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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 $H$, $T:K\to K$ a continuous pseudocontractive map. Let $\left\{{a}_{n}\right\}$, $\left\{{b}_{n}\right\}$, $\left\{{c}_{n}\right\}$, $\left\{{a}_{n}^{{}^{\text{'}}}\right\}$, $\left\{{b}_{n}^{{}^{\text{'}}}\right\}$ and $\left\{{c}_{n}^{{}^{\text{'}}}\right\}$ be real sequences in [0,1] satisfying appropriate conditions. For arbitrary ${x}_{1}\in K$, define the sequence ${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$ iteratively by ${x}_{n+1}={a}_{n}{x}_{n}+{b}_{n}T{y}_{n}+{c}_{n}{u}_{n}$; ${y}_{n}={a}_{n}^{{}^{\text{'}}}{x}_{n}+{b}_{n}^{{}^{\text{'}}}T{x}_{n}+{c}_{n}^{{}^{\text{'}}}{v}_{n}$, $n\ge 1,$ where $\left\{{u}_{n}\right\}$, $\left\{{v}_{n}\right\}$ are arbitrary sequences in $K$. Then, ${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$ converges strongly to a fixed point of $T$. A related result deals with the convergence of ${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$ 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 for nonlinear operators on topological linear spaces 47H05 Monotone operators (with respect to duality) and generalizations 47J05 Equations involving nonlinear operators (general) 47H06 Accretive operators, dissipative operators, etc. (nonlinear)