Skip to main content
SpringerLink
Log in

On a Generalized Ky Fan Inequality and Asymptotically Strict Pseudocontractions in the Intermediate Sense

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

In this paper, two iterative algorithms are considered for a generalized Ky Fan inequality and a fixed point problem of asymptotically strict pseudocontractions in the intermediate sense. Strong and weak convergence theorems are established in real Hilbert spaces.

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. Burachik, R.S., Lopez, J.O.: A convergence result for an outer approximation scheme. Comput. Appl. Math. 22, 397–409 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  2. Izmaelov, A.F., Solodov, M.V.: An active set Newton method for mathematical program with complementary constraints. SIAM J. Optim. 19, 1003–1027 (2008)

    Article  MathSciNet  Google Scholar 

  3. Maingé, P.E., Moudafi, A.: Strong convergence of an iterative method for hierarchical fixed-point problems. Pac. J. Optim. 3, 529–538 (2007)

    MathSciNet  MATH  Google Scholar 

  4. Yamada, I., Ogura, N.: Hybrid steepest descent method for the variational inequality problem over the fixed point set of certain quasi-nonexpansive mappings. Numer. Funct. Anal. Optim. 25, 619–655 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bauschke, H.H., Borwein, J.M.: On projection algorithms for solving convex feasibility problems. SIAM Rev. 38, 367–426 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  6. Combettes, P.L.: The convex feasibility problem in image recovery. In: Hawkes, P. (ed.) Advances in Imaging and Electron Physics. vol. 95, pp. 155–270. Academic Press, New York (1996).

    Google Scholar 

  7. Kotzer, T., Cohen, N., Shamir, J.: Image restoration by a novel method of parallel projection onto constraint sets. Optim. Lett. 20, 1772–1774 (1995)

    Google Scholar 

  8. Youla, D.C.: Mathematical theory of image restoration by the method of convex projections. In: Stark, H. (ed.) Image Recovery: Theory and Applications, pp. 29–77. Academic Press, New York (1987)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  10. Marino, G., Xu, H.K.: Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces. J. Math. Anal. Appl. 329, 336–346 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. Reich, S.: Weak convergence theorems for nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 67, 274–276 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  13. Genel, A., Lindenstrass, J.: An example concerning fixed points. Isr. J. Math. 22, 81–86 (1975)

    Article  MATH  Google Scholar 

  14. Martinez-Yanes, C., Xu, H.K.: Strong convergence of the CQ method for fixed point iteration processes. Nonlinear Anal. 64, 2400–2411 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. Nakajo, K., Takahashi, W.: Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. J. Math. Anal. Appl. 279, 372–379 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  16. Qin, X., Cho, S.Y., Kang, S.M.: On hybrid projection methods for asymptotically quasi-φ-nonexpansive mappings. Appl. Math. Comput. 215, 3874–3883 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Qin, X., Cho, Y.J., Kang, S.M., Zhou, H.: Convergence theorems of common fixed points for a family of Lipschitz quasi-pseudocontractions. Nonlinear Anal. 71, 685–690 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Qin, X., Cho, Y.J., Kang, S.M., Zhou, H.: Convergence of a modified Halpern-type iteration algorithm for quasi-φ-nonexpansive mappings. Appl. Math. Lett. 22, 1051–1055 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Qin, X., Su, Y.: Strong convergence theorems for relatively nonexpansive mappings in a Banach space. Nonlinear Anal. 67, 1958–1965 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  20. Takahashi, W., Takeuchi, Y., Kubota, R.: Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 341, 276–286 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Haugazeau, Y.: Sur les inéquations variationnelles et la minimisation de fonctionnelles convexes. Ph.D. Thesis, Université de Paris (1968)

  22. Kumam, P.: A hybrid approximation method for equilibrium and fixed point problems for a monotone mapping and a nonexpansive mapping. Nonlinear Anal. 2, 1245–1255 (2008)

