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Iterative solution of nonlinear equations of accretive and pseudocontractive types. (English) Zbl 1054.47056
The authors prove the following theorem without the assumption of boundedness of $\{x_n\}$. Theorem 1: Let $E$ be a real uniformly smooth Banach space. Let $A: D(A)= E\to 2^E$ be an accretive operator that satisfies the range condition and $A^{-1}(0)\ne\emptyset$. Suppose that $\{\lambda_n\}$ and $\{\theta_n\}$ are real sequences in $(0,1]$ satisfying the following conditions: (i) $\lim_{n\to\infty}\theta_n= 0$; (ii) $\sum^\infty_{n=1} \lambda_n\theta_n= \infty$, $\lim_{n\to\infty} {b(\lambda_n)\over \theta_n}= 0;$\par(iii) $\lim_{n\to\infty} {\theta_{n-1}/\theta_n- 1\over \lambda_n\theta_n}= 0$. Let $z\in E$ arbitrary and the sequence $\{x_n\}$ be generated from $x_0\in E$ by $x_{n+1}= x_n- \lambda_n(x_n- z))$, $u_n\in Ax_n$, for all positive integers $n$. Suppose that $\{u_n\}$ is bounded. Then there exists $d> 0$ such that, whenever $\lambda_n\le d$ and $b(\lambda_n)/\theta_n\le d^2$ for all $n\ge 0$, $\{x_n\}$ converges strongly to a zero of $A$. As a consequence of this result, they also prove the following theorem. Theorem 2: Let $K$ be a closed convex subset of a real uniformly smooth Banach space $E$. Let $T: K\to K$ be a bounded continuous pseudocontractive map with $F(T)\ne\emptyset$. Let $\{\lambda_n\}$ and $\{\theta_n\}$ be real squences satisfying\par\hskip17mm (i) $\lim_{n\to\infty}\theta_n= 0$;\par\hskip17mm (ii) $\sum^\infty_{n=1}\lambda_n\theta_n= \infty$, $\lambda_n(1+ \theta_n)\le 1$, $\lim_{n\to\infty} {b(\lambda_n)\over \theta_n}= 0$;\par\hskip17mm (iii) $\lim_{n\to\infty} {\theta_{n-1}/\theta_n- 1\over \lambda_n\theta_n}= 0$. Let $z\in K$ arbitrary and the sequence $\{x_n\}$ be generated from $x_0\in K$ by $x_{n+1}= x_n- \lambda_n((I- T)x_n+ \theta_n(x_n- z))$ for all positive integers $n$. Then there exists $d> 0$ such that, whenever $\lambda_n\le d$ and $b(\lambda_n)/\theta_n\le d^2$ for all $n\ge 0$, $\{x_n\}$ converges strongly to a fixed point of $T$. This theorem extends the class of Lipschitzian pseudocontractive maps to the class of bounded continuous pseudocontractive maps, which generalizes results of {\it C. E. Chidume} [Proc. Am. Math. Soc. 129, 2245--2251 (2001; Zbl 0979.47038)].

47J25Iterative procedures (nonlinear operator equations)
47H06Accretive operators, dissipative operators, etc. (nonlinear)
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
Full Text: DOI
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