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O’Hara, John G.; Pillay, Paranjothi; Xu, Hong-Kun

Iterative approaches to finding nearest common fixed points of nonexpansive mappings in Hilbert spaces. (English) Zbl 1052.47049

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 54, No. 8, 1417-1426 (2003).
In this paper, the authors establish the result that the iteration scheme \(x_{n+1}=\lambda_{n+1}y+(1-\lambda_{n+1})T_{n+1}x_n\) for infinitely/finitely many nonexpansive mappings \(T_i\) in a Hilbert space converges to \(Py\), where \(P\) is the projection onto the intersection of the fixed point sets of the \(T_i\)’s. This generalizes the result of T. Shimizu and W. Takahashi [J. Math. Anal. Appl. 211, 71–83 (1997; Zbl 0883.47075)], a complementary result to a result of H. H. Bauschke [J. Math. Anal. Appl. 202, 150–159 (1996; Zbl 0956.47024)], by introducing a condition on the sequence of parameters (\(\lambda_n\)). This condition improves P.-L. Lions’ condition [C. R. Acad. Sci., Paris, Sér. A 284, 1357–1359 (1977; Zbl 0349.47046)].
Reviewer: Palaniappan Vijayaraju (Chennai)

Cited in 6 Reviews
Cited in 56 Documents

MSC:

47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
65J15 Numerical solutions to equations with nonlinear operators

Keywords:

iterative approach; convex feasibility problem; common fixed point; nearest point projection; nonexpansive mapping

Citations:

Zbl 0883.47075; Zbl 0956.47024; Zbl 0349.47046
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\textit{J. G. O'Hara} et al., Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 54, No. 8, 1417--1426 (2003; Zbl 1052.47049)
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References:

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