Skip to main content
SpringerLink
Log in

Convergence theorems for semigroup-type families of non-self mappings

  • Published:
Rendiconti del Circolo Matematico di Palermo Aims and scope Submit manuscript

Abstract

Motivated by a paper Chidume and Zegeye [Strong convergence theorems for common fixed points of uniformly L-Lipschitzian pseudocontractive semi-groups, Applicable Analysis, 86 (2007), 353–366], we prove several strong convergence theorems for a family (not necessarily a semigroup) ℱ = {T(t): t ∈ G} of nonexpansive or pseudocontractive non-self mappings in a reflexive strictly convex Banach space with a uniformly Gâteaux differentiable norm, where G is an unbounded subset of ℝ+. Our results extend and improve the corresponding ones byMatsushita and Takahashi [Strong convergence theorems for nonexpansive nonself-mappings without boundary conditions,Nonlinear Analysis, 68 (2008), 412–419],Morales and Jung [Convergence of paths for pseudo-contractive mappings in Banach spaces, Proceedings of American Mathematical Society, 128 (2000), 3411–3419], Song [Iterative approximation to common fixed points of a countable family of nonexpansive mappings, Applicable Analysis, 86 (2007), 1329–1337], Song and Xu [Strong convergence theorems for nonexpansive semigroup in Banach spaces, Journal of Mathematical Analysis and Applications, 338 (2008), 152–161], Wong, Sahu, and Yao [Solving variational inequalities involving nonexpansive type mappings, Nonlinear Analysis, (2007) doi:10.1016/j.na. 2007.11.025] in the context of a non-semigroup family of non-self mappings.

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. Alber, Ya.I., Guerre-Delabriere S.: Principles of weakly contractive maps in Hilbert spaces, in: I. Gohberg, Yu. Lyubich (Eds.): New Results in Operator Theory, Advances and Applications, 98, Birkhauser, Basel (1997), 7–22

    Google Scholar 

  2. Barbu, V., Precupanu, Th.: Convexity and Optimization in Banach spaces, Editura Academiei R. S. R., Bucharest (1978)

    Google Scholar 

  3. Browder, F.E.: Convergence of approximants to fixed points of nonexpansive nonlinear mappings in Banach spaces, Arch. Rational Mech. Anal., 24 (1967), 82–90

    Article  MATH  MathSciNet  Google Scholar 

  4. Chang, S.S.: On Chidume’s open questions and approximation solutions of multi-valued strongly accretive mapping equation in Banach spaces, J. Math. Anal. Appl., 216 (1997), 94–111

    Article  MATH  MathSciNet  Google Scholar 

  5. Chidume, C.E., Zegeye, H.: Strong convergence theorems for common fixed points of uniformly L-Lipschitzian pseudocontractive semi-groups, Appl. Anal., 86 (2007), 353–366

    Article  MATH  MathSciNet  Google Scholar 

  6. Cioranescu, L.:, Geometry of Banach spaces, duality mappings and nonlinear problems, Kluwer Aacademic Publishers, Dordrecht (1990)

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  8. Deimling, K.: Zeros of accretive operators, Manuscripta Math., 13 (1974), 365–374

    Article  MATH  MathSciNet  Google Scholar 

  9. Deimling, K.: Nonlinear Functional Analysis, Spring-Verlag, Berlin (1985)

    MATH  Google Scholar 

  10. Goebel, K., Reich, S.: Uniform convexity, hyperbolic geometry, and nonexpansive mappings, Marcel Dekker, Inc., (1984)

  11. Goebel, K., Kirk, W.A.: A fixed point theorem for asymptotically nonexpansivemappings, Proc. Amer.Math. Soc., 35 (1972), 171–174

    Article  MATH  MathSciNet  Google Scholar 

  12. Ha, K.S., Jung, J.S.: Strong convergence theorems for accretive operators in Banach space, J. Math. Anal. Appl., 147 (1990), 330–339

