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Iterative solution of nonlinear equations involving set-valued uniformly accretive operators. (English) Zbl 1060.47511

Summary: Let \(E\) be a real normed linear space and let \(A:E \mapsto 2^E\) be a uniformly continuous and uniformly quasi-accretive multivalued map with nonempty closed values such that the range of \((I-A)\) is bounded and the inclusion \(f\in Ax\) has a solution \(x^*\in E\). It is proved that the Ishikawa and Mann type iteration processes converge strongly to \(x^*\). Further, if \(T:E\mapsto 2^E\) is a uniformly continuous and uniformly hemicontractive set-valued map with bounded range and a fixed point \(x^*\in E\), it is proved that both the Mann and Ishikawa type iteration processes converge strongly to \(x^*\). The strong convergence of these iteration processes with errors is also proved.

MSC:

47J25 Iterative procedures involving nonlinear operators
47H06 Nonlinear accretive operators, dissipative operators, etc.
65J15 Numerical solutions to equations with nonlinear operators
47H04 Set-valued operators
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