An iterative scheme for equilibrium problems and fixed point problems of strict pseudo-contraction mappings. (English) Zbl 1167.47307

Summary: We propose an iterative scheme for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a strict pseudo-contraction mapping in the setting of real Hilbert spaces. We establish some weak and strong convergence theorems of the sequences generated by our proposed scheme. Our results combine the ideas of G. Marino and H. K. Xu [J. Math. Anal. Appl. 329, No. 1, 336–346 (2007; Zbl 1116.47053)] and S. Takahashi and W. Takahashi [J. Math. Anal. Appl. 331, No. 1, 506–515 (2007; Zbl 1122.47056)]. In particular, necessary and sufficient conditions for the strong convergence of our iterative scheme are obtained.


47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems
47J20 Variational and other types of inequalities involving nonlinear operators (general)
65J15 Numerical solutions to equations with nonlinear operators
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] Byrne, C., A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse problems, 20, 103-120, (2004) · Zbl 1051.65067
[3] Ceng, L.C.; Yao, J.C., A hybrid iterative scheme for mixed equilibrium problems and fixed point problems, J. comput. appl. math., (2007) · Zbl 1128.49014
[4] Combettes, P.L.; Hirstoaga, S.A., Equilibrium programming in Hilbert spaces, J. nonlinear convex anal., 6, 117-136, (2005) · Zbl 1109.90079
[5] Flam, S.D.; Antipin, A.S., Equilibrium programming using proximal-like algorithms, Math. program., 78, 29-41, (1997) · Zbl 0890.90150
[6] Flores-Bazan, F., Existence theory for finite-dimensional pseudomonotone equilibrium problems, Acta appl. math., 77, 249-297, (2003) · Zbl 1053.90110
[7] Geobel, K.; Kirk, W.A., Topics on metric fixed-point theory, (1990), Cambridge University Press Cambridge, England
[8] Hadjisavvas, N.; Koml√≥si, S.; Schaible, S., Handbook of generalized convexity and generalized monotonicity, (2005), Springer-Verlag Berlin, Heidelberg, New York · Zbl 1070.26002
[9] Hadjisavvas, N.; Schaible, S., From scalar to vector equilibrium problems in the quasimonotone case, J. optim. theory appl., 96, 297-309, (1998) · Zbl 0903.90141
[10] Mann, W.R., Mean value methods in iteration, Proc. amer. math. soc., 4, 506-510, (1953) · Zbl 0050.11603
[11] Marino, G.; Xu, H.K., Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces, J. math. anal. appl., 329, 336-346, (2007) · Zbl 1116.47053
[12] Moudafi, A., Viscosity approximation methods for fixed-point problems, J. math. anal. appl., 241, 46-55, (2000) · Zbl 0957.47039
[13] Reich, S., Weak convergence theorems for nonexpansive mappings in Banach spaces, J. math. anal. appl., 67, 274-276, (1979) · Zbl 0423.47026
[14] Tada, A.; Takahashi, W., Strong convergence theorem for an equilibrium problem and a nonexpansive mapping, (), 609-617 · Zbl 1122.47055
[15] 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
[16] Tan, K.K.; Xu, H.K., Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. math. anal. appl., 178, 301-308, (1993) · Zbl 0895.47048
[17] Wittmann, R., Approximation of fixed points of nonexpansive mappings, Arch. math., 58, 486-491, (1992) · Zbl 0797.47036
[18] Xu, H.K., Iterative algorithms for nonlinear operators, J. London math. soc., 66, 240-256, (2002) · Zbl 1013.47032
[19] Zeng, L.C., A note on approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. math. anal. appl., 226, 245-250, (1998) · Zbl 0916.47047
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