Nonlinear Ergodic Theorem for Commutative Families of Positively Homogeneous Nonexpansive Mappings in Banach Spaces and Applications

Recently, two retractions (projections) which are different from the metric projection and the sunny nonexpansive retraction in a Banach space were introduced. In this paper, using nonlinear analytic methods and new retractions, we prove a nonlinear ergodic theorem for a commutative family of positively homogeneous and nonexpansive mappings in a uniformly convex Banach space. The limit points are characterized by using new retractions. In the proof, we use the theory of invariant means essentially. We apply our nonlinear ergodic theorem to get some nonlinear ergodic theorems in Banach spaces.