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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 =α n Ay n +(1-α n )T n y n ,x n+1 =α n Ax n +(1-α 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.

65J15Equations with nonlinear operators (numerical methods)
47J25Iterative procedures (nonlinear operator equations)
54H25Fixed-point and coincidence theorems in topological spaces