Chidume, C. E.; Shahzad, Naseer Strong convergence of an implicit iteration process for a finite family of nonexpansive mappings. (English) Zbl 1090.47055 Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 62, No. 6, 1149-1156 (2005). In [Numer. Funct. Anal. Optim. 22, 767–773 (2001; Zbl 0999.47043)], H.-K. Xu and 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. Reviewer: Edward Prempeh (Kumasi) Cited in 7 ReviewsCited in 53 Documents MSC: 47J25 Iterative procedures involving nonlinear operators 47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. 47H10 Fixed-point theorems 47J05 Equations involving nonlinear operators (general) Keywords:implicit iteration process; strong convergence; nonexpensive map; common fixed point Citations:Zbl 0999.47043 PDFBibTeX XMLCite \textit{C. E. Chidume} and \textit{N. Shahzad}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 62, No. 6, 1149--1156 (2005; Zbl 1090.47055) Full Text: DOI References: [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.; 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.