Ishikawa-type iterative process with errors
Ishikawa-type and Mann-type iterative processes with errors for \(m\)-accretive operators
For the solutions of equations \(x + Tx = f\) involving a mapping \(T : D \subset E \rightarrow E\) defined on a subset \(D\) of a Banach space \(E\) the authors consider the following Ishikawa-type iterative process with errors NEWLINE\[NEWLINE\begin{aligned} p_{n+1} &= \alpha_n x_n + \beta_n T Q y_n + \gamma_n u_n,\\ y_n &= \widehat{\alpha_n} x_n + \widehat{\beta_n} Tx_n + \widehat{\gamma_n} x_n,\\ x_{n+1} &= Q p_{n+1},\qquad n \geq 0, \end{aligned}NEWLINE\]NEWLINE starting from \(x_0 \in D\), where \(Q:E \rightarrow E\) is a retraction of \(E\) onto \(D\) and \(\alpha_n + \beta_n + \gamma_n = \widehat{\alpha_n} + \widehat{\beta_n} + \widehat{\gamma_n}\), \(\forall n \geq 0\). The specialization \(\widehat{\beta_n} = \widehat{\gamma_n} = 0\) gives the corresponding Mann-type process. The convergence of the processes is proved for the case of \(m\)-accretive -- but not necessarily Lipschitzian -- operators \(T\) on a closed domain \(D\) under the assumption that the space \(E\) is uniformly convex and uniformly smooth. Moreover, convergence of the processes is shown for equations \(x - \lambda Tx = f\) with an \(m\)-dissipative operator \(T\) on a closed domain \(D\) which, once again, need not be Lipschitzian. This extends various results in the literature.
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