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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 $A:E↦{2}^{E}$ be a uniformly continuous and uniformly quasi-accretive multivalued map with nonempty closed values such that the range of $\left(I-A\right)$ 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 $T:E↦{2}^{E}$ 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 (nonlinear operator equations) 47H06 Accretive operators, dissipative operators, etc. (nonlinear) 65J15 Equations with nonlinear operators (numerical methods) 47H04 Set-valued operators