Ces`aro and Barycentric Approximation Schemes for Coincidence Points of g-Nonexpansive Mappings

Abstract

We study quantitative approximation schemes for coincidence points of two mappings satisfying a g-nonexpansiveness condition. Starting from the approximate coincidence sequence considered in [3], we develop two complementary procedures for constructing approximate coincidence points and identify conditions under which their weak cluster points are genuine coincidence points. The first procedure is based on Cesàro averaging and applies when the mappings are affine and weak-to-weak continuous. The second is based on a barycentric construction associated with the nonexpansive mapping induced on the range of g. It applies when this range has suitable closedness, boundedness, and convexity properties, without requiring affinity. Thus, the two methods address different geometric settings: the first does not require convexity of the range of g, whereas the second does not require the mappings to be affine. As an application, we consider nonlinear Hammerstein operators on L²([a,b]). Under a natural Lipschitz condition, these operators are nonexpansive, and the barycentric method applies in the affine case. An explicit rank-one example illustrates the construction and shows that the coincidence set associated with the canonical pair (g²,g) can be strictly larger than the fixed-point set of g.