MODIFIED MANN'S ALGORITHM BASED ON THE CQ METHOD FOR PSEUDO-CONTRACTIVE MAPPINGS

  • Yao, Yonghong (Department of Mathematics, Tianjin Polytechnic University) ;
  • Zhou, Haiyun (Department of Mathematics, Shijiazhuang Mechanical Engineering College) ;
  • Liou, Yeong-Cheng (Department of Information Management, Cheng Shiu University)
  • 투고 : 2010.01.10
  • 심사 : 2010.02.25
  • 발행 : 2010.09.30

초록

IIn this paper, we suggest and analyze a modified Mann's algorithm based on the CQ method for pseudo-contractive mappings in Hilbert spaces. Further, we prove a strong convergence theorem according to the proposed algorithm for pseudo-contractive mappings.

키워드

참고문헌

  1. F.E. Browder and W.V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert spaces, J. Math. Anal. Appl.20(1967), 197-228 https://doi.org/10.1016/0022-247X(67)90085-6
  2. C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Problems 20(2004), 103-120. https://doi.org/10.1088/0266-5611/20/1/006
  3. T.H. Kim and H.K. Xu, Strong convergence of modified Mann iterations, Nonlinear Anal. 61(2005), 51-60. https://doi.org/10.1016/j.na.2004.11.011
  4. C. Matinez-Yanes and H.K. Xu, Strong convergence of the CQ method for fixed point processes, Nonlinear Anal. 64(2006), 2400-2411. https://doi.org/10.1016/j.na.2005.08.018
  5. K. Nakajo and W. Takahashi, Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl. 279(2003), 372-379. https://doi.org/10.1016/S0022-247X(02)00458-4
  6. S. Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 67(1979), 274-276. https://doi.org/10.1016/0022-247X(79)90024-6
  7. N. Shioji and W. Takahashi, Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces, Proc. Amer. Math. Soc. 125(1997), 3641-3645. https://doi.org/10.1090/S0002-9939-97-04033-1
  8. K. Deimling, Zeros of accretive operators, Manuscripta Math. 13(1974), 365-374. https://doi.org/10.1007/BF01171148
  9. M.O. Osilike and A. Udomene, Demiclosedness principle and convergence theorems for strictly pseudo-contractive mappings of Browder-Petryshyn type, J. Math. Anal. Appl. 256(2001), 431-445. https://doi.org/10.1006/jmaa.2000.7257
  10. C.E. Chidume and M. Chika, Fixed point iteration for pseudo-contractive maps, Proc. Amer. Math. Soc. 127(1999), 1163-1170. https://doi.org/10.1090/S0002-9939-99-05050-9
  11. C.E. Chidume, Iterative approximation of fixed points of Lipschitz pseudo-contractive maps, Proc. Amer. Math. Soc. 129(2001), 2245-2251 https://doi.org/10.1090/S0002-9939-01-06078-6
  12. C.E. Chidume and H. Zegeye, Approximate fixed point sequences and convergence theorems for Lipschitz pseudo-contractive maps, Proc. Amer. Math. Soc. 132(2003), 831-840.
  13. C.H. Morales and J.S. Jung, Convergence of paths for pseudo-contractive mappings in Banach spaces, Proc. Amer. Math. Soc. 128(2000), 3411-3419. https://doi.org/10.1090/S0002-9939-00-05573-8
  14. J. Schu, Approximating fixed points of Lipschitzian pseudo-contractive mappings, Houston J. Math. 19(1993), 107-115.
  15. A. Udomene, Path convergence, approximation of fixed points and variational solutions of Lipschitz pseudo-contractions in Banach spaces, Nonlinear Anal. 67(2007), 2403-2414. https://doi.org/10.1016/j.na.2006.09.001
  16. C.H. Morales, On the fixed-point theory for local k-pseudo-contractions, Proc. Amer. Math. Soc. 81(1981), 71-74.
  17. L.C. Zeng, N.C. Wong and J.C. Yao, Strong convergence theorems for strictly pseudo-contractive mappings of Browder-Petryshyn type, Taiwanese J. Math. 10(2006), 837-849.
  18. Y. Yao, Y.C. Liou and R. Chen, Strong convergence of an iterative algorithm for pseudo-contractive mapping in Banach spaces, Nonlinear Anal. 67(2007), 3311-3317. https://doi.org/10.1016/j.na.2006.10.013
  19. C.E. Chidume and S.A. Mutangadura, An example on the Mann iteration method for Lipschitz pseudo-contractions, Proc. Amer. Math. Soc. 129(2001), 2359-2363. https://doi.org/10.1090/S0002-9939-01-06009-9
  20. G. Marino and H.K. Xu, Weak and strong convergence theorems for strictly pseudo-contractions in Hilbert spaces, J. Math. Anal. Appl. 329(2007), 336-349. https://doi.org/10.1016/j.jmaa.2006.06.055
  21. H.Y. Zhou, Convergence theorems of common fixed points for a finite family of Lipschitz pseudo-contractions in Banach spaces, Nonlinear Anal. 68(2008), 2977-2983. https://doi.org/10.1016/j.na.2007.02.041
  22. M.A. Noor, Some developments in general variational inequalities, Appl. Math. Comput. 152(2004), 199-277. https://doi.org/10.1016/S0096-3003(03)00558-7
  23. M.A. Noor, Principles of Variational Inequalities, Lap-Lambert Academic Publishing AG and Co, Saarbrucken, Germany, 2009.
  24. Y. Yao, Y.C. Liou and S.M. Kang, An iterative algorithm for asymptotically nonexpansive mappings, J. Appl. Math. and Informatics, 28(2010), No. 1-2, 75-86.
  25. J. Kang and Y. Su, Approximation methods for finite family of nonspreading mappings and nonexpansive mappings in Hilert spaces, J. Appl. Math. and Informatics, 28(2010), No. 1-2, 87-98.
  26. R. Ahmad and F. Usman, System of mixed variational inequalities in reflexive Banach spaces, J. Appl. Math. and Informatics, 27(2009), No. 3-4, 693-702.