zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
A viscosity approximation method for equilibrium problems, fixed point problems of nonexpansive mappings and a general system of variational inequalities. (English) Zbl 1203.47064
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 α and μ-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:
47J25Iterative procedures (nonlinear operator equations)
47H09Mappings defined by “shrinking” properties
90C33Complementarity and equilibrium problems; variational inequalities (finite dimensions)
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 · doi:10.1016/j.na.2006.08.032
[2]Blum E., Oettli W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123–145 (1994)
[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 · doi:10.1016/0022-247X(67)90085-6
[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 · doi:10.1007/s00186-007-0207-4
[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 · doi:10.1016/j.na.2008.04.035
[6]Combettes P.L., Hirstoaga S.A.: Equilibrium programming using proximal-like algorithms. Math. Prog. 78, 29–41 (1997) · doi:10.1016/S0025-5610(96)00071-8
[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)
[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 · doi:10.1023/A:1008643727926
[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 · doi:10.1007/s10957-005-7564-z
[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 · doi:10.1016/S0898-1221(00)00179-6
[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 · doi:10.1016/j.amc.2007.07.075
[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)
[14]Shimoji K., Takahashi W.: Strong convergence to common fixed points of infinite nonexpansive mappings and applications. Taiwanese J. Math. 5, 387–404 (2001)
[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 · doi:10.1016/j.jmaa.2004.11.017
[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 · doi:10.1016/S0895-7177(00)00218-1
[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 · doi:10.1016/j.jmaa.2006.08.036
[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 · doi:10.1023/A:1025407607560
[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)
[21]Verma R.U.: Iterative algorithms and a new system of nonlinear quasivariational inequalities. Adv. Nonlinear Var. Inequal. 4(1), 117–124 (2001)
[22]Wittmann R.: Approximation of fixed points of nonexpansive mappings. Arch. Math. 58, 486–491 (1992) · Zbl 0797.47036 · doi:10.1007/BF01190119
[23]Xu H.K.: Viscosity approximation methods for nonexpansive mappings. J. Math. Anal. Appl. 298, 279–291 (2004) · Zbl 1061.47060 · doi:10.1016/j.jmaa.2004.04.059
[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)
[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 · doi:10.1016/j.amc.2006.08.062
[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 · doi:10.1007/s10957-004-1182-z
[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)