Su, Yongfu; Qin, Xiaolong Monotone CQ iteration processes for nonexpansive semigroups and maximal monotone operators. (English) Zbl 1220.47122 Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 68, No. 12, 3657-3664 (2008). Summary: K. Nakajo and W. Takahashi [J. Math. Anal. Appl. 279, No. 2, 372–379 (2003; Zbl 1035.47048)] proved strong convergence theorems for nonexpansive mappings, nonexpansive semigroups and the proximal point algorithm for zero-points 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 [loc.cit.]using the monotone CQ method, and to prove strong convergence theorems. The Cauchy sequence method is used, so we proceed without use of the demiclosedness principle and Opial’s condition, and other weak topological techniques. Cited in 19 Documents MSC: 47J25 Iterative procedures involving nonlinear operators 47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. 47H10 Fixed-point theorems Keywords:strong convergence; CQ method; nonexpansive mapping; nonexpansive semigroup; proximal point algorithm Citations:Zbl 1035.47048 PDFBibTeX XMLCite \textit{Y. Su} and \textit{X. Qin}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 68, No. 12, 3657--3664 (2008; Zbl 1220.47122) Full Text: DOI References: [1] Takahashi, W., Nonlinear Functional Analysis (2000), Yokohama Publishers: Yokohama Publishers Yokohama [2] Byrne, C., A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Problems, 20, 103-120 (2004) · Zbl 1051.65067 [3] Podilchuk, C. I.; Mammone, R. J., Image recovery by convex projections using a least-squares constraint, J. Opt. Soc. Amer. A, 7, 517-521 (1990) [4] Sezan, M. I.; Stark, H., Applications of convex projection theory to image recovery in tomography and related areas, (Stark, H., Image Recovery Theory and Applications (1987), Academic Press: Academic Press Orlando), 415-562 [5] Youla, D., Mathematical theory of image restoration by the method of convex projections, (Stark, H., Image Recovery Theory and Applications (1987), Academic Press: Academic Press Orlando), 29-77 [6] Youla, D., On deterministic convergence of iterations of relaxed projection operators, J. Vis. Commun. Image Represent., 1, 12-20 (1990) [7] Reich, S., Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 67, 274-276 (1979) · Zbl 0423.47026 [8] Nakajo, K.; Takahashi, W., Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl., 279, 372-379 (2003) · Zbl 1035.47048 [9] Kamimura, S.; Takahashi, W., Approximating solutions of maximal monotone operators in Hilbert spaces, J. Approx. Theory, 106, 226-240 (2000) · Zbl 0992.47022 [10] Martinet, B., Regularisation d’inequations variationnelles par approximations successives, Rev. Francaise. Inform. Rech. Oper., 4, 154-159 (1970) · Zbl 0215.21103 [11] Rockafellar, R. T., Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14, 877-898 (1976) · Zbl 0358.90053 [12] Wittmann, R., Approximation of fixed points of nonexpansive mappings, Arch. Math., 58, 486-491 (1992) · Zbl 0797.47036 [13] Shimizu, T.; Takahashi, W., Strong convergence to common fixed points of families of nonexpansive mappings, J. Math. Anal. Appl., 211, 71-83 (1997) · Zbl 0883.47075 [14] Marino, G.; Xu, H. K., Convergence of generalized proximal point algorithms, Commun. Appl. Anal., 3, 791-808 (2004) · Zbl 1095.90115 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.