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)
An extragradient-like approximation method for variational inequality problems and fixed point problems. (English) Zbl 1124.65056
Summary: The purpose of this paper is to investigate the problem of finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of a variational inequality problem for a monotone, Lipschitz continuous mapping. The authors introduce an extragradient-like approximation method based on the so-called extragradient method and a viscosity approximation method. A strong convergence theorem is proved for two iterative sequences generated by this method.

MSC:
65K10Optimization techniques (numerical methods)
49J40Variational methods including variational inequalities
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
49M25Discrete approximations in calculus of variations
47H09Mappings defined by “shrinking” properties
WorldCat.org
Full Text: DOI
References:
[1] Browder, F. E.; Petryshyn, W. V.: Construction of fixed points of nonlinear mappings in Hilbert space. Journal of mathematical analysis and applications 20, 197-228 (1967) · Zbl 0153.45701
[2] Korpelevich, G. M.: The extragradient method for finding saddle points and other problems. Ekonomika i matematicheskie metody 12, 747-756 (1976) · Zbl 0342.90044
[3] Liu, F.; Nashed, M. Z.: Regularization of nonlinear ill-posed variational inequalities and convergence rates. Set-valued analysis 6, 313-344 (1998) · Zbl 0924.49009
[4] Goebel, K.; Kirk, W. A.: Topics on metric fixed-point theory. (1990) · Zbl 0708.47031
[5] Rockafellar, R. T.: On the maximality of sums of nonlinear monotone operators. Transactions of the American mathematical society 149, 75-88 (1970) · Zbl 0222.47017
[6] Xu, H. K.: Iterative algorithms for nonlinear operators. Journal of the London mathematical society 66, 240-256 (2002) · Zbl 1013.47032
[7] Takahashi, W.: Nonlinear functional analysis. (2000) · Zbl 0997.47002
[8] Takahashi, W.; Toyoda, M.: Weak convergence theorems for nonexpansive mappings and monotone mappings. Journal of optimization theory and applications 118, 417-428 (2003) · Zbl 1055.47052
[9] Suzuki, T.: Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals. Journal of mathematical analysis and applications 305, 227-239 (2005) · Zbl 1068.47085
[10] Nadezhkina, N.; Takahashi, W.: Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings. Journal of optimization theory and applications 128, 191-201 (2006) · Zbl 1130.90055
[11] Zeng, L. C.; Yao, J. C.: Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems. Taiwanese journal of mathematics 10, No. 5, 1293-1303 (2006) · Zbl 1110.49013
[12] Xu, H. K.: Viscosity approximation methods for nonexpansive mappings. Journal of mathematical analysis and applications 298, 279-291 (2004) · Zbl 1061.47060