DOI QR코드

DOI QR Code

Convergence of an Iterative Algorithm for Systems of Variational Inequalities and Nonlinear Mappings in Banach Spaces

JEONG, JAE UG

  • Received : 2015.04.27
  • Accepted : 2015.07.13
  • Published : 2015.12.23

Abstract

In this paper, we consider the problem of convergence of an iterative algorithm for a general system of variational inequalities, a nonexpansive mapping and an ${\eta}$-strictly pseudo-contractive mapping. Strong convergence theorems are established in the framework of real Banach spaces.

Keywords

Nonexpansive mapping;strictly pseudo-contractive mapping;variational inequality;uniformly convex

References

  1. K. Aoyama, H. Iiduka and W. Takahashi, Weak convergence of an iterative sequence for accretive operators in Banach spaces, Fixed Point Theory Appl., 2006, 2006: 35390.
  2. R. E. Bruck, Properties of fixed point sets of nonexpansive mappings in Banach spaces, Trans. Amer. Math. Soc., 179(1973), 251-262. https://doi.org/10.1090/S0002-9947-1973-0324491-8
  3. C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Probl., 20(2008), 103-120.
  4. R. Chen, P. K. Lin and Y. Song, An approximation method for strictly pseudo-contractive mappings, Nonlinear Anal., 64(2006), 2527-2535. https://doi.org/10.1016/j.na.2005.08.031
  5. Y. J. Cho, H. Y. Zhou and G. Guo, Weak and strong convergence theorems for three-step iterations with errors for asymptotically nonexpansive mappings, Comput. Math. Appl., 47(2004), 707-717. https://doi.org/10.1016/S0898-1221(04)90058-2
  6. A. Kangtunyakarn, Fixed point theory for nonlinear mappings in Banach spaces and applications, Fixed Point Theory Appl., 2014, 2014: 108. https://doi.org/10.1186/1687-1812-2014-108
  7. S. Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., bf 67(1979), 274-276. https://doi.org/10.1016/0022-247X(79)90024-6
  8. S. Reich, Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal. Appl., 75(1980), 287-292. https://doi.org/10.1016/0022-247X(80)90323-6
  9. 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. https://doi.org/10.1016/j.jmaa.2004.11.017
  10. W. Takahashi, Nonlinear Functional Analysis-Fixed Point Theory and its Application, Yokohama Publishers Inc., Yokohama, 2000 ( in Japanese).
  11. Y. Takahashi, K. Hashimoto and M. Kato, On sharp uniform convexity, smoothness and strong type, cotype inequalities, J. Nonlinear Convex Anal., 3(2002), 267-281.
  12. A. VanderLugt, Optical Signal Processing, Wiley, New York, 2005.
  13. H. K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal., 16(1991), 1127-1138. https://doi.org/10.1016/0362-546X(91)90200-K
  14. H. K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc., 66(2002), 240-256. https://doi.org/10.1112/S0024610702003332
  15. H. 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
  16. H. Zhou, Convergence theorems for $\lambda$-strict pseudo-contractions in 2-uniformly smooth Banach spaces, Nonlinear Anal., 69(2008), 3160-3173. https://doi.org/10.1016/j.na.2007.09.009

Acknowledgement

Supported by : Dong-eui University