, Volume 15, Issue 2, pp 281–295 | Cite as

An iterative algorithm for finding a common solution of fixed points and a general system of variational inequalities for two inverse strongly accretive operators

  • Phayap Katchang
  • Poom Kumam


In this paper, we introduce an iterative scheme based on a viscosity approximation method with a modified extragradient method for finding a common solutions of a general system of variational inequalities for two inverse-strongly accretive operator and solutions of fixed point problems involving the nonexpansive mapping in Banach spaces. Consequently, we obtain new strong convergence theorems in the frame work of Banach spaces. Our results extend and improve the recent results of Qin et al. (J Comput Appl Math 233:231–240, 2009) and many others.


Inverse-strongly accretive Fixed point Iteration Banach space Variational inequality Viscosity approximation Nonexpansive mapping Extragradient method 

Mathematics Subject Classification (2000)

Primary 47H05 47H10 47J25 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aoyama, K., Iiduka, H., Takahashi, W.: Weak convergence of an iterative sequence for accretive operators in Banach spaces. In: Fixed Point Theory and Applications, vol. 2006, article 35390, pp. 1–13Google Scholar
  2. 2.
    Browder F.E.: Fixed point theorems for noncompact mappings in Hilbert spaces. Proc. Natl. Acad. Sci. 53, 1272–1276 (1965)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Bruck R.E.: Properties of fixed point sets of nonexpansive mappings in Banach spaces. Trans. Am. Math. Soc. 179, 251–262 (1973)MathSciNetMATHCrossRefGoogle Scholar
  4. 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. Methods Oper. Res. 67, 375–390 (2008)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    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)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Cho Y.J., Yao Y., Zhou H.: Strong convergence of an iterative algorithm for accretive operators in Banach spaces. J. Comput. Anal. Appl. 10(1), 113–125 (2008)MathSciNetMATHGoogle Scholar
  7. 7.
    Hao Y.: Iterative algorithms for inverse-strongly accretive mappings with applications. J. Appl. Math. Comput. 31, 193–202 (2009)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Katchang P., Khamlae Y., Kumam P.: A viscosity iterative schemes for inverse-strongly accretive operators in Banach spaces. J. Comput. Anal. Appl. 12(3), 678–686 (2010)MathSciNetMATHGoogle Scholar
  9. 9.
    Kitahara S., Takahashi W.: Image recovery by convex combinations of sunny nonexpansive retractions. Methods Nonlinear Anal. 2, 333–342 (1993)MathSciNetMATHGoogle Scholar
  10. 10.
    Osilike M.O., Igbokwe D.I.: Weak and strong convergence theorems for fixed points of pseudocontractions and solutions of monotone type operator equations. Comput. Math. Appl. 40, 559–567 (2000)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Qin X., Cho S.Y., Kang S.M.: Convergence of an iterative algorithm for systems of variational inequalities and nonexpansive mappings with applications. J. Comput. Appl. Math. 233, 231–240 (2009)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Reich S.: Asymptotic behavior of contractions in Banach spaces. J. Math. Anal. Appl. 44(1), 57–70 (1973)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Reich S.: Strong convergence theorems for resolvents of accretive operators in Banach spaces. J. Math. Anal. Appl. 75, 287–292 (1980)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Suzuki T.: Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals. J. Math. Anal. Appl. 305(1), 227–239 (2005)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Xu H.K.: Inequalities in Banach spaces with applications. Nonlinear Anal. 16, 1127–1138 (1991)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Xu H.K.: Viscosity approximation methods for nonexpansive mappings. J. Math. Anal. Appl. 298, 279–291 (2004)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Yao Y., Noor M.A., Noor K.I., Liou Y.-C., Yaqoob H.: Modified extragradient methods for a system of variational inequalities in Banach spaces. Acta Appl. Math. 110, 1211–1224 (2010)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceKing Mongkut’s University of Technology Thonburi (KMUTT)BangkokThailand

Personalised recommendations