Monotone CQ iteration processes for nonexpansive semigroups and maximal monotone operators

Nakajo and Takahashi [K. Nakajo, W. Takahashi, Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl. 279 (2003) 372–379] proved strong convergence theorems for nonexpansive mappings, nonexpansive semigroups and the proximal point algorithm for zero-point of monotone operators in Hilbert spaces by the CQ iteration method. The purpose of this paper is to modify the CQ iteration method of K. Nakajo and W. Takahashi using the monotone CQ method, and to prove strong convergence theorems. In the proof process of this article, the Cauchy sequence method is used, so we proceed without use of the demiclosedness principle and Opial’s condition, and other weak topological techniques.