×

Weak convergence theorems for a system of mixed equilibrium problems and nonspreading mappings in a Hilbert space. (English) Zbl 1189.47067

From the summary: We introduce an iterative sequence and prove a weak convergence theorem for finding a solution of a system of mixed equilibrium problems and the set of fixed points of a quasi-nonexpansive mapping in Hilbert spaces. Moreover, we apply our result to obtain a weak convergence theorem for finding a solution of a system of mixed equilibrium problems and the set of fixed points of a nonspreading mapping. Using this result, we improve and unify several results in fixed point problems and equilibrium problems.

MSC:

47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
PDF BibTeX XML Cite
Full Text: DOI EuDML

References:

[1] Kohsaka, F; Takahashi, W, Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces, Archiv der Mathematik, 91, 166-177, (2008) · Zbl 1149.47045
[2] Mann, WR, Mean value methods in iteration, Proceedings of the American Mathematical Society, 4, 506-510, (1953) · Zbl 0050.11603
[3] Dotson, WG, On the Mann iterative process, Transactions of the American Mathematical Society, 149, 65-73, (1970) · Zbl 0203.14801
[4] Iemoto, S; Takahashi, W, Approximating common fixed points of nonexpansive mappings and nonspreading mappings in a Hilbert space, Nonlinear Analysis: Theory, Methods & Applications, 71, 2082-2089, (2009) · Zbl 1239.47054
[5] Blum, E; Oettli, W, From optimization and variational inequalities to equilibrium problems, The Mathematics Student, 63, 123-145, (1994) · Zbl 0888.49007
[6] Moudafi, A, From alternating minimization algorithms and systems of variational inequalities to equilibrium problems, Communications on Applied Nonlinear Analysis, 16, 31-35, (2009) · Zbl 1178.49020
[7] Verma, RU, Projection methods, algorithms, and a new system of nonlinear variational inequalities, Computers & Mathematics with Applications, 41, 1025-1031, (2001) · Zbl 0995.47042
[8] Ceng, L-C; Yao, J-C, A hybrid iterative scheme for mixed equilibrium problems and fixed point problems, Journal of Computational and Applied Mathematics, 214, 186-201, (2008) · Zbl 1143.65049
[9] Yao, Y; Liou, Y-C; Yao, J-C, A new hybrid iterative algorithm for fixed-point problems, variational inequality problems, and mixed equilibrium problems, (2008) · Zbl 1203.47087
[10] Ceng, L-C; Wang, C-Y; Yao, J-C, Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities, Mathematical Methods of Operations Research, 67, 375-390, (2008) · Zbl 1147.49007
[11] Osilike, MO; Igbokwe, DI, Weak and strong convergence theorems for fixed points of pseudocontractions and solutions of monotone type operator equations, Computers & Mathematics with Applications, 40, 559-567, (2000) · Zbl 0958.47030
[12] Goebel K, Kirk WA: Topics in Metric Fixed Point Theory, Cambridge Studies in Advanced Mathematics. Volume 28. Cambridge University Press, Cambridge, UK; 1990:viii+244. · Zbl 0708.47031
[13] 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
[14] Xu, H-K, Viscosity approximation methods for nonexpansive mappings, Journal of Mathematical Analysis and Applications, 298, 279-291, (2004) · Zbl 1061.47060
[15] Flåm, SD; Antipin, AS, Equilibrium programming using proximal-like algorithms, Mathematical Programming, 78, 29-41, (1997) · Zbl 0890.90150
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.