[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Springer Nature Link
Log in

The general iterative methods for nonexpansive semigroups in Banach spaces

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

Abstract

Let E be a real reflexive strictly convex Banach space which has uniformly Gâteaux differentiable norm. Let \({\mathcal{S} = \{T(s): 0 \leq s < \infty\}}\) be a nonexpansive semigroup on E such that \({Fix(\mathcal{S}) := \cap_{t\geq 0}Fix( T(t) ) \not= \emptyset}\) , and f is a contraction on E with coefficient 0 <  α <  1. Let F be δ-strongly accretive and λ-strictly pseudo-contractive with δ + λ >  1 and \({0 < \gamma < \min\left\{\frac{\delta}{\alpha}, \frac{1-\sqrt{ \frac{1-\delta}{\lambda} }}{\alpha} \right\} }\) . When the sequences of real numbers {α n } and {t n } satisfy some appropriate conditions, the three iterative processes given as follows :

$${\left.\begin{array}{ll}{x_{n+1} = \alpha_n \gamma f(x_n) + (I - \alpha_n F)T(t_n)x_n,\quad n\geq 0,}\\ {y_{n+1} = \alpha_n \gamma f(T(t_n)y_n) + (I - \alpha_n F)T(t_n)y_n,\quad n\geq 0,}\end{array}\right.}$$

and

$$ z_{n+1} = T(t_n)( \alpha_n \gamma f(z_n) + (I - \alpha_n F)z_n),\quad n\geq 0 $$

converge strongly to \({\tilde{x}}\) , where \({\tilde{x}}\) is the unique solution in \({Fix(\mathcal{S})}\) of the variational inequality

$${ \langle (F - \gamma f)\tilde {x}, j(x - \tilde{x}) \rangle \geq 0,\quad x\in Fix(\mathcal{S}).}$$

Our results extend and improve corresponding ones of Li et al. (Nonlinear Anal 70:3065–3071, 2009) and Chen and He (Appl Math Lett 20:751–757, 2007) and many others.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Shioji N., Takahashi W.: Strong convergence theorems for asymptotically nonexpansive semigroups in Hilbert spaces. Nonlinear Anal. 34, 87–99 (1998)

    Article  Google Scholar 

  2. Shimizu T., Takahashi W.: Strong convergence to common fixed points of families of nonexpansive mappings. J. Math. Anal. Appl. 211, 71–83 (1997)

    Article  Google Scholar 

  3. Chen R., Song Y.: Convergence to common fixed point of nonexpansive semigroups. J. Comput. Appl. Math. 200, 566–575 (2007)

    Article  Google Scholar 

  4. Suzuki T.: On strong convergence to common fixed points of nonexpansive semigroups in Hilbert spaces. Proc. Am. Math. Soc. 131, 2133–2136 (2002)

    Article  Google Scholar 

  5. Benavides T.D., Acedo G.L., Xu H.K.: Construction of sunny nonexpansive retractions in Banach spaces. Bull. Aust. Math. Soc. 66(1), 9–16 (2002)

    Article  Google Scholar 

  6. Xu H.K.: A strong convergence theorem for contraction semigroups in Banach spaces. Bull. Aust. Math. Soc. 72, 371–379 (2005)

    Article  Google Scholar 

  7. Chen R., He H.: Viscosity approximation of common fixed points of nonexpansive semigroups in Banach space. Appl. Math. Lett. 20, 751–757 (2007)

    Article  Google Scholar 

  8. Chen R., He H.: Modified Mann Iterations for Nonexpansive Semigroups in Banach Space. Acta Mathematica Sinica 26(1), 193–202 (2010)

    Article  Google Scholar 

  9. Deutsch F., Yamada I.: Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings. Numer. Funct. Anal. Optim. 19, 33–56 (1998)

    Article  Google Scholar 

  10. Xu H.K.: An iterative approach to quadratic optimization. J. Optim. Theory Appl. 116, 659–678 (2003)

    Article  Google Scholar 

  11. Xu H.K.: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 66, 240–256 (2002)

    Article  Google Scholar 

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

    Book  Google Scholar 

  13. Giannessi F., Maugeri A., Pardalos Panos M. (eds): Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models. Kluwer, Dordrecht (2002)

    Google Scholar 

  14. Moudafi A.: Viscosity approximation methods for fixed-points problems. J. Math. Anal. Appl. 241, 46–55 (2000)

    Article  Google Scholar 

  15. Xu H.K.: Approximations to fixed points of contraction semigroups in Hilbert spaces. Numer. Funct. Anal. Optim. 19, 157–163 (1998)

    Article  Google Scholar 

  16. Marino G., Xu H.K.: A general iterative method for nonexpansive mapping in Hilbert spaces. J. Math. Anal. Appl. 318, 43–52 (2006)

    Article  Google Scholar 

  17. Li S., Li L., Su Y.: General iterative methods for a one-parameter nonexpansive semigroup in Hilbert space. Nonlinear Anal. 70, 3065–3071 (2009)

    Article  Google Scholar 

  18. Zeidler E.: Nonlinear Functional Analysis and Its Applications. III. Springer, New York, NY, USA (1985)

    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. Zhou H.: Convergence theorems for λ-strict pseudo-contractions in 2-uniformly smooth Banach spaces. Nonlinear Anal. 69, 3160–3173 (2008)

    Article  Google Scholar 

  21. Aleyner A., Censor Y.: Best approximation to common fixed points of a semigroup of nonexpansive operators. J. Nonlinear Convex Anal. 6(1), 137–151 (2005)

    Google Scholar 

  22. Aleyner A., Reich S.: An explicit construction of sunny nonexpansive retractions in Banach spaces. Fixed Point Theory Appl. 3, 295–305 (2005)

    Google Scholar 

  23. Bruck R.E.: Nonexpansive retracts of Banach spaces. Bull. Am. Math. Soc. 76, 384–386 (1970)

    Article  Google Scholar 

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

    Article  Google Scholar 

  25. Aoyama K., Kimura Y., Takahashi W., Toyoda M.: Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space. Nonlinear Anal. 67, 2350–2360 (2007)

    Article  Google Scholar 

  26. Piri, H., Vaezi, H.: Strong convergence of a generalized iterative method for semigroups of nonexpansive mappings in Hilbert spaces. Fixed Point Theory Appl. Article ID 907275, 16 pp. doi:10.1155/2010/907275. (2010)

  27. Megginson R.E.: An Introduction to Banach Space Theory. Springer, New York (1998)

    Book  Google Scholar 

  28. Takahashi, W.: Nonlinear Functional Analysis- Fixed Point Theory and its Applications. Yokohama Publishers Inc., Yokohama, (Japanese) (2000)

  29. Li X.N., Gu J.S.: Strong convergence of modified Ishikawa iteration for a nonexpansive semigroup in Banach spaces. Nonlinear Anal. 73, 1085–1092 (2010)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok, 65000, Thailand

    Rattanaporn Wangkeeree & Rabian Wangkeeree

Authors
  1. Rattanaporn Wangkeeree
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rabian Wangkeeree
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rattanaporn Wangkeeree.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wangkeeree, R., Wangkeeree, R. The general iterative methods for nonexpansive semigroups in Banach spaces. J Glob Optim 55, 417–436 (2013). https://doi.org/10.1007/s10898-011-9835-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-011-9835-6

Keywords

Mathematics Subject Classification (2000)

Search

Navigation