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Strong convergence of an iterative method for pseudo-contractive and monotone mappings

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Abstract

In this paper, we introduce an iterative process which converges strongly to a common element of fixed points of pseudo-contractive mapping and solutions of variational inequality problem for monotone mapping. As a consequence, we provide an iteration scheme which converges strongly to a common element of set of fixed points of finite family continuous pseudo-contractive mappings and solutions set of finite family of variational inequality problems for continuous monotone mappings. Our theorems extend and unify most of the results that have been proved for this class of nonlinear mappings.

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References

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

    Google Scholar 

  2. Borwein J.M.: Fifty years of maximal monotonicity. Optim. Lett. 4, 473–490 (2010)

    Article  Google Scholar 

  3. 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 

  4. Browder F.E.: Nonlinear monotone operators and convex sets in Banach spaces. Bull. Am. Math. Soc. 71, 780–785 (1965)

    Article  Google Scholar 

  5. Browder F.E.: The fixed point theory of multi-valued mappings in topological vector spaces. Math. Ann. 177, 283–301 (1968)

    Article  Google Scholar 

  6. Bruck R.E.: On the weak convergence ofan ergodic iteration for the solution of variational inequalities for monotone operators in Hilbert space. J. Math. Anal. Appl. 61, 159–164 (1977)

    Article  Google Scholar 

  7. Halpern B.: Fixed points of nonexpanding maps. Bull. Am. Math. Soc. 73, 957–961 (1967)

    Article  Google Scholar 

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

    Article  Google Scholar 

  9. Iiduka H., Takahashi W., Toyoda M.: Approximation of solutions of variational inequalities for monotone mappings. Pan Am. Math. J. 14, 49–61 (2004)

    Google Scholar 

  10. Lions P.L.: Approximation de points fixes de contractions. C. R. Acad. Sci. Sér. A-B Paris 284, 1357–1359 (1977)

    Google Scholar 

  11. Lions J.L., Stampacchia G.: Variational inequalities. Comm. Pure Appl. Math. 20, 493–517 (1967)

    Article  Google Scholar 

  12. Liu F., Nashed M.Z.: Regularization of nonlinear ill-posed variational inequalities and convergence rates. Set-Valued Anal. 6, 313–344 (1998)

    Article  Google Scholar 

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

    Article  Google Scholar 

  14. Nakajo K., Takahashi W.: Strong and weak convergence theorems by an improved splitting method. Comm. Appl. Nonlinear Anal. 9, 99–107 (2002)

    Google Scholar 

  15. Nilsrakoo W., Saejung S.: Equilibrium problems and Moudafi’s viscosity approximation methods in Hilbert spaces. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 17, 195–213 (2010)

    Google Scholar 

  16. Noor M.A., Noor K.I., Al-Said E.: Iterative methods for solving general quasi-variational inequalities. Optim. Lett. 4, 513–530 (2010)

    Article  Google Scholar 

  17. Petrusel A., Yao J.C.: Viscosity approximation to common fixed points of families of nonexpansive mappings with generalized contractions mappings. Nonlinear Anal. 69, 1100–1111 (2008)

    Article  Google Scholar 

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

    Article  Google Scholar 

  19. Su Y., Shang M., Qin X.: An iterative method of solution for equilibrium and optimization problems. Nonlinear Anal. 69, 2709–2719 (2008)

    Article  Google Scholar 

  20. Takahashi W.: Nonlinear variational inequalities and fixed point theorems. J. Math. Soc. Japan 28, 168–181 (1976)

    Article  Google Scholar 

  21. Takahashi W.: Nonlinear complementarity problem and systems of convex inequalities. J. Optim. Theory Appl. 24, 493–508 (1978)

    Article  Google Scholar 

  22. Takahashi W.: Convex Analysis and Approximation of Fixed Points. Yokohama Publishers, Yokohama (2000)

    Google Scholar 

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

    Article  Google Scholar 

  24. 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  Google Scholar 

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

    Article  Google Scholar 

  26. Yao Y., Yao J.C.: On modified iterative method for nonexpansive mappings and monotone mappings. Appl. Math. Comput. 186, 1551–1558 (2007)

    Article  Google Scholar 

  27. Zegeye, H.: An iterative approximation for a common fixed point of two pseudo-contractive mappings. ISRN Math. Anal. 2011, 14. doi:10.5402/2011/621901

  28. Zegeye H., Shahzad N.: Approximation methods for a common fixed point of finite family of nonexpansive mappings. Numer. Funct. Anal. 28(11-12), 1405–1419 (2007)

    Article  Google Scholar 

  29. Zegeye H., Shahzad N.: Viscosity approximation methods for a common fixed point of finite family of nonexpansive mappings. Appl. Math. Comput. 191, 155–163 (2007)

    Article  Google Scholar 

  30. Zegeye H., Shahzad N.: Strong convergence for monotone mappings and relatively weak nonexpansive mappings. Nonlinear Anal. 70, 2707–2716 (2009)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Botswana, Pvt. Bag 00704, Gaborone, Botswana

    Habtu Zegeye

  2. Department of Mathematics, King Abdul Aziz University, P.O.B. 80203, Jeddah, 21589, Saudi Arabia

    Naseer Shahzad

Authors
  1. Habtu Zegeye
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  2. Naseer Shahzad
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Correspondence to Naseer Shahzad.

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Zegeye, H., Shahzad, N. Strong convergence of an iterative method for pseudo-contractive and monotone mappings. J Glob Optim 54, 173–184 (2012). https://doi.org/10.1007/s10898-011-9755-5

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  • DOI: https://doi.org/10.1007/s10898-011-9755-5

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