A hybrid approximation method for equilibrium and fixed point problems for a monotone mapping and a nonexpansive mapping. (English) Zbl 1163.49003

Summary: The purpose of this paper is to present an iterative scheme by a hybrid method for finding a common element of the set of fixed points of a nonexpansive mapping, the set of solutions of an equilibrium problem and the set of solutions of the variational inequality for \(\alpha \)-inverse-strongly monotone mappings in the framework of a Hilbert space. We show that the iterative sequence converges strongly to a common element of the above three sets under appropriate conditions. Additionally, the idea of our results are applied to find a zero of a maximal monotone operator and a strictly pseudocontractive mapping in a real Hilbert space.


49J40 Variational inequalities
47H10 Fixed-point theorems
47H05 Monotone operators and generalizations
49M30 Other numerical methods in calculus of variations (MSC2010)
47J20 Variational and other types of inequalities involving nonlinear operators (general)
Full Text: DOI


[1] Blum, E.; Oettli, W., From optimization and variational inequalities to equilibrium problems, Math. Student, 63, 123-145 (1994) · Zbl 0888.49007
[2] Combettes, P. L.; Hirstoaga, S. A., Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal., 6, 117-136 (2005) · Zbl 1109.90079
[3] Flam, S. D.; Antipin, A. S., Equilibrium progamming using proximal-link algorithms, Math. Program., 78, 29-41 (1997) · Zbl 0890.90150
[4] Genel, A.; Lindenstrass, J., An example concerning fixed points, Israel. J. Math., 22, 81-86 (1975) · Zbl 0314.47031
[5] Goebel, K.; Kirk, W. A., Topics in Metric Fixed Point Theory (1990), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0708.47031
[6] Iiduka, H.; Takahashi, W., Strong convergence theorems for nonexpansive mapping and inverse-strong monotone mappings, Nonlinear Anal., 61, 341-350 (2005) · Zbl 1093.47058
[7] Kirk, W. A., Fixed point theorem for mappings which do not increase distance, Amer. Math. Monthly, 72, 1004-1006 (1965) · Zbl 0141.32402
[8] Lia, L.; Song, W., A hybrid of the extragradient method and proximal point algorithm for inverse strongly monotone operators and maximal monotone operators in Banach spaces, Nonlinear Anal.: Hybrid Systems, 1, 398-413 (2007) · Zbl 1117.49011
[9] Mann, W. R., Mean value methods in iteration, Proc. Amer. Math. Soc., 4, 506-510 (1953) · Zbl 0050.11603
[10] Moudafi, A.; Thera, M., (Proximal and Dynamical Approaches to Equilibrium Problems. Proximal and Dynamical Approaches to Equilibrium Problems, Lecture Note in Economics and Mathematical Systems, vol. 477 (1999), Springer-Verlag: Springer-Verlag New York), 187-201 · Zbl 0944.65080
[11] Nakajo, K.; Takahashi, W., Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl., 279, 372-379 (2003) · Zbl 1035.47048
[12] Opial, Z., Weak convergence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc., 73, 591-597 (1967) · Zbl 0179.19902
[13] Reich, S., Weak convergence theorems for nonexpansive mappings, J. Math. Anal. Appl., 67, 274-276 (1979) · Zbl 0423.47026
[14] Rockafellar, R. T., On the maximality of sums of nonlinear monotone operators, Trans. Amer. Math. Soc., 149, 75-88 (1970) · Zbl 0222.47017
[15] Rockafellar, R. T., Monotone operators and proximal point algorithm, SIAM J. Control Optim., 14, 877-898 (1976) · Zbl 0358.90053
[16] Takahashi, S.; Takahashi, W., Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl., 331, 506-515 (2007) · Zbl 1122.47056
[17] Takahashi, W., Nonlinear Functional Analysis (2000), Yokohama Publishers: Yokohama Publishers Yokohama · Zbl 0997.47002
[18] Tada, A.; Takahashi, W., Weak and strong convergence theorems for a nonexpansive mappings and an equilibrium problem, J. Optim. Theory Appl., 133, 359-370 (2007) · Zbl 1147.47052
[19] Takahashi, W.; Toyoda, M., Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 118, 417-428 (2003) · Zbl 1055.47052
[20] 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
[21] Zeng, L. C.; Schaible, S.; Yao, J. C., Iterative algorithm for generalized set-valued strongly nonlinear mixed variational-like inequalities, J. Optim. Theory Appl., 124, 725-738 (2005) · Zbl 1067.49007
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