## Iterative approximation of fixed points of nonexpansive mappings.(English)Zbl 1095.47034

Let $$K$$ be a nonempty closed convex subset of a real Banach space $$E$$ which has a uniformly Gâteaux differentiable norm and let $$T: K\rightarrow K$$ be a nonexpansive mapping with nonempty set of all fixed points $$F(T)$$. For a sequence $$\{\alpha_n\}$$ of real numbers in $$[0,1]$$ and an arbitrary $$u\in K$$, the sequence $$\{x_n\}$$ in $$K$$ defined by $$x_0\in K$$ and $x_{n+1}=\alpha_n u+ (1-\alpha_n) T x_n,\,\,n\geq 0,$ was introduced by B. Halpern [Bull. Am. Math. Soc. 73, 957–961 (1967; Zbl 0177.19101)] and subsequently studied by several authors in order to approximate the fixed points of $$T$$, see, e.g., the reviewer’s recent monograph [V. Berinde, “Iterative approximation of fixed points” (Efemeride, Baia Mare) (2002; Zbl 1036.47037)].
In the present paper, the authors use the same kind of iterative scheme but with the averaged map $$S$$, given by $$Sx:=(1-\delta)x+\delta Tx,\,\,\delta\in (0,1)$$, instead of $$T$$, and prove a strong convergence theorem for approximating the fixed points of $$T$$.

### MSC:

 47J25 Iterative procedures involving nonlinear operators 47H10 Fixed-point theorems 54H25 Fixed-point and coincidence theorems (topological aspects)

### Citations:

Zbl 0177.19101; Zbl 1036.47037
Full Text:

### References:

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