The Mild Modification of the (DL)-Condition and Fixed Point Theorems for Some Generalized Nonexpansive Mappings in Banach Spaces
Kanok Chuikamwong, Naknimit Akkasriworn · 2012
A Banach space X is said to satisfy property (D) if there exists α ∈ [0,1) such that for any nonempty weakly compact convex subset E of X, any sequence {xn }⊂ E which is regular relative to E, and any sequence {yn }⊂ A(E,{xn}) which is regular relative to E, we have r(E,{yn}) ≤ αr (E,{xn}). A this property is the mild modification of the (DL)-condition. Let X be a Banach space satisfying property (D) and let E be a weakly compact convex subset of X .I fT : E → E is a mapping satisfying condition (E) and (Cλ) for some λ ∈ (0,1). We study the existence of a fixed point for this mapping.