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Applied Mathematics and Mechanics

, Volume 16, Issue 6, pp 583–592 | Cite as

Iterative construction of solutions to nonlinear equations of lipschitzian and local strongly accretive operators

  • Zeng Luchuan
Article

Abstract

In this paper, we investigate the Ishikawa iteration process in a p-uniformly smooth Banach space X. Let T: X→X be a Lipschitzian and local strongly accretive operator and the set sol(T) of solutions of the equation Tx=f be nonempty. We show that sol(T) is a singleton and the Ishikawa sequence converges strongly to the unique solution of the equation Tx=f. In addition, whenever T is a Lipschitzian and local pseudocontractive mapping from a nonempty convex subset K of X and the set F(T) of fixed points of T is nonempty, we prove that F(T) is a singleton and the Ishikawa sequence converges strongly to the unique fixed point of T. Our results are the improvements and extension of the results of Deng and Ding(4) and Tan and Xu(5).

Key words

local strongly accretive local strictly pseudocontractive p-uniformly smooth Banach space 

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References

  1. [1]
    C. E. Chidume, Iterative approximation of fixed points of Lipschitzian strictly pseudocontractive mappings,Proc. Amer. Math. Soc.,99 (1987), 283–288.MATHMathSciNetCrossRefGoogle Scholar
  2. [2]
    C. E. Chidume, An iterative process for nonlinear Lipschitzian strongly accretive mapping inL ρ spaces,J. Math. Anal. Appl.,151 (1990), 453–461.MATHMathSciNetCrossRefGoogle Scholar
  3. [3]
    X. Weng, Fixed point iteration for local strictly pseudocontractive mapping,Proc. Amer. Math. Soc.,113 (1991), 727–731.MATHMathSciNetCrossRefGoogle Scholar
  4. [4]
    L. Deng and X. P. Ding, Iterative process for Lipschitz local strictly pseudocontractive mappings,Appl. Math. and Mech. (English Ed.),15, 2 (1994), 119–123.MATHMathSciNetGoogle Scholar
  5. [5]
    K. K. Tan and H. K. Xu, Iterative solutions to nonlinear equations of strongly accretive operators in Banach space,J. Math. Anal. Appl.,178 (1993), 9–21.MATHMathSciNetCrossRefGoogle Scholar
  6. [6]
    T. C. Lim, H. K. Xu. and Z. B. Xu, SomeL ρ inequalities and their applications to fixed point theory and approximation theory, inProgress in Approximation Theory, Eds. by P. Nevai and A. Pinkus, Academic Press (1991), 609–624.Google Scholar
  7. [7]
    H. K. Xu, Inequalities in Banach spaces with applications,Nonlinear Anal.,16 (1991). 1127–1138.MATHMathSciNetCrossRefGoogle Scholar
  8. [8]
    J. Diestel, Geometry of Banach spaces-selected topics,Lecture Notes in Mathematics, Vol.485, Springer-Verlag (1975).Google Scholar

Copyright information

© Shanghai University of Technology (SUT) 1995

Authors and Affiliations

  • Zeng Luchuan
    • 1
  1. 1.Department of MathematicsShanghai Normal UniversityShanghaiP. R. China

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