A viscosity approximation method for equilibrium problems, fixed point problems of nonexpansive mappings and a general system of variational inequalities. (English) Zbl 1203.47064

Plubtieng, S.; Thammathiwat, T.

J. Glob. Optim. 46, No. 3, 447-464 (2010).

Summary: We introduce and study a new iterative scheme for finding the common element of the set of common fixed points of a sequence of nonexpansive mappings, the set of solutions of an equilibrium problem and the set of solutions of the general system of variational inequality for \(\alpha \) and \(\mu \)-inverse-strongly monotone mappings. We show that the sequence converges strongly to a common element of the above three sets under some parameters controlling conditions. This main theorem extends a recent result of L.-C. Ceng, C.-Y. Wang and J.-C. Yao [Math. Methods Oper. Res. 67, No. 3, 375–390 (2008; Zbl 1147.49007)] and many others.

MSC:

47J25	Iterative procedures involving nonlinear operators
47H09	Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
90C33	Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)

Citations:

Zbl 1147.49007

References:

[1]	Aoyama K., Kimura Y., Takahashi W., Toyoda M.: Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space. Nonlinear Anal. 67, 2350–2360 (2007) · Zbl 1130.47045
[2]	Blum E., Oettli W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123–145 (1994) · Zbl 0888.49007
[3]	Browder F.E., Petryshyn W.V.: Construction of fixed points of nonlinear mappings in Hilbert space. J. Math. Anal. Appl. 20, 197–228 (1967) · Zbl 0153.45701
[4]	Ceng L.C., Wang C.Y., Yao J.C.: Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities. Math. Meth. Oper. Res. 67, 375–390 (2008) · Zbl 1147.49007
[5]	Chang S.S., Joseph Lee H.W., Chan C.K.: A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization. Nonlinear Anal. 70, 3307–3319 (2009) · Zbl 1198.47082
[6]	Combettes P.L., Hirstoaga S.A.: Equilibrium programming using proximal-like algorithms. Math. Prog. 78, 29–41 (1997)
[7]	Geobel, K., Kirk, W.A.: Topics in metric fixed point theory Cambridge Stud. Adv. Math., vol. 28. Cambridge University Press (1990)
[8]	Kumam, W., Kumam, P.: Hybrid iterative scheme by a relaxed extragradient method for solutions of equilibrium problems and a general system of variational inequalities with application to optimization. Nonlinear. Anal. (2009). doi: 10.1016/j.nahs.2009.05.007(2009) · Zbl 1175.49009
[9]	Liu F., Nashed M.Z.: Regularization of nonlinear Ill-posed variational inequalities and convergence rates. Set-Valued Anal. 6, 313–344 (1998) · Zbl 0924.49009
[10]	Nadezhkina N., Takahashi W.: Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 128, 191–201 (2006) · Zbl 1130.90055
[11]	Osilike M.O., Igbokwe D.I.: Weak and strong convergence theorems for fixed points of pseudocontractions and solutions of monotone type operator equations. Comp. Math. Appl. 40, 559–567 (2000) · Zbl 0958.47030
[12]	Plubtieng S., Punpaeng R.: A new iterative method for equilibrium problems and fixed point problems of nonexpansive mappings and monotone mappings. Appl. Math. Comput. 197, 548–558 (2008) · Zbl 1154.47053
[13]	Plubtieng S., Thammathiwat T.: A viscosity approximation method for finding a common fixed point of nonexpansive and firmly nonexpansive mappings in Hilbert spaces. Thai. J. Math. 6(2), 377–390 (2008) · Zbl 1184.47047
[14]	Shimoji K., Takahashi W.: Strong convergence to common fixed points of infinite nonexpansive mappings and applications. Taiwanese J. Math. 5, 387–404 (2001) · Zbl 0993.47037
[15]	Suzuki T.: Strong convergence of krasnoselskii and manns type sequences for one-parameter nonexpansive semigroups without bochner integrals. J. Math. Anal. Appl. 305, 227–239 (2005) · Zbl 1068.47085
[16]	Takahashi W.: Weak and strong convergence theorems for families of nonexpansive mappings and their applications. Ann. Univ. Marjae Curie-Sklodowska Sect. 51, 277–292 (1997)
[17]	Takahashi W., Shimoji K.: Convergence theorems for nonexpansive mappings and feasibility problems. Math. Comput. Model. 32, 1463–1471 (2000) · Zbl 0971.47040
[18]	Takahashi S., Takahashi W.: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl. 311(1), 506–515 (2007) · Zbl 1122.47056
[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]	Verma R.U.: On a new system of nonlinear variational inequalities and associated iterative algorithms. Math. Sci. Res., Hot-Line 3(8), 65–68 (1999) · Zbl 0970.49011
[21]	Verma R.U.: Iterative algorithms and a new system of nonlinear quasivariational inequalities. Adv. Nonlinear Var. Inequal. 4(1), 117–124 (2001) · Zbl 1014.47050
[22]	Wittmann R.: Approximation of fixed points of nonexpansive mappings. Arch. Math. 58, 486–491 (1992) · Zbl 0797.47036
[23]	Xu H.K.: Viscosity approximation methods for nonexpansive mappings. J. Math. Anal. Appl. 298, 279–291 (2004) · Zbl 1061.47060
[24]	Yao J.C., Chadli O.: Pseudomonotone complementarity problems and variational inequalities. In: Crouzeix, JP., Haddjissas, N., Schaible, S. (eds) Handbook of generalized convexity and monotonicity, pp. 501–558. Springer, Netherlands (2005) · Zbl 1106.49020
[25]	Yao Y., Yao J.C.: On modified iterative method for nonexpansive mappings and monotone mappings. Appl. Math. Comput. 186(2), 1551–1558 (2007) · Zbl 1121.65064
[26]	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
[27]	Zeng L.C., Yao J.C.: Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems. Taiwanese J. Math. 10, 1293–1303 (2006) · Zbl 1110.49013