Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem. (English) Zbl 1147.47052
The authors introduce two iterative sequences for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of an equilibrium problem in a Hilbert space. Then they show that one of the sequences converges strongly and the other converges weakly.
Reviewer: Jinhai Chen (Hongkong)
MSC:
47J25 | Iterative procedures involving nonlinear operators |
47H10 | Fixed-point theorems |
49J40 | Variational inequalities |
47H09 | Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. |
91B50 | General equilibrium theory |
Keywords:
equilibrium problems; nonexpansive mappings; firmly nonexpansive mappings; weak and strong convergence; monotonicityReferences:
[1] | Takahashi, W., Convex Analysis and Approximation of Fixed Points (2000), Yokohama: Yokohama Publishers, Yokohama · Zbl 1089.49500 |
[2] | Takahashi, W., Nonlinear Functional Analysis (2000), Yokohama: Yokohama Publishers, Yokohama · Zbl 0997.47002 |
[3] | Blum, E.; Oettli, W., From optimization and variational inequalities to equilibrium problems, Math. Stud., 63, 123-145 (1994) · Zbl 0888.49007 |
[4] | Flam, S. D.; Antipin, A. S., Equilibrium programming using proximal-like algorithms, Math. Program., 78, 29-41 (1997) · Zbl 0890.90150 · doi:10.1007/BF02614504 |
[5] | Moudafi, A.; Thera, M., Proximal and dynamical approaches to equilibrium problems, Lecture Notes in Economics and Mathematical Systems, vol. 477, 187-201 (1999), New York: Springer, New York · Zbl 0944.65080 |
[6] | Combettes, P. L.; Hirstoaga, S. A., Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal., 6, 117-136 (2005) · Zbl 1109.90079 |
[7] | Mann, W. R., Mean value methods in iteration, Proc. Am. Math. Soc., 4, 506-510 (1953) · Zbl 0050.11603 · doi:10.2307/2032162 |
[8] | Nakajo, K.; Takahashi, W., Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl., 279, 372-379 (2003) · Zbl 1035.47048 · doi:10.1016/S0022-247X(02)00458-4 |
[9] | Takahashi, W.; Toyoda, M., Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 113, 417-428 (2003) · Zbl 1055.47052 · doi:10.1023/A:1025407607560 |
[10] | Opial, Z., Weak convergence of the sequence of successive approximation for nonexpansive mappings, Bull. Am. Math. Soc., 73, 591-597 (1967) · Zbl 0179.19902 · doi:10.1090/S0002-9904-1967-11761-0 |
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