Approximating fixed point of({\lambda},{\rho})-firmly nonexpansive mappings in modular function spaces

In this paper, we first introduce an iterative process in modular function spaces and then extend the idea of a {\lambda}-firmly nonexpansive mapping from Banach spaces to modular function spaces. We call such mappings as ({\lambda},{\rho})-firmly nonexpansive mappings. We incorporate the two ideas to approximate fixed points of ({\lambda},{\rho})-firmly nonexpansive mappings using the above mentioned iterative process in modular function spaces. We give an example to validate our results

Porosity Results for Sets of Strict Contractions on Geodesic Metric Spaces

We consider a large class of geodesic metric spaces, including Banach spaces, hyperbolic spaces and geodesic $\mathrm{CAT}(\kappa)$-spaces, and investigate the space of nonexpansive mappings on either a convex or a star-shaped subset in these settings. We prove that the strict contractions form a negligible subset of this space in the sense that they form a $\sigma$-porous subset. For separable metric spaces we show that a generic nonexpansive mapping has Lipschitz constant one at typical points of its domain. These results contain the case of nonexpansive self-mappings and the case of nonexpansive set-valued mappings as particular cases.Comment: 35 pages; accepted version of the manuscript; accepted for publication in Topological Methods in Nonlinear Analysi

A note on strong convergence to common fixed points of nonexpansive mappings in Hilbert spaces

The aim of this paper is to investigate the links between ${\cal T}_C$-class algorithms, CQ Algorithm and shrinking projection methods. We show that strong convergence of these algorithms are related to coherent ${\cal T}_C$-class sequences of mapping. Some examples dealing with nonexpansive finite set of mappings and nonexpansive semigroups are given. They extend some existing theorems

Best proximity pair results for relatively nonexpansive mappings in geodesic spaces

Given $A$ and $B$ two nonempty subsets in a metric space, a mapping $T : A \cup B \rightarrow A \cup B$ is relatively nonexpansive if $d(Tx,Ty) \leq d(x,y) \text{for every} x\in A, y\in B.$ A best proximity point for such a mapping is a point $x \in A \cup B$ such that $d(x,Tx)=\text{dist}(A,B)$. In this work, we extend the results given in [A.A. Eldred, W.A. Kirk, P. Veeramani, Proximal normal structure and relatively nonexpansive mappings, Studia Math., 171 (2005), 283-293] for relatively nonexpansive mappings in Banach spaces to more general metric spaces. Namely, we give existence results of best proximity points for cyclic and noncyclic relatively nonexpansive mappings in the context of Busemann convex reflexive metric spaces. Moreover, particular results are proved in the setting of CAT(0) and uniformly convex geodesic spaces. Finally, we show that proximal normal structure is a sufficient but not necessary condition for the existence in $A \times B$ of a pair of best proximity points