On nonlinear contraction semigroups in sup norm spaces

Introduction to Hyperconvex Spaces

“…The nonlinear theory is still developing. The recent interest into these spaces goes back to the results of Sine [54] and Soardi [57] who proved independently that fixed point property for nonexpansive mappings holds in bounded hyperconvex spaces. Since then many interesting results have been shown to hold in hyperconvex spaces.…”

“…Sine [54] and Soardi [57] results are at the origin of the recent interest to hyperconvex metric spaces. Both Sine and Soardi showed that nonexpansive mappings defined on a bounded hyperconvex metric space have fixed points.…”

Metric fixed point theory on hyperconvex spaces: recent progress

“…This survey was mainly devoted to the study of metric hyperconvexity and its impact on metric fixed point theory which began in late seventies of the twentieth century [60,64]. It is already a decade after the Handbook on Metric Fixed Point Theory appeared, and many new facts on hyperconvex metric spaces and fixed points have been stated in these past 10 years.…”

“…The study of fixed point theory for nonexpansive mappings on hyperconvex spaces was initiated, independently, by Sine [60] and Soardi [60], and very fundamental properties about hyperconvex spaces as well as the structure of the fixed point set of nonexpansive mappings were shown by Baillon in [9]. After the fundamental work of Sine on hyperconvexity and fixed points, which may be compiled in the series of works [35,50,[60][61][62][63], many authors continued stating new facts during the past decade of the twentieth century.…”

“…After the fundamental work of Sine on hyperconvexity and fixed points, which may be compiled in the series of works [35,50,[60][61][62][63], many authors continued stating new facts during the past decade of the twentieth century. The aforementioned chapter [21] in the Handbook of Metric Fixed Point Theory was devoted to collect many of those results.…”