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Monotone $CQ$ algorithm of fixed points for weak relatively nonexpansive mappings and applications. (English) Zbl 1199.47295
Summary: S.-Y. Matsushita and 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.
##### MSC:
 47J25 Iterative procedures (nonlinear operator equations) 65J15 Equations with nonlinear operators (numerical methods) 47H09 Mappings defined by “shrinking” properties 47H10 Fixed point theorems for nonlinear operators on topological linear spaces 47H05 Monotone operators (with respect to duality) and generalizations