Skip to main content
SpringerLink
Log in

A general iteration scheme for variational inequality problem and common fixed point problems of nonexpansive mappings in q-uniformly smooth Banach spaces

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

In this paper, we introduce a 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 systems of variational inequalities for two inverse strongly accretive mappings in a q-uniformly smooth Banach space. Then, we prove a strong convergence theorem for the iterative sequence generated by the proposed iterative algorithm under very mild conditions. The methods in the paper are novel and different from those in the early and recent literature. Our results can be viewed as improvement, supplementation, development and extension of the corresponding results in some references to a great extent.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Reich S.: Asymptotic behavior of contractions in Banach spaces. J. Math. Anal. Appl. 44, 57–70 (1973)

    Article  Google Scholar 

  2. James R.C.: Orthogonality and linear functionals in normed linear spaces. Trans. Am. Math. Soc. 61, 265–292 (1947)

    Article  Google Scholar 

  3. Aoyama K., Kimura Y., Takahashi W., Toyoda M.: Approximation of common fixed point of a countable family of nonexpansive mapping in a Banach space. Nonlinear Anal. Theory Methods Appl. 67, 2350–2360 (2007)

    Article  Google Scholar 

  4. Stampacchi G.: Formes bilineaires coercivites sur les ensembles convexes. C. R. Acad. Sci. Paris 258, 4413–4416 (1964)

    Google Scholar 

  5. Korpelevich G.M.: An extragradient method for finding addle points and for other problems. Ekon. Mat. Metody 12, 747–756 (1976)

    Google Scholar 

  6. Takahashi W., Toyoda M.: Weak convergence theorems for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 118, 417–428 (2003)

    Article  Google Scholar 

  7. 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)

    Article  Google Scholar 

  8. 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)

    Article  Google Scholar 

  9. Cai, G., Bu, S.Q.: Modified extragradient methods for variational inequality problems and fixed point problems for an infinite family of nonexpansive mappings in Banach spaces. J. Glob. Optim. doi:10.1007/s10898-012-9883-6

  10. 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. (2010). doi:10.1155/2010/916806

  11. Bruck R.E.: Properties of fixed point sets of nonexpansive mappings in Banach spaces. Trans. Am. Math. Soc. 179, 251–262 (1973)

    Article  Google Scholar 

  12. Xu H.K.: Iterative algorithm for nonlinear operators. J. Lond. Math. Soc. 2, 1–17 (2002)

    Google Scholar 

  13. Kamimura S., Takahashi W.: Strong convergence of a proximal-type algorithmin a Banach space. SIAM J. Optim. 13, 938–945 (2002)

    Article  Google Scholar 

  14. Xu H.K.: Inequalities in Banach spaces with applications. Nonlinear Anal. 16, 1127–1138 (1991)

    Article  Google Scholar 

  15. Mitrinovic` D.S.: Analytic inequalities. Springer, New York (1970)

    Google Scholar 

  16. Pongsakorn, S., Poom, K.: Iterative methods for variational inequality problems and fixed point problems of a countable family of strict pseudo-contractions in a q-uniformly smooth Banach space. Fixed Point Theory Appl. (2012). doi:10.1186/1687-1812-2012-65

  17. Qin X.L., 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)

    Google Scholar 

  18. 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)

    Article  Google Scholar 

  19. Browder F.E., Petryshyn W.V.: Construction of fixed points of nonlinear mappings in Hilbert spaces. J. Math. Anal. Appl. 20, 197–228 (1967)

    Article  Google Scholar 

  20. Suzuki T.: Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals. J. Math. Anal. Appl. 305, 227–239 (2005)

    Article  Google Scholar 

  21. Song, Y.L., Hu, C.S.: Strong convergence theorems of a new general iterative process with Meir–Keeler contractions for a countable family of λ i-strict pseudocontractions in q-uniformly smooth Banach spaces. Fixed Point Theory Appl. (2010). doi:10.1155/2010/354202

  22. Yao Y., Noor M.A.: On viscosity iterative methods for variational inequalities. J. Math. Anal. Appl. 325, 776–787 (2007)

    Article  Google Scholar 

  23. Hao Y.: Strong convergence of an iterative method for inverse strongly accretive operators. J. Inequal. Appl. 2008, 420989 (2008). doi:10.1155/2008/42098

    Article  Google Scholar 

  24. Noor, M.A., Noor, I.K.: On general mixed variational inequalities. Acta Appl. Math. (2009). doi:10.1007/s10440-008-9402.4

  25. Jung J.S., Morales C.: The Mann process for perturbed m-accretive operators in Banach spaces. Nonlinear Anal. 46, 231–243 (2001)

    Article  Google Scholar 

  26. Shioji N., Takahashi W.: Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces. Proc. Am. Math. Soc. 12, 3461–3465 (1997)

    Google Scholar 

  27. Zhou H.Y.: Convergence theorems of common fixed points for a finite family of Lipschitz pseudocontractions in Banach spaces. Nonlinear Anal. 68, 2977–2983 (2008)

    Article  Google Scholar 

  28. Yao Y., Liou Y.C., Kang S.M.: Strong convergence of an iterative algorithm on an in? nite countable family of nonexpansive mappings. Appl. Math. Comput. 208, 211–218 (2009)

    Article  Google Scholar 

  29. Pardalos P.M., Rassias T.M., Khan A.A.: Nonlinear Analysis and Variational Problems. Springer, Berlin (2010)

    Book  Google Scholar 

  30. Kangtunyakarn, A.: Convergence theorem of common fixed points for a family of nonspreadingmappings in Hilbert space. Optim. Lett. (2012). doi:10.1007/s11590-011-0326-y

  31. Song, Y.L., Zeng, L.C.: Strong convergence of a new general iterative method for variational inequality problems in Hilbert spaces. Fixed Point Theory Appl. (2012). doi:10.1186/1687-1812-2012-46

  32. Noor M.A.: Projection-splitting algorithms for general mixed variational inequalities. J. Comput. Anal. Appl. 4, 47–61 (2002)

    Google Scholar 

  33. Jitpeera, T., Katchang, P., Kumam, P.: A viscosity of Cearo mean approximation method for a mixed equilibrium, variational inequality, and fixed point problems. Fixed Point Theory Appl. Article ID 945051, p. 24 (2011). doi:10.1155/2011/945051

  34. Chang S.S.: Set-valued variational inclusion in Banach spaces. J. Math. Anal. Appl. 248, 438–454 (2000)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China

    Yanlai Song & Luchuan Ceng

  2. Scientific Computing Key Laboratory, Shanghai University, Shanghai, 200234, China

    Luchuan Ceng

Authors
  1. Yanlai Song
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Luchuan Ceng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Luchuan Ceng.

Additional information

This research was supported by the National Science Foundation of China (11071169) and the Innovation Program of Shangha Municipal Education Commission (09ZZ133).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Song, Y., Ceng, L. A general iteration scheme for variational inequality problem and common fixed point problems of nonexpansive mappings in q-uniformly smooth Banach spaces. J Glob Optim 57, 1327–1348 (2013). https://doi.org/10.1007/s10898-012-9990-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-012-9990-4

Keywords

Mathematics Subject Classification (2010)

Search

Navigation