Approximating fixed points of nonexpansive mappings. (English) Zbl 0856.47032

Reich, Simeon

Panam. Math. J. 4, No. 2, 23-28 (1994).

Let \(C\) be a closed convex subset of a uniformly smooth Banach space \(X\), \(T: C\to C\) a nonexpansive mapping with a fixed point, \(x_0\) a point in \(C\), and \(\{k_n\}\) an increasing sequence in \([0, 1)\). It is shown that if \(X\) has a weakly sequentially continuous duality map, \(\lim_{n\to \infty} k_n= 1\), and \(\sum^\infty_{n= 1} (1- k_n)= \infty\), then the sequence \(\{x_n\}\) defined by \(x_n= (1- k_n) x_0+ k_n Tx_{n- 1}\), \(n\geq 1\), converges strongly to a fixed point of \(T\). This is an extension to a Banach space setting of a result previously known only for Hilbert space.

Reviewer: J.Appell (Würzburg)

MSC:

47H09	Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10	Fixed-point theorems
47J25	Iterative procedures involving nonlinear operators
65J15	Numerical solutions to equations with nonlinear operators

Keywords:

fixed point; iterative method; nonexpansive mapping; strong and weak convergence; convex subset; uniformly smooth Banach space; weakly sequentially continuous duality map