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Iterative approximations of fixed points and solutions for strongly accretive and strongly pseudo-contractive mappings in Banach spaces. (English) Zbl 0933.47040
Consider a map $$T : K \to K$$, where $$K$$ is a bounded closed convex subset of a Banach space $$X$$. For $$x_0 \in K$$ the Ishikawa iteration is defined by $x_{n+1} = (1-\alpha_n)x_n + \alpha_n T y_n,\quad y_n = (1-\beta_n)x_n + \beta_n T x_n, \quad n=0,1,2,\ldots,$ where $$\{ \alpha_n \}$$, $$\{ \beta_n \}$$ are two real sequences in $$[0,1]$$. Let the set $$F(T)$$ of fixed points of $$T$$ in $$K$$ be nonempty. For a strongly pseudo-contractive $$T$$ the authors prove the convergence of the Ishikawa iteration to a fixed point under certain assumptions on $$\alpha_n,\;\beta_n$$ and a smoothness condition on $$T$$ respectively $$X$$. There are also results for strongly accretive $$T$$ and for the Mann iteration $x_{n+1} = (1-\alpha_n)x_n + \alpha_n T x_n, \quad n=0,1,2,\ldots .$

##### MSC:
 47J25 Iterative procedures involving nonlinear operators 65J15 Numerical solutions to equations with nonlinear operators 47H06 Nonlinear accretive operators, dissipative operators, etc.
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##### References:
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