Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

AN ITERATION SCHEMES FOR NONEXPANSIVE MAPPINGS AND VARIATIONAL INEQUALITIES

  • Wang, Hong-Jun (College of Mathematics and Information Science Henan Normal University) ;
  • Song, Yi-Sheng (College of Mathematics and Information Science Henan Normal University)
  • Received : 2010.02.11
  • Published : 2011.09.30
Download PDF
⟨ Previous Next ⟩

Abstract

An iterative algorithm is provided to find a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of some variational inequality in a Hilbert space. Using this result, we consider a strong convergence result for finding a common fixed point of a nonexpansive mapping and a strictly pseudocontractive mapping. Our results include the previous results as special cases and can be viewed as an improvement and refinement of the previously known results.

Keywords

References

  1. Ya. I. Alber and A. N. Iusem, Extension of subgradient techniques for nonsmooth opti- mization in Banach spaces, Set-valued Anal. 9 (2001), no. 4, 315-335. https://doi.org/10.1023/A:1012665832688
  2. Y. Alber, S. Reich, and J. C. Yao, Iterative methods for solving fixed-point problems with nonself-mappings in Banach spaces, Abstr. Appl. Anal. 2003 (2003), no. 4, 193-216. https://doi.org/10.1155/S1085337503203018
  3. A. Bnouhachem, M. A. Noor, and Z. Hao, Some new extragradient iterative methods for variational inequalities, Nonlinear Anal. 70 (2009), no. 3, 1321-1329. https://doi.org/10.1016/j.na.2008.02.014
  4. F. E. Browder, Fixed point theorems for noncompact mappings in Hilbert space, Proc. Nat. Acad. Sci. U.S.A. 53 (1965), 1272-1276. https://doi.org/10.1073/pnas.53.6.1272
  5. F. E. Browder, Nonlinear monotone operators and convex sets in Banach spaces, Bull. Amer. Math. Soc. 71 (1965), 780-785. https://doi.org/10.1090/S0002-9904-1965-11391-X
  6. F. E. Browder, The fixed point theory of multi-valued mappings in topological vector spaces, Math. Ann. 177 (1968), 283-301. https://doi.org/10.1007/BF01350721
  7. F. E. Browder and W. E. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl. 20 (1967), 197-228. https://doi.org/10.1016/0022-247X(67)90085-6
  8. R. E. Bruck, On the weak convergence of an ergodic iteration for the solution of vari- ational inequalities for monotone operators in Hilbert space, J. Math. Anal. Appl. 61 (1977), no. 1, 159-164. https://doi.org/10.1016/0022-247X(77)90152-4
  9. B. Halpen, Fixed points of nonexpansive maps, Bull. Amer. Math. Soc. 73 (1967), 957- 961. https://doi.org/10.1090/S0002-9904-1967-11864-0
  10. Z. Huang and M. A. Noor, Some new unified iteration schemes with errors for nonex- pansive mappings and variational inequalities, Appl. Math. Comput. 194 (2007), no. 1, 135-142. https://doi.org/10.1016/j.amc.2007.04.056
  11. H. Iiduka and W. Takahashi, Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings, Nonlinear Anal. 61 (2005), no. 3, 341-350. https://doi.org/10.1016/j.na.2003.07.023
  12. M. A. Noor, Some developments in general variational inequalities, Appl. Math. Com- put. 152 (2004), no. 1, 199-277. https://doi.org/10.1016/S0096-3003(03)00558-7
  13. M. A. Noor, General variational inequalities and nonexpansive mappings, J. Math. Anal. Appl. 331 (2007), no. 2, 810-822. https://doi.org/10.1016/j.jmaa.2006.09.039
  14. M. A. Noor and A. Bnouhachem, On an iterative algorithm for general variational inequalities, Appl. Math. Comput. 185 (2007), no. 1, 155-168. https://doi.org/10.1016/j.amc.2006.07.018
  15. M. A. Noor and Z. Huang, Three-step methods for nonexpansive mappings and varia- tional inequalities, Appl. Math. Comput. 187 (2007), no. 2, 680-685. https://doi.org/10.1016/j.amc.2006.08.088
  16. M. A. Noor and Z. Huang, Wiener-Hopf equation technique for variational inequalities and nonexpansive mappings, Appl. Math. Comput. 191 (2007), no. 2, 504-510. https://doi.org/10.1016/j.amc.2007.02.117
  17. X. Qin, S. Y. Cho, and S. M. Kang, Convergence of an iterative algorithm for systems of variational inequalities and nonexpansive mappings with applications, J. Comput. Appl. Math. 233 (2009), no. 2, 231-240. https://doi.org/10.1016/j.cam.2009.07.018
  18. X. Qin and M. A. Noor, General Wiener-Hopf equation technique for nonexpansive mappings and general variational inequalities in Hilbert spaces, Appl. Math. Comput. 201 (2008), no. 1-2, 716-722. https://doi.org/10.1016/j.amc.2008.01.007
  19. S. Reich, Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal Appl. 75 (1980), no. 1, 287-292. https://doi.org/10.1016/0022-247X(80)90323-6
  20. S. Reich, Approximating zeros of accretive operators, Proc. Amer. Math. Soc. 51 (1975), no. 2, 381-384. https://doi.org/10.1090/S0002-9939-1975-0470762-1
  21. B. E. Rhoades, Some theorems on weakly contractive maps, Nonlinear Anal. 47 (2001), no. 4, 2683-2693. https://doi.org/10.1016/S0362-546X(01)00388-1
  22. Y. Song and R. Chen, Convergence theorems of iterative algorithms for continuous pseudocontractive mappings, Nonlinear Anal. 67 (2007), no. 2, 486-497. https://doi.org/10.1016/j.na.2006.06.009
  23. Y. Song and R. Chen, Strong convergence theorems on an iterative method for a family of finite non- expansive mappings, Appl. Math. Comput. 180 (2006), no. 1, 275-287. https://doi.org/10.1016/j.amc.2005.12.013
  24. Y. Song and R. Chen, Viscosity approximation methods for nonexpansive nonself-mappings, J. Math. Anal. Appl. 321 (2006), no. 1, 316-326. https://doi.org/10.1016/j.jmaa.2005.07.025
  25. Y. Song and R. Chen, Iterative approximation to common fixed points of nonexpansive mapping se- quences in re exive Banach spaces, Nonlinear Anal. 66 (2007), no. 3, 591-603. https://doi.org/10.1016/j.na.2005.12.004
  26. Y. Song, R. Chen, and H. Zhou, Viscosity approximation methods for nonexpansive mapping sequences in Banach spaces, Nonlinear Anal. 66 (2007), no. 5, 1016-1024. https://doi.org/10.1016/j.na.2006.01.001
  27. G. Stampacchia, Formes bilineaires coercitives sur les ensembles convexes, C. R. Acad. Sci. Paris 258 (1964), 4413-4416.
  28. 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), no. 1, 227-239. https://doi.org/10.1016/j.jmaa.2004.11.017
  29. W. Takahashi, Nonlinear variational inequalities and fixed point theorems, J. Math. Soc. Japan 28 (1976), no. 1, 168-181. https://doi.org/10.2969/jmsj/02810168
  30. W. Takahashi,, Nonlinear complementarity problem and systems of convex inequalities, J. Op- tim. Theory Appl. 24 (1978), no. 3, 499-506. https://doi.org/10.1007/BF00932892
  31. W. Takahashi and M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl. 118 (2003), no. 2, 417-428. https://doi.org/10.1023/A:1025407607560