Yao, Yonghong; Yao, Jen-Chih On modified iterative method for nonexpansive mappings and monotone mappings. (English) Zbl 1121.65064 Appl. Math. Comput. 186, No. 2, 1551-1558 (2007). Let \(C\) be a nonempty, closed, convex subset of a real Hilbert space \(H\) and let \(A:C\to H\) be an inverse-strongly monotone mapping. Moreover, let \(S:C\to C\) be nonexpansive. The authors study a new class of iteration schemes for finding a fixed-point \(x^*\) of \(S\) that also fulfills the variational inequality \(\langle Ax^*, v-x^*\rangle \geq 0\) for all \(v\in C\). Results on the strong convergence of the sequence of iterates are proven. Also the case \(A=I-T\) with a pseudocontractive mapping \(T:C\to C\) is considered. Reviewer: Etienne Emmrich (Berlin) Cited in 4 ReviewsCited in 122 Documents MSC: 65J15 Numerical solutions to equations with nonlinear operators 47H05 Monotone operators and generalizations 47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. 47J25 Iterative procedures involving nonlinear operators Keywords:nonexpansive mapping; monotone mapping; fixed point; variational inequality; iteration; convergence; Hilbert space × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Browder, F. E.; Petryshyn, W. V., Construction of fixed points of nonlinear mappings in hilbert space, Journal of Mathematical Analysis and Applications, 20, 197-228 (1967) · Zbl 0153.45701 [2] Korpelevich, G. M., An extragradient method for finding saddle points and for other problems, Ekonomika i Matematicheskie Metody, 12, 747-756 (1976) · Zbl 0342.90044 [3] Liu, F.; Nashed, M. Z., Regularization of nonlinear Ill-posed variational inequalities and convergence rates, Set-Valued Analysis, 6, 313-344 (1998) · Zbl 0924.49009 [4] Nadezhkina, N.; Takahashi, W., Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings, Journal of Optimization Theory and Applications, 128, 191-201 (2006) · Zbl 1130.90055 [5] Osilike, M. O.; Igbokwe, D. I., Weak and strong convergence theorems for fixed points of pseudocontractions and solutions of monotone type operator equations, Computers and Mathematics with Applications, 40, 559-567 (2000) · Zbl 0958.47030 [6] Suzuki, T., Strong convergence of krasnoselskii and mann’s type sequences for one-parameter nonexpansive semigroups without bochner integrals, Journal of Mathematical Analysis and Applications, 305, 227-239 (2005) · Zbl 1068.47085 [7] Takahashi, W.; Toyoda, M., Weak convergence theorems for nonexpansive mappings and monotone mappings, Journal of Optimization Theory and Applications, 118, 417-428 (2003) · Zbl 1055.47052 [8] Xu, H. K., Viscosity approximation methods for nonexpansive mappings, Journal of Mathematical Analysis and Applications, 298, 279-291 (2004) · Zbl 1061.47060 [9] Yao, J. C., Variational inequalities with generalized monotone operators, Mathematics of Operations Research, 19, 691-705 (1994) · Zbl 0813.49010 [10] Yao, J. C.; Chadli, O., Pseudomonotone complementarity problems and variational inequalities, (Crouzeix, J. P.; Haddjissas, N.; Schaible, S., Handbook of Generalized Convexity and Monotonicity (2005)), 501-558 · Zbl 1106.49020 [11] Zeng, L. C.; Schaible, S.; Yao, J. C., Iterative algorithm for generalized set-valued strongly nonlinear mixed variational-like inequalities, Journal of Optimization Theory and Applications, 124, 725-738 (2005) · Zbl 1067.49007 [12] Zeng, L. C.; Yao, J. C., Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems, Taiwanese Journal of Mathematics, 10, 1293-1303 (2006) · Zbl 1110.49013 [13] Zeng, L. C.; Wong, N. C.; Yao, J. C., Strong convergence theorems for strictly pseudocontractive mappings of Browder-Petryshyn type, Taiwanese Journal of Mathematics, 10, 837-850 (2006) · Zbl 1159.47054 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.