Fixed points of nonexpanding maps

Introduction.This paper is concerned with nonexpanding maps from the unit ball of a real Hubert space into itself.Browder [l] has established that such maps always possess at least one fixed point.We shall develop a method, which resembles the simple iterative method, for approximating fixed points of such maps.In fact, we shall generate a sequence, {x n }, by the recursive formula x n+ i = k n +if(xn) where ƒ is the map in question and {k n } is a sequence of real numbers.Our main result is Theorem 3 which states sufficient conditions on k n to insure the strong convergence of x n to a fixed point of ƒ.Definitions and preliminary observations.Let H be a Hubert space with inner product denoted by ( , ) and norm by || ||.Let B be the unit ball, B={xeH\\\x\\^l}.A map ƒ: B-±B is nonexpanding if ll/(*W(y)ll âlk-yll for an Xl y eB.Assume that ƒ : B~-*B is nonexpanding.It is not difficult to establish that the set F of fixed points must be convex.Using the con-