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Monotone $CQ$ algorithm of fixed points for weak relatively nonexpansive mappings and applications. (English) Zbl 1199.47295
Summary: {\it S.-Y.\thinspace Matsushita} and {\it W.  Takahashi} [J. Approx. Theory 134, No. 2, 257--266 (2005; Zbl 1071.47063)] proved a strong convergence theorem for relatively nonexpansive mappings in a Banach space by using the hybrid method ($CQ$ method) in mathematical programming. The purpose of this paper is to modify this method and to prove strong convergence theorems for weak relatively nonexpansive mappings and maximal monotone operators in Banach spaces. The convergence rate of our monotone CQ method is faster than the hybrid method of [op. cit.]. In addition, the Cauchy sequence method is used in this paper without using the Kadec-Klee property.

47J25Iterative procedures (nonlinear operator equations)
65J15Equations with nonlinear operators (numerical methods)
47H09Mappings defined by “shrinking” properties
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
47H05Monotone operators (with respect to duality) and generalizations
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