We prove that recent results of Wang (2007) concerning the iterative approximation of fixed points of nonexpansive mappings using a hybrid iteration method in Hilbert spaces can be extended to arbitrary Banach spaces without the strong monotonicity assumption imposed on the hybrid operator.
References
-
Wang, L: An iteration method for nonexpansive mappings in Hilbert spaces. Fixed Point Theory and Applications. 2007, 8 pages (2007)
-
Xu, HK, Kim, TH: Convergence of hybrid steepest-descent methods for variational inequalities. Journal of Optimization Theory and Applications. 119(1), 185–201 (2003)
-
Yamada, I: The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings. In: Butnariu D, Censor Y, Reich S (eds.) Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications (Haifa, 2000), Studies in Computational Mathematics, vol. 8, pp. 473–504. North-Holland, Amsterdam, The Netherlands (2001)
-
Opial, Z: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bulletin of the American Mathematical Society. 73, 591–597 (1967). Publisher Full Text
-
Jung, JS: Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces. Journal of Mathematical Analysis and Applications. 302(2), 509–520 (2005). Publisher Full Text
-
Osilike, MO, Aniagbosor, SC: Weak and strong convergence theorems for fixed points of asymptotically nonexpansive mappings. Mathematical and Computer Modelling. 32(10), 1181–1191 (2000). Publisher Full Text
-
Osilike, MO, Aniagbosor, SC, Akuchu, BG: Fixed points of asymptotically demicontractive mappings in arbitrary Banach spaces. Panamerican Mathematical Journal. 12(2), 77–88 (2002)
-
Deng, L: Convergence of the Ishikawa iteration process for nonexpansive mappings. Journal of Mathematical Analysis and Applications. 199(3), 769–775 (1996). Publisher Full Text