## 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.

### 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