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Convergence of the Mann iteration algorithm for a class of pseudocontractive mappings. (English) Zbl 1193.65085
Summary: Let $K$ be a nonempty, closed and convex subset of a real Banach space $E$. Let $T: K\to K$ be a strictly pseudocontractive map in the sense of Browder and Petryshyn. For a fixed $x_0\in K$, define a sequence $\{x_n\}$ by $$x_{n+1}=(1-\alpha_n)x_n+\alpha_nTx_n,$$ where $\{\alpha_n\}$ is a real sequence defined in $[0,1]$ satisfying the following conditions: (i) $\sum^\infty_{n=1}\alpha_n=\infty$, (ii) $\sum^\infty_{n=1}\alpha^2_n<\infty$. Then $\liminf_{n\to\infty}\Vert x_n-Tx_n\Vert=0$. If, in addition, $T$ is demicompact, then $\{x_n\}$ converges strongly to some fixed point of $T$.

##### MSC:
 65J15 Equations with nonlinear operators (numerical methods) 47H10 Fixed-point theorems for nonlinear operators on topological linear spaces 47H09 Mappings defined by “shrinking” properties
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##### References:
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