    MathSciNet  MATH  Google Scholar 

  23. Tada, A., Takahashi, W.: Weak and strong convergence theorem for an equilibrium problem and a nonexpansive mapping. J. Optim. Theory Appl. 133, 359–370 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  25. Fan, K.: A minimax inequality and applications. In: Shisha, O. (ed.) Inequality III, pp. 103–113. Academic Press, New York (1972)

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  27. Brézis, H., Nirenberg, L., Stampacchia, G.: A remark on Ky Fan’s minimax principle. Boll. Unione Mat. Ital. 6(4), 293–300 (1972)

    MATH  Google Scholar 

  28. Bianchi, M., Schaible, S.: Generalized monotone bifunctions and equilibrium problems. J. Optim. Theory Appl. 90, 31–43 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  29. Konnov, I.V., Yao, J.C.: On the generalized vector variational inequality problem. J. Math. Anal. Appl. 206, 42–58 (1997)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  31. Chang, S.S., Lee, H.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  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  33. Moudafi, A.: Second-order differential proximal methods for equilibrium problems. J. Inequal. Pure Appl. Math. 4, 18 (2003)

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  35. Kirk, W.A.: Fixed point theorems for non-Lipschitzian mappings of asymptotically nonexpansive type. Isr. J. Math. 17, 339–346 (1974)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  37. Xu, H.K.: Existence and convergence for fixed points of mappings of asymptotically nonexpansive type. Nonlinear Anal. 16, 1139–1146 (1991)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  39. Qihou, L.: Convergence theorems of the sequence of iterates for asymptotically demicontractive and hemicontractive mappings. Nonlinear Anal. 26, 1835–1842 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  40. Schu, J.: Iterative construction of fixed points of asymptotically nonexpansive mappings. J. Math. Anal. Appl. 158, 407–413 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  41. Cho, Y.J., Qin, X., Kang, J.I.: Convergence theorems based on hybrid methods for generalized equilibrium problems and fixed point problems. Nonlinear Anal. 71, 4203–4214 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  42. Inchan, I., Nammanee, K.: Strong convergence theorems by hybrid method for asymptotically k-strict pseudo-contractive mapping in Hilbert space. Nonlinear Anal. 3, 380–385 (2009)

    MathSciNet  MATH  Google Scholar 

  43. Kumam, K., Petrot, N., Wangkeeree, R.: A hybrid iterative scheme for equilibrium problems and fixed point problems of asymptotically k-strict pseudo-contractions. J. Comput. Appl. Math. 233, 2013–2026 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  44. Qin, X., Cho, Y.J., Kang, S.M.: Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces. J. Comput. Appl. Math. 225, 20–30 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  45. Qin, X., Cho, S.Y., Kang, S.M.: Strong convergence of shrinking projection methods for quasi-φ-nonexpansive mappings and equilibrium problems. J. Comput. Appl. Math. 234, 750–760 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  46. Takahashi, W., Zembayashi, K.: Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces. Nonlinear Anal. 70, 45–57 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  47. Jung, J.S.: Strong convergence of iterative methods for k-strictly pseudo-contractive mappings in Hilbert spaces. Appl. Math. Comput. 215, 3746–3753 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  48. Rockafellar, R.T.: On the maximality of sums of nonlinear monotone operators. Trans. Am. Math. Soc. 149, 75–88 (1970)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  50. Tan, K.K., Xu, H.K.: Approximating fixed points of nonexpansive mappings by the Ishikawa iterative process. J. Math. Anal. Appl. 178, 301–308 (1993)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Information Sciences, North China University of Water Resources and Electric Power, Zhengzhou, 450011, China

    Xiaolong Qin

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

    Lai-Jiu Lin

  3. Department of Mathematics and the RINS, Gyeongsang National University, Jinju, 660-701, Korea

    Shin Min Kang

Authors
  1. Xiaolong Qin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lai-Jiu Lin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Shin Min Kang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shin Min Kang.

Additional information

Communicated by Jen-Chih Yao.

The authors are grateful to the referees for their valuable comments and suggestions.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Qin, X., Lin, LJ. & 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). https://doi.org/10.1007/s10957-011-9853-z

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-011-9853-z

Keywords

Search

Navigation