Skip to main content
SpringerLink
Log in

Weak and strong convergence theorems for asymptotically pseudo-contraction mappings in the intermediate sense in Hilbert spaces

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

Abstract

In this paper, we prove both weak and strong convergence theorems for finding a common element of the solution set for a generalized equilibrium problem, the fixed point set of an asymptotically k-strict pseudo-contraction mapping in the intermediate sense, and the solution set of the variational inequality for a monotone and Lipschitz-continuous mapping by using a new hybrid extragradient method. Our results generalize and improve related results in the literatures.

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. Takahashi S., Takahashi W.: Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mappings in Hilbert space. Nonlinear Anal. 69, 1025–1033 (2008)

    Article  Google Scholar 

  2. Blum E., Oettli W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123–145 (1994)

    Google Scholar 

  3. Chadli O., Wong N.C., Yao J.C.: Equilibrium problems with applications to eigenvalue problems. J. Optim. Theory Appl. 117, 245–266 (2003)

    Article  Google Scholar 

  4. Ceng L.C., Yao J.C.: Hybrid viscosity approximation schemes for equilibrium problems and fixed point problems of infinitely many nonexpansive mappings. Appl. Math. Comput. 198, 729–741 (2008)

    Article  Google Scholar 

  5. Chang S.S., Lee W.J., Chan C.K.: A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization. Nonlinear Anal. 70, 3307–3319 (2009)

    Article  Google Scholar 

  6. Moudafi, A., Théra, M.: Proximal and Dynamical Approaches to equilibrium problems. In: Lecture Notes in Economics and Mathematical Systems, vol. 477, pp. 187–201. Springer, Berlin (1999)

  7. Iiduka H., Takahashi W.: Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings. Nonlinear Anal. 61, 341–350 (2005)

    Article  Google Scholar 

  8. Colao V., Marino G., Xu H.K.: A iterative method for finding common solutions of equilibrium and fixed point problems. J. Math. Anal. Appl. 344, 340–352 (2008)

    Article  Google Scholar 

  9. Combettes P.L., Hirstoaga A.: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 6, 117–136 (2005)

    Google Scholar 

  10. Takahashi S., Takahashi W.: Viscosity approximation methods for equilibrium problem and fixed point problems in Hilbert space. J. Math. Anal. Appl. 331, 506–515 (2007)

    Article  Google Scholar 

  11. Peng J.W., Yao J.C.: A new hybrid-extragradient method for generalized mixed equilibrium problems, fixed point problems and Variational inequality problems. Taiwan. J. Math. 12, 1401–1432 (2008)

    Google Scholar 

  12. Goebel K., Kirk W.A.: A fixed point theorem for asymptotically nonexpansive mappings. Proc. Am. Math. Soc. 35, 171–174 (1972)

    Article  Google Scholar 

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

    Article  Google Scholar 

  14. Kim T.H., Xu H.K.: Convergence of the modified Mann’s iteration method for asymptotically strict pseudo-contractions. Nonlinear Anal. 68, 2828–2836 (2008)

    Article  Google Scholar 

  15. Bruck R.E., Kuczumow T., Reich S.: Convergence of iterates of asymptotically nonexpansive mappings in Banach Space with the uniform opial property. Colloq. Math. 65, 169–179 (1993)

    Google Scholar 

  16. Sahu D.R., Xu H.K., Yao J.C.: Asymptotically strict pseudocontractive in the intermediate sense. Nonlinear Anal. 70, 3502–3511 (2009)

    Article  Google Scholar 

  17. Mann W.R.: Mean value methods in iteration. Proc. Am. Math. Soc. 4, 506–510 (1953)

    Article  Google Scholar 

  18. Schu J.: Weak and strong convergence to fixed points of asymptotically nonexpansive mappings. Bull. Aust. Math. Soc. 43, 153–159 (1991)

    Article  Google Scholar 

  19. Ceng L.C., Ansari Q.H., Yao J.C.: Mann-type steepest-descent and modified hybrid steepest-descent methods for variational inequalities in Banach spaces. Numer. Funct. Anal. Optim. 29, 987–1033 (2008)

