On the Convergence of Some Iteration Processes in Uniformly Convex Banach Spaces

For the approximation of fixed points of a nonexpansive operator T in a uniformly convex Banach space E the convergence of the Mann-Toeplitz iteration x„+1 = a„T(x„) + (1 a„)xñ is studied. Strong convergence is established for a special class of operators T. Via regularization this result can be used for general nonexpansive operators, if E possesses a weakly sequentially continuous duality mapping. Furthermore strongly convergent combined regularization-iteration methods are presented. Throughout this note, let (E, | • |) be a uniformly convex Banach space, let C be a nonempty closed convex subset of E. Let T; C -* C denote a nonexpansive operator, i.e. |7Xx) T(y)\ < |x — y\ holds for all x, y E C. To approximate a fixed point of T we define the following iterative method (Mann-Toeplitz process) by x,GC, xn+x = a„T(xn) + (l-a„)xn, an G[0, 1] (n > 1). (1) We make the assumptions that S^a,, = oo and an G [0, b] with b E (0, 1) for almost all positive integers n. Let us recall that any mapping J: E^>E* which fulfills (J(u),u) = \J(u)\-\u\, \J(u)\ = \u\ for all u E E is termed a duality mapping. We verify easily (see also [3, Theorem 8.9]) that the nonexpansive operator T satisfies (x-yT(x) + T(y),J(x-y))>0 for all x, y E C. This means that the operator S:= I — T is accretive. Now we call an operator S; C^E [<p(\x\) <p(\y\)] -[\x\ \y\] \/x,y E C. (2) If S satisfies the stronger condition Received by the editors November 17, 1976 and, in revised form, August 22, 1977. AMS (A/OS) subject classifications (1970). Primary 47H15, 65J05.