East Asian mathematical journal

The Youngnam Mathematical Society (영남수학회)
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VISCOSITY METHODS OF APPROXIMATION FOR A COMMON SOLUTION OF A FINITE FAMILY OF ACCRETIVE OPERATORS

  • Chen, Jun-Min (College of Mathematics and Computer Hebei University) ;
  • Zhang, Li-Juan (College of Mathematics and Computer Hebei University) ;
  • Fan, Tie-Gang (College of Mathematics and Computer Hebei University)
  • Received : 2010.01.31
  • Accepted : 2011.01.03
  • Published : 2011.01.31
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Abstract

In this paper, we try to extend the viscosity approximation technique to find a particular common zero of a finite family of accretive mappings in a Banach space which is strictly convex reflexive and has a weakly sequentially continuous duality mapping. The explicit viscosity approximation scheme is proposed and its strong convergence to a solution of a variational inequality is proved.

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References

  1. Browder F. E., Convergence theorems or sequences of nonlinear operators in Banach space, Math. Z. 100 (1967), 202-225.
  2. Ciovenescu I., Geometry of Banach space, Duality Mapping and Nonlinear Problems, Kluner Academic Publishers, 1990.
  3. Chen Junmin, Zhang Lijuan and Fan Tiegang, Viscosity Approximation Methods for Nonexpansive Mappings and Monotone Mappings, J. Math. Anal. Appl. 334 (2007), 1450-1461. https://doi.org/10.1016/j.jmaa.2006.12.088
  4. Gossez J. P. and Lami Dozo E., Some geometric properties related to the fixed point theory for nonexpansive mappings, Pacific J. Math. 40 (1972), 565-573. https://doi.org/10.2140/pjm.1972.40.565
  5. Hu Lianggen and Liu Liwei, A new iterative algorithm for common solutions of a finite family of accretive operators, Nonlinear Anal. 70 (2009), 2344-2351. https://doi.org/10.1016/j.na.2008.03.016
  6. Jung J. S., Strong convergence of an iterative method for finding common zeros of a finite family of accretive operators, Commun. Korean Math. Soc. 24(3) (2009), 381-393. https://doi.org/10.4134/CKMS.2009.24.3.381
  7. Moudafi A., Viscosity Approximation Methods for Fixed-Points Problems, J. Math. Anal. Appl. 241 (2000), 46-55. https://doi.org/10.1006/jmaa.1999.6615
  8. Paul-Emile Mainge, Viscosity methods for zeros of accretive operators, J. Approximation Theory 140 (2006), 127-140. https://doi.org/10.1016/j.jat.2005.11.017
  9. Suziki T., Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces, Fixed Point Theory Appl. 1 (2005), 103-123.
  10. Takahashi W., Nonlinear Function Analysis-Fixed Point Theory and its Applications, Yokohama Publishers, Yokohama, 2000 (in Japanese).
  11. Xu H. K., An iterative approach to quadratic optimization, J. Optim. Theory Appl. 116 (2003), 659-678. https://doi.org/10.1023/A:1023073621589
  12. Xu H. K., Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl. 298 (2004), 279-291. https://doi.org/10.1016/j.jmaa.2004.04.059
  13. Zegeye H. and Shahzad N., Strong convergence theorems for a common zero of a family of m-accretive mappings, Nonlinear Anal. 66 (2007), 1161-1169. https://doi.org/10.1016/j.na.2006.01.012