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