Strong convergence of modified Mann iterations. (English) Zbl 1091.47055

Let \(X\) be a real Banach space with a norm \(\|\cdot\|\) and let \(C\) be a nonempty, closed and convex subset of \(X\). A mapping \(T:C\to C\) is nonexpansive provided that \(\| Tx- Ty\|\leq\| x-y\|\) for all \(x,y\in C\). Assume that \(T\) has at least one fixed point in \(C\). The authors consider the following iteration sequence \(\{x_n\}\) for \(T: x_0= x\in C\), \(y_n= \alpha_nx_n+ (1-\alpha_n)Tx_n\), \(x_{n+1}= \beta_n u+(1- \beta_n)y_n\), where \(u\) is an arbitrary fixed element in \(C\) and \(\{\alpha_n\}\), \(\{\beta_n\}\) are two sequences in the interval \((0,1)\) converging to \(0\) and such that \(\sum\alpha_n= \sum \beta_n= \infty\). Moreover, \(\sum|\alpha_{n+1}- \alpha_n|< \infty\), \(\sum|\beta_{n+1}- \beta_n|< \infty\). Under the assumption that \(X\) is uniformly smooth, it is shown that the sequence \(\{x_n\}\) converges strongly to a fixed point of \(T\). An analogous result is proved for accretive operators.


47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
46B25 Classical Banach spaces in the general theory
47H06 Nonlinear accretive operators, dissipative operators, etc.
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