Abstract
In this paper, we introduce an iteration process of finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of a variational inequality problem for an inverse strongly-monotone mapping, and then obtain a weak convergence theorem. Using this result, we obtain a weak convergence theorem for a pair of a nonexpansive mapping and a strictly pseudocontractive mapping. Further, we consider the problem of finding a common element of the set of fixed points of a nonexpansive mapping and the set of zeros of an inverse strongly-monotone mapping.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Browder, F. E., and Petryshyn, W.V., Construction of Fixed Points of Nonlinear Mappings in Hilbert Space, Journal of Mathematical Analysis and Applications, Vol. 20, pp. 197-228, 1967.
Liu, F., and Nashed, M.Z., Regularization of Nonlinear Ill-Posed Variational Inequalities and Convergence Rates, Set-Valued Analysis, Vol. 6, pp. 313-344, 1998.
Takahashi, W., Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, Japan, 2000.
Takahashi, W., and Tamura, T., Convergence Theorems for a Pair of Nonexpansive Mappings, Journal of Convex Analysis, Vol. 5, pp. 45-56, 1998.
Yamada, I., The Hybrid Steepest-Descent Method for the Variational Inequality Problem over the Intersection of Fixed-Point Sets of Nonexpansive Mappings, Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, Edited by D. Butnariu, Y. Censor, and S. Reich, Kluwer Academic Publishers, Dordrecht, Holland, pp. 473-504, 2001.
Opial, Z., Weak Convergence of the Sequence of Successive Approximations for Nonexpansive Mappings, Bulletin of the American Mathematical Society, Vol. 73, pp. 591-597, 1967.
Browder, F. E., Fixed-Point Theorems for Noncompact Mappings in Hilbert Space, Proceedings of the National Academy of Sciences of the USA, Vol. 53, pp. 1272-1276, 1965.
Rockafellar, R. T., On the Maximality of Sums of Nonlinear Monotone Operators, Transactions of the American Mathematical Society, Vol. 149, pp. 75-88, 1970.
Schu, J., Weak and Strong Convergence to Fixed Points of Asymptotically Nonexpansive Mappings, Bulletin of the Australian Mathematical Society, Vol. 43, pp. 153-159, 1991.
Rhoades, B.E., Fixed-Point Iterations Using Infinite Matrices, Transactions of the American Mathematical Society, Vol. 196, pp. 161-176, 1974.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Takahashi, W., Toyoda, M. Weak Convergence Theorems for Nonexpansive Mappings and Monotone Mappings. Journal of Optimization Theory and Applications 118, 417–428 (2003). https://doi.org/10.1023/A:1025407607560
Issue Date:
DOI: https://doi.org/10.1023/A:1025407607560