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Strong convergence of an implicit iteration process for a finite family of nonexpansive mappings. (English) Zbl 1090.47055
In [Numer. Funct. Anal. Optim. 22, 767--773 (2001; Zbl 0999.47043)], {\it H.-K. Xu} and {\it R. G. Ori} introduced an implicit iteration process for a finite family of nonexpansive mapping and proved the weak convergence of this process to a common fixed point of a finite family of nonexpansive maps in a Hilbert space. They posed the question: What assumptions would have to be made on the finite family of nonexpansive mappings and/or their parameters $\{ \alpha _{n}\} $ to guarantee the strong convergence of the sequence $\{ x_{n}\} $ generated by their implicit iteration process? The authors prove the strong convergence of the sequence $\{ x_{n}\} $ in a much more general uniformly convex Banach space under the condition that one member of the finite family of nonexpansive mappings is semi-compact or any one of the contractive assumptions of Proposition 3.4 of their paper holds.

47J25Iterative procedures (nonlinear operator equations)
47H09Mappings defined by “shrinking” properties
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
47J05Equations involving nonlinear operators (general)
Full Text: DOI
[1] C.E. Chidume, Iterative algorithms for nonexpansive mappings and some of their generalizations, Nonlinear Anal. and Appl., R.P. Agarwal, et al. (Eds.), Kluwer Academic Publishers, 2003, pp. 383 -- 429. · Zbl 1057.47003
[2] Kirk, W. A.: On nonlinear mappings of strongly semicontractive type. J. math. Anal. appl. 27, 409-412 (1969) · Zbl 0183.15103
[3] Petryshyn, W. V.: Fixed point theorems for various classes of 1-set-contractive and 1-ball-contractive mappings in Banach spaces. Trans. amer. Math. soc. 182, 323-352 (1973) · Zbl 0277.47033
[4] Shahzad, N.: Approximating fixed points of non-self nonexpansive mappings in Banach spaces. Nonlinear anal. 61, 1031-1039 (2005) · Zbl 1089.47058
[5] Tan, K. K.; Xu, H. K.: Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process. J. math. Anal. appl. 178, 301-308 (1993) · Zbl 0895.47048
[6] Xu, H. K.: Inequalities in Banach spaces with applications. Nonlinear anal. 16, 1127-1138 (1991) · Zbl 0757.46033
[7] Xu, H. K.; Ori, R.: An implicit iterative process for nonexpansive mappings. Numer. funct. Anal. optim. 22, 767-773 (2001) · Zbl 0999.47043
[8] Zhou, Y.; Chang, S. S.: Convergence of implicit iteration process for a finite family of asymptotically nonexpansive mappings in Banach spaces. Numer. funct. Anal. appl. 23, 911-921 (2002) · Zbl 1041.47048