The Methods of Hilbert Spaces and Structure of the Fixed-Point Set of Lipschitzian Mapping

The purpose of this paper is to prove, by asymptotic center techniques and the methods of Hilbert spaces, the following theorem. Let H be a Hilbert space, let C be a nonempty bounded closed convex subset of H, and let M an,k n,k≥1 be a strongly ergodic matrix. If T : C → C is a lipschitzian mapping such that lim infn→∞infm 0,1,... ∑∞ k 1 an,k · ‖Tk m‖ 2 < 2, then the set of fixed points Fix T {x ∈ C : Tx x} is a retract of C. This result extends and improves the corresponding results of 7, Corollary 9 and 8, Corollary 1 .