Glasnik Matematicki, Vol. 45, No.2 (2010), 513-523.

CONVERGENCE THEOREMS OF ZEROS OF A FINITE FAMILY OF m-ACCRETIVE OPERATORS IN BANACH SPACES

Yan Hao and Sun Young Cho

School of Mathematics, Physics and Information Science, Zhejiang Ocean University, Zhoushan 316004, China
e-mail: zjhaoyan@yahoo.cn

Department of Mathematics, Gyeongsang National University, Chinju 660-701, Korea
e-mail: ooly61@yahoo.co.kr

Abstract.   In this paper, we continue to study convergence problems for a Ishikawa-like iterative process for a finite family of m-accretive mappings. Strong convergence theorems are established in uniformly smooth Banach spaces.

2000 Mathematics Subject Classification.   47H09, 47J25.

Key words and phrases.   Accretive mapping, pseudo-contractive mapping, iterative process, zero.

Full text (PDF) (free access)

DOI: 10.3336/gm.45.2.16

References:

  1. F. E. Browder, Fixed point theorems for noncompact mappings in Hilbert space, Proc. Nat. Acad. Sci. USA 53 (1965), 1272-1276.
    MathSciNet    

  2. V. Berinde, Iterative approximation of fixed points, Editura Efemeride, Baia Mare, 2002.
    MathSciNet    

  3. R. E. Bruck Jr., A strongly convergent iterative solution of 0 in U(x) for a maximal monotone operator U in Hilbert space, J. Math. Anal. Appl. 48 (1974), 114-126.
    MathSciNet     CrossRef

  4. R. E. Bruck, Nonexpansive projections on subsets of Banach spaces, Pacific J. Math. 47 (1973), 341-355.
    MathSciNet     CrossRef

  5. C. E. Chidume and H. Zegeye, Iterative solution of 0 in Ax for m-accretive operator A in certain Banach spaces, J. Math. Anal. Appl. 269 (2002), 421-430.
    MathSciNet     CrossRef

  6. Y. J. Cho and X. Qin, Viscosity approximation methods for a family of m-accretive mappings in reflexive Banach spaces, Positivity 12 (2008), 483-494.
    MathSciNet     CrossRef

  7. Y. J. Cho and X. Qin, Convergence of a general iterative method for nonexpansive mappings in Hilbert spaces, J. Comput. Appl. Math. 228 (2009), 458-465.
    MathSciNet     CrossRef

  8. Y. J. Cho, S. M. Kang and X. Qin, Approximation of common fixed points of an infinite family of nonexpansive mappings in Banach spaces, Comput. Math. Appl. 56 (2008), 2058-2064.
    MathSciNet     CrossRef

  9. L. C. Ceng, A. R. Khan, Q. H. Ansari and J.-C. Yao, Strong convergence of composite iterative schemes for zeros of m-accretive operators in Banach spaces, Nonlinear Anal. 70 (2009), 1830-1840.
    MathSciNet     CrossRef

  10. T. Dominguez Benavides, G. Lobez Acedo and H.-K. Xu, Iterative solutions for zeros of accretive operators, Math. Nachr. 248/249 (2003), 62-71.
    MathSciNet     CrossRef

  11. J. S. Jung, Convergence of composite iterative methods for finding zeros of accretive operators, Nonlinear Anal. 71 (2009), 1736-1746.
    MathSciNet     CrossRef

  12. T. H. Kim and H. K. Xu, Strong convergence of modified Mann iterations, Nonlinear Appl. 61 (2005), 51-60.
    MathSciNet     CrossRef

  13. W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953), 506-510.
    MathSciNet     CrossRef

  14. X. Qin, Y. J. Cho, J. I. Kang and S. M. Kang, Strong convergence theorems for an infinite family of nonexpansive mappings in Banach spaces, J. Comput. Appl. Math. 230 (2009), 121-127.
    MathSciNet     CrossRef

  15. X. Qin and Y. Su, Approximation of zero point of accretive operator in Banach spaces, J. Math. Anal. Appl. 329 (2007), 415-424.
    MathSciNet     CrossRef

  16. X. Qin, Y. Su and M. Shang, Strong convergence of the composite Halpern iteration, J. Math. Anal. Appl. 339 (2008), 996-1002.
    MathSciNet     CrossRef

  17. X. Qin and Y. Su, Strong convergence theorems for relatively nonexpansive mappings in a Banach space, Nonlinear Anal. 67 (2007), 1958-1965.
    MathSciNet     CrossRef

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

  19. S. Reich, Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal. Appl. 75 (1980), 287-292.
    MathSciNet     CrossRef

  20. S. Reich, Asymptotic behavior of contractions in Banach spaces, J. Math. Anal. Appl. 44 (1973), 57-70.
    MathSciNet     CrossRef

  21. T. Suzuki, Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals, J. Math. Anal. Appl. 305 (2005), 227-239.
    MathSciNet     CrossRef

  22. M. Shang, Y. Su and X. Qin, Strong convergence theorems for a finite family of nonexpansive mappings, Fixed Point Theory Appl. 2007 (2007), Art. ID 76971, 9pp.
    MathSciNet    

  23. W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, 2000.
    MathSciNet    

  24. H. K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc. (2) 66 (2002), 240-256.
    MathSciNet     CrossRef

  25. Y. Yao, R. Chen and J. C. Yao, Strong convergence and certain control conditions for modified Mann iteration, Nonlinear Anal. 68 (2008), 1687-1693.
    MathSciNet    

  26. H. Zegeyea and N. Shahzad, Strong convergence theorems for a common zero of a finite family of m-accretive mappings, Nonlinear Anal. 66 (2007), 1161-1169.
    MathSciNet     CrossRef

  27. E. Zeidler, Nonlinear functional analysis and its applications, Part II: Monotone Operators, Springer-Verlag, Berlin, 1985.
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