    Article  MATH  MathSciNet  Google Scholar 

  13. Halpern B.: Fixed points of nonexpandingmaps, Bull. Amer. Math. Soc., 73 (1967) 957–961

    Article  MATH  MathSciNet  Google Scholar 

  14. Halpern, B.R., Bergman, G.M.: A fixed-point theorem for inward and outward maps, Trans. Amer. Math. Soc., 130 (1968), 353–358

    Article  MATH  MathSciNet  Google Scholar 

  15. Jung, J.S., Sahu, D.R.: Convergence of approximating paths to solutions of variational inequalities involving non-Lipschitzian mappings, J. KoreanMath. Soc., 45 (2008), 377–392

    MATH  MathSciNet  Google Scholar 

  16. Martin, R.H.: Differential equations on closed subsets of a Banach space, Trans. Amer. Math. Soc., 179 (1973), 399–414

    Article  MATH  MathSciNet  Google Scholar 

  17. Matsushita, S., Takahashi, W.: Strong convergence theorems for nonexpansive nonself-mappings without boundary conditions, Nonlinear Anal., 68 (2008), 412–419

    Article  MATH  MathSciNet  Google Scholar 

  18. Megginson, R.E.: An Introduction to Banach Space Theory, Springer-Verlag New Tork, Inc, (1998)

  19. Morales, C.H., Jung J.S.: Convergence of paths for pseudo-contractive mappings in Banach spaces, Proc. Amer. Math. Soc., 128 (2000), 3411–3419

    Article  MATH  MathSciNet  Google Scholar 

  20. Reich, S.: Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal. Appl., 75 (1980), 287–292

    Article  MATH  MathSciNet  Google Scholar 

  21. Rhoades, B.E.: Some theorems on weakly contractive maps, Nonlinear Anal., 47 (2001) 2683–2693

    Article  MATH  MathSciNet  Google Scholar 

  22. Sahu, D.R.: Fixed points of demicontinuous nearly Lipschitzian mappings in Banach spaces, Comment. Math. Univ. Carolin., 46 (2005), 653–666

    MATH  MathSciNet  Google Scholar 

  23. Sahu, D.R., Beg, I.: Weak and strong convergence of fixed points of nearly asymptotically nonexpansivemappings, Int. J. Mod. Math., 3 (2008), 135–151

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  25. Song, Y.: Iterative approximation to common fixed points of a countable family of nonexpansive mappings, Appl. Anal., 86 (2007), 1329–1337

    Article  MATH  MathSciNet  Google Scholar 

  26. Song, Y., Chen, R.: Viscosity approximation methods for nonexpansive nonself-mappings, J. Math. Anal. Appl., 321 (2006) 316–326

    Article  MATH  MathSciNet  Google Scholar 

  27. Song, Y., Chen, R.: Iterative approximation to common fixed points of nonexpansivemapping sequences in reflexive Banach spaces, Nonlinear Anal., 66 (2007), 591–603

    Article  MATH  MathSciNet  Google Scholar 

  28. Song, Y., Xu, S.: Strong convergence theorems for nonexpansive semigroup in Banach spaces, J. Math. Anal. Appl., 338 (2008), 152–161

    Article  MATH  MathSciNet  Google Scholar 

  29. Takahashi, W.: Nonlinear functional analysis. Fixed point theory and its applications, Yokohama Publishers, Yokohama (2000)

    MATH  Google Scholar 

  30. Wong, N.C., Sahu, D.R., Yao, J.C.: Solving variational inequalities involving nonexpansive type mappings, Nonlinear Anal., (2007) doi:10.1016/j.na.2007.11.025

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

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Banaras Hindu University, Varanasi, 221005, India

    Dayaram Sahu

  2. Department of Mathematics, National University of Ireland Galway, Dublin, Ireland

    Donal O’Regan

Authors
  1. Dayaram Sahu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Donal O’Regan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Donal O’Regan.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sahu, D., O’Regan, D. Convergence theorems for semigroup-type families of non-self mappings. Rend. Circ. Mat. Palermo 57, 305–329 (2008). https://doi.org/10.1007/s12215-008-0023-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12215-008-0023-6

Keywords

Mathematics Subject Classification (2000)

Search

Navigation