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Approximation of a zero point of accretive operator in Banach spaces. (English) Zbl 1115.47055
The authors consider the composite approximation scheme \[ \begin{gathered} y_n=\beta_nx_n+(1-\beta_n)J_{r_n}x_n, \\ x_{n+1}=\alpha_nu+(1-\alpha_n)y_n, \end{gathered} \] where \(J_{r_n}\) is the resolvent of the \(m\)-accretive operator \(A\) acting in the Banach space \(E\), \(C\subset E\) is closed convex, and \(u\in C\). By assuming that \(E\) is uniformly convex, \(A^{-1}(0)\neq\varnothing\) and the sequences \((\alpha_n)_n\), \((\beta_n)_n\) and \((r_n)_n\) satisfy some technical conditions, the authors show that the sequence \((x_n)_n\) converges to a zero point of \(A\). The same conclusion holds true if \(E\) is reflexive and has a weakly continuous duality map \(J_{\varphi}\) with gauge \(\varphi\).

47J25 Iterative procedures involving nonlinear operators
47H06 Nonlinear accretive operators, dissipative operators, etc.
Full Text: DOI
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