y in Y . The pair (Y,X) is said to have the extension property for nonexpansive mappings, if every nonexpansive mapping from an arbitrary subset S of Y into X can be extended as a nonexpansive mapping to the whole Y into X . A metric space X is said to be hyperconvex if the pair (Y,X)