We study the existence of fixed points in the context of uniformly convex geodesic metric spaces, hyperconvex spaces and Banach spaces for single and multivalued mappings satisfying conditions that generalize the concept of nonexpansivity. Besides, we use the fixed point theorems proved here to give common fixed point results for commuting mappings.
In this survey we present an exposition of the development during the last decade of metric fixed point theory on hyperconvex metric spaces. Therefore we mainly cover results where the conditions on the mappings are metric. We will recall results about proximinal nonexpansive retractions and their impact into the theory of best approximation and best proximity pairs. A central role in this survey will be also played by some recent developments on R-trees. Finally, some considerations and new results on the extension of compact mappings will be shown.
Mathematics Subject Classification
In this work we prove a new strong convergence result of the regularized successive approximation method given bywhere lim n→∞ q n = 0 andfor T a total asymptotically nonexpansive mapping, i.e., T is such thatwhere k 1 n and k 2 n are real null convergent sequences and φ : R + → R + is continuous and such that φ(0) = 0 and lim t→∞ φ(t) t ≤ C for a certain constant C > 0. Among other features, our results essentially generalize existing results on strong convergence for T nonexpansive and asymptotically nonexpansive. The convergence and stability analysis is given for both self-and nonself-mappings.Running title: Strong convergence to fixed points
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.