## 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.

### MSC:

 47H10 Fixed-point theorems 47H05 Monotone operators and generalizations 47J05 Equations involving nonlinear operators (general) 47H06 Nonlinear accretive operators, dissipative operators, etc.
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