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Fixed point iteration for pseudocontractive maps. (English) Zbl 0913.47052
Summary: Let $K$ be a compact convex subset of a real Hilbert space $H$, $T:K\rightarrow K$ a continuous pseudocontractive map. Let $\{a_{n}\}$, $\{b_{n}\}$, $\{c_{n}\}$, $\{a_{n}^{'}\}$, $\{b_{n}^{'}\}$ and $\{c_{n}^{'}\}$ be real sequences in [0,1] satisfying appropriate conditions. For arbitrary $x_{1}\in K$, define the sequence $\{x_{n}\}_{n=1}^{\infty}$ iteratively by $x_{n+1} = a_{n}x_{n} + b_{n}Ty_{n} + c_{n}u_{n}$; $y_{n} = a_{n}^{'}x_{n} + b_{n}^{'}Tx_{n} + c_{n}^{'}v_{n}$, $n\geq 1,$ where $\{u_{n}\}$, $\{v_{n}\}$ are arbitrary sequences in $K$. Then, $\{x_{n}\}_{n=1}^{\infty}$ converges strongly to a fixed point of $T$. A related result deals with the convergence of $\{x_{n}\}_{n=1}^{\infty}$ to a fixed point of $T$ when $T$ is Lipschitz and pseudocontractive. Our theorems also hold for the slightly more general class of continuous hemicontractive nonlinear maps.

47H10Fixed-point theorems for nonlinear operators on topological linear spaces
47H05Monotone operators (with respect to duality) and generalizations
47J05Equations involving nonlinear operators (general)
47H06Accretive operators, dissipative operators, etc. (nonlinear)
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