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2009 | 7 | 2 | 335-347
Tytuł artykułu

An extragradient iterative scheme by viscosity approximation methods for fixed point problems and variational inequality problems

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
In this paper, we introduce a new iterative process for finding the common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality problem for an α-inverse-strongly-monotone, by combining an modified extragradient scheme with the viscosity approximation method. We prove a strong convergence theorem for the sequences generated by this new iterative process.
Wydawca
Czasopismo
Rocznik
Tom
7
Numer
2
Strony
335-347
Opis fizyczny
Daty
wydano
2009-06-01
online
2009-05-24
Twórcy
  • Department of Applied Mathematics, Babeş-Bolyai University Cluj-Napoca, Cluj-Napoca, Romania, petrusel@math.ubbcluj.ro
autor
Bibliografia
  • [1] Browder F.E., Petryshyn W.V., Construction of fixed points of nonexpansive mappings in Hilbert space, J. Math. Anal. Appl., 1967, 20, 197–228 http://dx.doi.org/10.1016/0022-247X(67)90085-6[Crossref]
  • [2] Ceng L.-C., Petruşel A., Yao J.-C., Weak convergence theorem by a modified extragradient method for nonexpansive mappings and monotone mappings, Fixed Point Theory, 2008, 9, 73–87
  • [3] Chen J., Zhang L., Fan T., Viscosity approximation methods for nonexpansive mappings and monotone mappings, J. Math. Anal. Appl., 2007, 334, 1450–1461 http://dx.doi.org/10.1016/j.jmaa.2006.12.088[Crossref]
  • [4] Goebel K., Kirk W.A., Topics on metric fixed point theory, Cambridge University Press, Cambridge, 1990
  • [5] Korpelevich G.M., An extragradient method for finding saddle points and for other problems, Ekonomika i Matematicheskie Metody, 1976, 12, 747–756
  • [6] Liu F., Nashed M.Z., Regularization of nonlinear ill-posed variational inequalities and convergence rates, Set-Valued Analysis, 1998, 6, 313–344 http://dx.doi.org/10.1023/A:1008643727926[Crossref]
  • [7] Moudafi A., Viscosity approximating methods for fixed point problems, J. Math. Anal. Appl., 2000, 241, 46–55 http://dx.doi.org/10.1006/jmaa.1999.6615[Crossref]
  • [8] Nadezhkina N., Takahashi W., Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings, J. Optim. Theory and Appl., 2006, 128, 191–201 http://dx.doi.org/10.1007/s10957-005-7564-z[Crossref]
  • [9] Noor M.A., Modified projection method for pseudomonotone variational inequalities, Applied Mathematics Letters, 2002, 15, 315–320 http://dx.doi.org/10.1016/S0893-9659(01)00137-9[Crossref]
  • [10] Opial Z., Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc., 1967, 73, 591–597 http://dx.doi.org/10.1090/S0002-9904-1967-11761-0[Crossref]
  • [11] Peng J.-W., Yao J.-C., A modified CQ method for equilibrium problems, fixed points and variational inequality, Fixed Point Theory, 2008, 9, 515–531
  • [12] Rockafellar R.T., On the maximality of sums of nonlinear monotone operators, Trans. Amer. Math. Soc., 1970, 149, 75–88 http://dx.doi.org/10.2307/1995660[Crossref]
  • [13] Schu J., Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bull. Austral. Math. Soc., 1991, 43, 153–159 http://dx.doi.org/10.1017/S0004972700028884[Crossref]
  • [14] Stampacchia G., Formes bilineaires coercivities sur les ensembles convexes, C.R. Acad. Sci. Paris, 1964, 258, 4413–4416
  • [15] Su Y., Shang M., Qin X., A general iterative scheme for nonexpansive mappings and inverse-strongly monotone mappings, J. Appl. Math. Comput., 2008, 28, 283–294 http://dx.doi.org/10.1007/s12190-008-0103-y[Crossref]
  • [16] Takahashi W., Toyoda M., Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 2003, 118, 417–428 http://dx.doi.org/10.1023/A:1025407607560[Crossref]
  • [17] Wang S., Guo B., Viscosity approximation methods for nonexpansive mappings and inverse-strongly monotone mappings in Hilbert spaces, J. Appl. Math. Comput., 2008, 28, 351–365 http://dx.doi.org/10.1007/s12190-008-0109-5[Crossref]
  • [18] Xu H.K., Viscosity approximating methods for nonexpansive mappings, J. Math. Anal. Appl., 2004, 298, 279–291 http://dx.doi.org/10.1016/j.jmaa.2004.04.059[Crossref]
  • [19] Xu H.K., Kim T.H., Convergence of hybrid steepest-descend methods for variational inequalities, J. Optim. Theory Appl., 2003, 119, 185–201 http://dx.doi.org/10.1023/B:JOTA.0000005048.79379.b6[Crossref]
  • [20] 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, 2006, 10, 1293–1303
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-009-0003-x
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