Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
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STRONG CONVERGENCE OF AN ITERATIVE ALGORITHM FOR SYSTEMS OF VARIATIONAL INEQUALITIES AND FIXED POINT PROBLEMS IN q-UNIFORMLY SMOOTH BANACH SPACES

  • Received : 2012.04.27
  • Accepted : 2012.06.15
  • Published : 2012.06.30
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Abstract

In this paper, we introduce a new iterative scheme to investigate the problem of nding a common element of nonexpansive mappings and the set of solutions of generalized variational inequalities for a $k$-strict pseudo-contraction by relaxed extra-gradient methods. Strong convergence theorems are established in $q$-uniformly smooth Banach spaces.

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References

  1. F.E. Browder and W.V. Petryshyn, Construction of xed 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. R.E. Bruck, Properties of xed point sets of nonexpansive mappings in Ba- nach spaces, Trans. Amer. Math. Soc. 179 (1973), 251-262. https://doi.org/10.1090/S0002-9947-1973-0324491-8
  3. C.E. Chidume and S. A. Mutangadura, An example on the Mann itera- tion method for Lipschitz pseudocontractions, Proc. Amer. Math. Soc. 129 (2001), 2359-2363. https://doi.org/10.1090/S0002-9939-01-06009-9
  4. B. Halpern, Fixed points of nonexpansive maps, Bull. Amer. Math. Soc. 73(1967), 957-961. https://doi.org/10.1090/S0002-9904-1967-11864-0
  5. S. Kitahara and W. Takahashi, Image recovery by convex combinations of sunny nonexpansive retractions, Topol. Methods Nonlinear Anal. 2 (1993), 333-342. https://doi.org/10.12775/TMNA.1993.046
  6. W.R. Mann, Mean value methods in iterations, Proc. Amer. Math. Soc. 14(1953), 506-510.
  7. G. Marino and H. K. Xu, Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces, J. Math. Anal. Appl. 329 (2007), 336-346. https://doi.org/10.1016/j.jmaa.2006.06.055
  8. K. Nakajo and W. Takahashi, Strong convergence theorems for nonexpan- sive mappings and nonexpansive semigroups, J. Math. Anal. Appl. 279 (2003), 372-379. https://doi.org/10.1016/S0022-247X(02)00458-4
  9. S. Reich, Asymptotic behavior of constractions in Banach spaces, J. Math. Anal. Appl. 44(1973), 57-70. https://doi.org/10.1016/0022-247X(73)90024-3
  10. 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
  11. O. Scherzer, Convergence criteria of iterative methods based on Landweber iteration for solving nonlinear problems, J. Math. Anal. Appl. 194 (1991), 911-933.
  12. T. Suzuki, Strong convergence of Krasnoselskii and Manns 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
  13. Y. Wang and R. Chen, Hybrid methods for accretive variational inequalities involving pseudocontractions in Banach spaces, Fixed Point Theory Appl. 63 (2011). doi:10.1186/1687-1812-2011-63.
  14. 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
  15. H.K. Xu, Iterative algorithms for nonlinear operators, J. Lond. Math. Soc. 66 (2002), 240-256. https://doi.org/10.1112/S0024610702003332
  16. H. Zhang and Y. Su, Convergence theorems for strict pseudo-contractions in q-uniformly smooth Banach spaces, Nonlinear Anal. 71 (2009), 4572-4580. https://doi.org/10.1016/j.na.2009.03.033
  17. H. Zhou, Convergence theorems for -strict pseudocontractions in 2- uniformly smooth Banach spaces, Nonlinear Anal. 69 (2008), 3160-3173. https://doi.org/10.1016/j.na.2007.09.009