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Convergence of Mann’s type iteration method for generalized asymptotically nonexpansive mappings. (English) Zbl 1471.65043

Summary: Let \(C\) be a nonempty, closed and convex subset of a real Hilbert space \(H\). Let \(T_{i}: C \to H\), \(i=1,2,\dots ,N\) be a finite family of generalized asymptotically nonexpansive mappings. It is our purpose, in this paper to prove strong convergence of Mann’s type method to a common fixed point of \(\{T_{i}:i=1,2,\dots ,N\}\) provided that the interior of common fixed points is nonempty. No compactness assumption is imposed either on \(T\) or on \(C\). As a consequence, it is proved that Mann’s method converges for a fixed point of nonexpansive mapping provided that interior of \(F(T)\neq \emptyset\). The results obtained in this paper improve most of the results that have been proved for this class of nonlinear mappings.

MSC:

65J15 Numerical solutions to equations with nonlinear operators
47H10 Fixed-point theorems
47J25 Iterative procedures involving nonlinear operators
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