Bożena Piątek

Bio: Bożena Piątek is an academic researcher from Silesian University of Technology. The author has contributed to research in topics: Fixed-point property & Geodesic. The author has an hindex of 7, co-authored 17 publications receiving 155 citations.

Fixed Points of Single- and Set-Valued Mappings in Uniformly Convex Metric Spaces with No Metric Convexity

Abstract: We study the existence of fixed points and convergence of iterates for asymptotic pointwise contractions in uniformly convex metric spaces. We also study the existence of fixed points for set-valued nonexpansive mappings in the same class of spaces. Our results do not assume convexity of the metric which makes a big difference when studying the existence of fixed points for set-valued mappings.

On cyclic Meir–Keeler contractions in metric spaces

Abstract: Cyclic Meir–Keeler contractions are considered under the recently introduced WUC and HW properties on pairs of subsets of metric spaces. We show that, in contrast with previous results in the theory, best proximity point theorems under these properties do not directly extend from cyclic contractions to cyclic Meir–Keeler contractions. We obtain, however, a positive result for cyclic Meir–Keeler contractions under additional properties which is shown to be an extension of already existing results for cyclic contractions. Moreover, we give examples supporting the necessity of our additional conditions.

Halpern iteration in CAT( κ ) spaces

Abstract: In this paper we show that an iterative sequence generated by the Halpern algorithm converges to a fixed point in the case of complete CAT(κ) spaces. Similar results for Hadamard manifolds were obtained in [Li, C., Lopez, G., Martin-Marquez, V.: Iterative algorithms for nonexpansive mappings on Hadamard manifolds. Taiwanese J. Math., 14, 541–559 (2010)], but we study a much more general case. Moreover, we discuss the Halpern iteration procedure for set-valued mappings.

On the gluing of hyperconvex metrics and diversities

Abstract: In this work we consider two hyperconvex diversities (or hyperconvex metric spaces) ( X; d X ) and ( Y; d Y ) with nonempty intersection are given and we wonder whether there is a natural way to glue them so that the new glued diversity (or metric space) remains being hyperconvex. We provide positive and negative answers in both situations.

Set-valued analysis

Abstract: This chapter relates the notions of mutations with the concept of graphical derivatives of set-valued maps and more generally links the above results of morphological analysis with some basic facts of set-valued analysis that we shall recall.

Fixed Point Theory

Abstract: Elementary Fixed Point Theorems * Theorem of Borsuk and Topological Transversality * Homology and Fixed Points * Leray-Schauder Degree and Fixed Point Index * The Lefschetz-Hopf Theory * Selected Topics * Index

Variational Inequalities for Set-Valued Vector Fields on Riemannian Manifolds: Convexity of the Solution Set and the Proximal Point Algorithm

Abstract: We consider variational inequality problems for set-valued vector fields on general Riemannian manifolds. The existence results of the solution, convexity of the solution set, and the convergence property of the proximal point algorithm for the variational inequality problems for set-valued mappings on Riemannian manifolds are established. Applications to convex optimization problems on Riemannian manifolds are provided.

On the Ishikawa iteration processes for multivalued mappings in some CAT ( κ ) spaces

Abstract: The purpose of this paper is to prove the strong convergence of the Ishikawa iteration processes for some generalized multivalued nonexpansive mappings in the framework of CAT(1) spaces. Our results extend the corresponding results given by Shahzad and Zegeye (Nonlinear Anal. 71:838-844, 2009), Puttasontiphot (Appl. Math. Sci. 4:3005-3018, 2010), Song and Cho (Bull. Korean Math. Soc. 48:575-584, 2011) and many others.