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Convergence theorems for implicit iteration process for a finite family of continuous pseudocontractive mappings. (English) Zbl 1086.47046

The authors establish a necessary and sufficient condition for the strong convergence to the common fixed point of a finite family \(\{Ti\}\) of pseudocontractive mappings, using an implicit iteration process for nonexpansive mappings, in arbitrary real Banach space. The iteration sequence \(\{x_n\}\) defined by \(x_n= \alpha_n x_{n-1}+ (1-\alpha_n) Tx_n\), \(n\geq 1\), where \(T_n= T_n\bmod M\), \(T_i: K\to K\) are strictly pseudocontractive mappings such that \(F= \bigcap^M_{i=1} F(T_i)\), with one map \(T\in \{T_i\}\) which is semicompact. An equivalent condition for the weak convergence of the sequence is also given. The proofs of the results are logical and fairly standard.

MSC:

47J25 Iterative procedures involving nonlinear operators
47H06 Nonlinear accretive operators, dissipative operators, etc.
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems
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References:

[1] Osilike, M. O., Implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps, J. Math. Anal. Appl., 294, 73-81 (2004) · Zbl 1045.47056
[2] Osilike, M. O.; Udomene, A., Demiclosedness principle results for strictly pseudocontractive mappings of Browder-Petryshyn type, J. Math. Anal. Appl., 256, 431-445 (2001) · Zbl 1009.47067
[3] Xu, H. K.; Ori, R. G., An implicit iteration process for nonexpansive mappings, Numer. Fuct. Anal. Optim., 22, 767-773 (2001) · Zbl 0999.47043
[4] Chang, S.-S.; Cho, Y. J.; Zhou, H., Iterative Methods for Nonlinear Operator Equations in Banach Spaces (2002), Nova Science Publishers: Nova Science Publishers New York
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