    Article  Google Scholar 

  20. Chidume C.E., Shahzad N., Zegeye H.: Convergence theorems for mappings which are asymptotically nonexpansive in the intermediate sense. Numer. Funct. Anal. Optim. 25, 239–257 (2004)

    Article  Google Scholar 

  21. Ceng L.C., Wong N.C., Yao J.C.: Fixed point solutions of variational inequalities for a finite family of asymptotically nonexpansive mappings without common fixed point assumption. Math. Appl. 56, 2312–2322 (2008)

    Google Scholar 

  22. Kim G.E., Kim T.H.: Mann and Ishikawa iterations with errors for non-lipschitzian mappings in Banach spaces. Comput. Math. Appl. 42, 1565–1570 (2001)

    Article  Google Scholar 

  23. Huang S.: Fixed points of a sequence of asymptotically nonexpansive mappings. Fixed Point Theory 9, 465–485 (2008)

    Google Scholar 

  24. Huang S.: Hybrid extragradient methods for asymptotically strict pseudo-contractions in the intermediate sense and variational inequality problems. Optimization 60, 739–754 (2011)

    Article  Google Scholar 

  25. Qin X.L., Lin L.J., Kang S.M.: On a generalized Ky Fan inequality and asymptotically strict pseudocontractions in the intermediate sense. J. Optim. Theory Appl. 150, 553–579 (2011)

    Article  Google Scholar 

  26. Peng J.W., Yao J.C.: A new hybrid-extragradient method for generalized mixed equilibrium problems and fixed point problems and variational inequality problems. Taiwan. J. Math. 12, 1401–1433 (2008)

    Google Scholar 

  27. Ceng L.C., Ansari Q.H., Yao J.C.: Strong and weak convergence theorems for asymptotically strict pseudocontractive mappings in intermediate sense. J. Nonlinear Convex Anal. 11, 283–308 (2010)

    Google Scholar 

  28. Ceng L.C., Petrusel A., Yao J.C.: Iterative approaches to solving equilibrium problems and fixed point problems of infinitely many nonexpansive mappings. J. Optim. Theory Appl. 143, 37–58 (2009)

    Article  Google Scholar 

  29. Ceng L.C., Petrusel A., Yao J.C.: Iterative approximation of fixed points for asymptotically strict pseudocontractive type mappings in the intermediate sense. Taiwan. J. Math. 15, 587–606 (2011)

    Google Scholar 

  30. Ceng L.C., Sahu D.R., Yao J.C.: Implicit iterative algorithms for asymptotically nonexpansive mappings in the intermediate sense and Lipschitz-continuous monotone mappings. J. Comput. Appl. Math. 233, 2902–2915 (2010)

    Article  Google Scholar 

  31. Inchan I., Nammance K.: Strong convergence theorems by hybrid method for asymptotically k-strict pseudo-contractive in Hilbert Space. Nonlinear Anal. Hybrid Syst. 3, 380–385 (2009)

    Article  Google Scholar 

  32. Takahashi W.: Introduction to Nonlinear and Convex Analysis. Yokohoma Publishers, Yokohoma (2009)

    Google Scholar 

  33. Tan K.K., Xu H.K.: The nonlinear ergodic theorem for asymptotically nonexpansive mappings in Banach spaces. Proc. Am. Math. Soc. 114, 399–404 (1992)

    Article  Google Scholar 

  34. Yao J.C.: Variational inequalities with generalized monotone operators. Math. Oper. Res. 19, 691–705 (1994)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, National Changhua University of Education, Changhua, 50058, Taiwan

    Lai-Jiu Lin & Chih-Sheng Chuang

  2. Department of Mathematics, National Cheng Kung University, Tainan, 701, Taiwan

    Chih-Sheng Chuang

  3. Department of Electronic Engineering, Nan Kai University of Technology, Nantour, 54243, Taiwan

    Zenn-Tsun Yu

Authors
  1. Lai-Jiu Lin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zenn-Tsun Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Chih-Sheng Chuang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lai-Jiu Lin.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lin, LJ., Yu, ZT. & Chuang, CS. Weak and strong convergence theorems for asymptotically pseudo-contraction mappings in the intermediate sense in Hilbert spaces. J Glob Optim 56, 165–183 (2013). https://doi.org/10.1007/s10898-012-9968-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-012-9968-2

Keywords

Search

Navigation