Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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CONVERGENCE THEOREMS FOR INVERSE-STRONGLY MONOTONE MAPPINGS AND QUASI-φ-NONEXPANSIVE MAPPINGS

  • Qin, Xiaolong (DEPARTMENT OF MATHEMATICS GYEONGSANG NATIONAL UNIVERSITY) ;
  • Kang, Shin-Min (DEPARTMENT OF MATHEMATICS THE RESEARCH INSTITUTE OF NATURAL SCIENCE GYEONGSANG NATIONAL UNIVERSITY) ;
  • Cho, Yeol-Je (DEPARTMENT OF MATHEMATICS EDUCATION THE RESEARCH INSTITUTE OF NATURAL SCIENCE GYEONGASAN NATIONAL UNIVERSITY)
  • Published : 2009.09.30
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Abstract

In this paper, we consider a hybrid projection algorithm for a pair of inverse-strongly monotone mappings and a quasi-$\phi4-nonexpansive mapping. Strong convergence theorems are established in the framework of Banach spaces.

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References

  1. Ya. I. Alber, Metric and generalized projection operators in Banach spaces: properties and applications, Theory and applications of nonlinear operators of accretive and monotone type, 15–50, Lecture Notes in Pure and Appl. Math., 178, Dekker, New York, 1996
  2. Ya. I. Alber and S. Reich, An iterative method for solving a class of nonlinear operator equations in Banach spaces, Panamer. Math. J. 4 (1994), no. 2, 39–54
  3. B. Beauzamy, Introduction to Banach Spaces and Their Geometry (Second edition), North-Holland Mathematics Studies, 68. Notas de Matematica [Mathematical Notes], 86. North-Holland Publishing Co., Amsterdam, 1985
  4. D. Butnariu, S. Reich, and A. J. Zaslavski, Asymptotic behavior of relatively nonexpansive operators in Banach spaces, J. Appl. Anal. 7 (2001), no. 2, 151–174
  5. D. Butnariu, S. Reich, and A. J. Zaslavski, Weak convergence of orbits of nonlinear operators in reflexive Banach spaces, Numer. Funct. Anal. Optim. 24 (2003), no. 5-6, 489–508 https://doi.org/10.1081/NFA-120023869
  6. Y. Censor and S. Reich, Iterations of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization, Optimization 37 (1996), no. 4, 323–339 https://doi.org/10.1080/02331939608844225
  7. I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Mathematics and its Applications, 62. Kluwer Academic Publishers Group, Dordrecht, 1990
  8. H. Iiduka and W. Takhashi, Strong convergence studied by a hybrid type method for monotone operators in a Banach space, Nonlinear Anal. 68 (2008), no. 12, 3679–3688 https://doi.org/10.1016/j.na.2007.04.010
  9. S. Kamimura and W. Takahashi, Strong convergence of a proximal-type algorithm in a Banach space, SIAM J. Optim. 13 (2002), no. 3, 938–945 https://doi.org/10.1137/S105262340139611X
  10. S. Y. Matsushita and W. Takahashi, A strong convergence theorem for relatively nonexpansive mappings in a Banach space, J. Approx. Theory 134 (2005), no. 2, 257–266 https://doi.org/10.1016/j.jat.2005.02.007
  11. S. Ohsawa and W. Takahashi, Strong convergence theorems for resolvents of maximal monotone operators in Banach spaces, Arch. Math. (Basel) 81 (2003), no. 4, 439–445 https://doi.org/10.1007/s00013-003-0508-7
  12. X. Qin and Y. Su, Strong convergence theorems for relatively nonexpansive mappings in a Banach space, Nonlinear Anal. 67 (2007), no. 6, 1958–1965 https://doi.org/10.1016/j.na.2006.08.021
  13. X. Qin, Y. J. Cho, and S. M. Kang, Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces, J. Comput. Appl. Math. 225 (2009), no. 1, 20–30 https://doi.org/10.1016/j.cam.2008.06.011
  14. S. Reich, A weak convergence theorem for the alternating method with Bregman distances, Theory and applications of nonlinear operators of accretive and monotone type, 313–318, Lecture Notes in Pure and Appl. Math., 178, Dekker, New York, 1996
  15. R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim. 14 (1976), no. 5, 877–898 https://doi.org/10.1137/0314056
  16. M. V. Solodov and B. F. Svaiter, Forcing strong convergence of proximal point iterations in a Hilbert space, Math. Program. 87 (2000), no. 1, Ser. A, 189–202 https://doi.org/10.1007/s101079900113
  17. Y. Su and X. Qin, Monotone CQ iteration processes for nonexpansive semigroups and maximal monotone operators, Nonlinear Anal. 68 (2008), no. 12, 3657–3664 https://doi.org/10.1016/j.na.2007.04.008
  18. W. Takahashi, Nonlinear Functional Analysis, Fixed point theory and its applications. Yokohama Publishers, Yokohama, 2000
  19. H. Zegeye and N. Shahzad, Strong convergence theorems for monotone mappings and relatively weak nonexpansive mappings, Nonlinear Anal. 70 (2009), no. 7, 2707–2716 https://doi.org/10.1016/j.na.2008.03.058

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