Abstract
In this paper, we introduce a new general iterative algorithm for finding a common element of the set of common fixed points of an infinite family of nonexpansive mappings and the set of solutions of a general variational inequality for two inverse-strongly accretive mappings in Banach space. We obtain some strong convergence theorems by a modified extragradient method under suitable conditions. Our results extend the recent results announced by many others.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Takahashi W.: Nonlinear Functional Analysis: Fixed Point Theory and Its Applications. Yokohama Publishers Inc., Yokohama (2002)
Cho Y.J., Kang S.M., Qin X.: Approximation of common fixed points of an infinite family of nonexpansive mappings in Banach spaces. Comput. Math. Appl. 56, 2058–2064 (2008)
Cho Y.J., Qin X.: Viscosity approximation methods for a finite family of m-accretive mappings in reflexive Banach spaces. Positivity 12, 483–494 (2008)
Qin X., Cho Y.J., Kang J.I., Kang S.M.: Strong convergence theorems for an infinite family of nonexpansive mappings in Banach spaces. J. Comput. Anal. Appl. 230(1), 121–127 (2009)
Yao Y., Liou Y.C., Kang S.M.: Strong convergence of an iterative algorithm on an infinite countable family of nonexpansive mappings. Appl. Math. Comput. 208, 211–218 (2009)
Marino G., Xu H.K.: A general iterative method for nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 318, 43–52 (2006)
Xu H.K.: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 2, 1–17 (2002)
Kumam P., Wattanawitoon K.: A general composite explicit iterative schemes of fixed point solutions of variational inequalities for nonexpansive semigroups. Math. Comput. Model. 53, 998–1006 (2011)
Li S., Li L., Su Y.: General iterative methods for a one-parameter nonexpansive semigroup in Hilbert spaces. Nonlinear Anal. 70, 3065–3071 (2009)
Plubtieng S., Wangkeeree R.: A general viscosity approximation method of fixed point solutions of variational inequalities for nonexpansive semigroups in Hilbert spaces. Bull. Korean Math. Soc. 45, 717–728 (2008)
Wangkeeree, R., Petrot, N., Wankeeree, R.: The general iterative methods for nonexpansive mappings in Banach spaces. J. Glob. Optim. doi:10.1007/s10898-010-9617-6
Ceng L.C., Yao J.C.: Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems. Taiwan J. Math. 10, 1293–1303 (2006)
Yao Y., Noor M.A.: On viscosity iterative methods for variational inequalities. J. Math. Anal. Appl. 325, 776–787 (2007)
Ceng L.C., Wang C., Yao J.C.: Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities. Math. Methods Oper. Res. 67, 375–390 (2008)
Yao Y., Noor M.A., Noor K.I., Liou Y.C.: Modified extragradient methods for a system of variational inequalities in Banach spaces. Acta Appl. Math. 110, 1211–1224 (2010)
Qin, X., Chang, S.S., Cho, Y.J., Kang, S.M.: Approximation of solutions to a system of variational inclusions in Banach spaces. J. Inequal. Appl., Article ID 916806, p. 16 (2010). doi:10.1155/2010/916806
Giannessi F., Maugeri A., Pardalos P.M.: Equilibrium Problems and Variational Models. Kluwer, Dordrecht (2001)
Pardalos P.M., Rassias T.M., Khan A.A.: Nonlinear Analysis and Variational Problems. Springer, Berlin (2010)
Eslamian, M.: Convergence theorems for nonspreading mappings and nonexpansive multivalued mappings and equilibrium problems. Optim. Lett. (2012). doi:10.1007/s11590-011-0438-4
Kangtunyakarn, A.: Convergence theorem of common fixed points for a family of nonspreading mappings in Hilbert space. Optim. Lett. (2012). doi:10.1007/s11590-011-0326-y
Reich S.: Asymptotic behavior of contractions in Banach spaces. J. Math. Anal. Appl. 44, 57–70 (1973)
Kitahara S., Takahashi W.: Image recovery by convex combinations of sunny nonexpansive retractions. Topol. Methods Nonlinear Anal. 2, 333–342 (1993)
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)
Xu H.K.: Inequalities in Banach spaces with applications. Nonlinear Anal. 16, 1127–1138 (1991)
Kamimura S., Takahashi W.: Strong convergence of a proximal-type algorithm in a Banach space. SIAM J. Optim. 13, 938–945 (2002)
Chang S.S.: On Chidumes open questions and approximate solutions of multivalued strongly accretive mapping equations in Banach spaces. J. Math. Anal. Appl. 216, 94–111 (1997)
Bruck R.E.: Properties of fixed point sets of nonexpansive mappings in Banach spaces. Trans. Am. Math. Soc. 179, 251–262 (1973)
Plubtieng S., Ungchittrakool K.: Approximation of common fixed points for a countable family of relatively nonexpansive mappings in a Banach space and applications. Nonlinear Anal. 72, 2896–2908 (2010)
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by the NSF of China (No.11171172).
Rights and permissions
About this article
Cite this article
Cai, G., Bu, S. Modified extragradient methods for variational inequality problems and fixed point problems for an infinite family of nonexpansive mappings in Banach spaces. J Glob Optim 55, 437–457 (2013). https://doi.org/10.1007/s10898-012-9883-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-012-9883-6