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Convergence comparison of several iteration algorithms for the common fixed point problems. (English) Zbl 1167.65029

Summary: We discuss the following viscosity approximations with weak contraction \(A\) for a non-expansive mapping sequence \(\{T_{n}\}\), \[ y_{n}=\alpha_{n}Ay_{n}+(1-\alpha_{n})T_{n}y_{n}, x_{n+1}=\alpha_{n}Ax_{n}+(1-\alpha_{n})T_{n}x_{n}. \] We prove that Browder’s and Halpern’s type convergence theorems [B.Halpern, Bull.Am.Math.Soc.73, 957–961 (1967; Zbl 0177.19101)] imply Moudafi’s viscosity approximations with weak contraction [A.Moudafi, J. Math.Anal.Appl.241, No.1, 46–55 (2000; Zbl 0957.47039)], and give an estimate of the convergence rate between Halpern’s type iteration and Moudafi’s viscosity approximations with weak contraction.

MSC:

65J15 Numerical solutions to equations with nonlinear operators
47J25 Iterative procedures involving nonlinear operators
54H25 Fixed-point and coincidence theorems (topological aspects)
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