Glasnik Matematicki, Vol. 46, No.1 (2011), 233-247.

APPROXIMATING COMMON SOLUTIONS OF VARIATIONAL INEQUALITIES BY ITERATIVE ALGORITHMS WITH APPLICATIONS

Xiaolong Qin, Sun Young Cho and Yeol Je Cho

School of Mathematics and Information Sciences, North China University of Water Resources and Electric Power, Zhengzhou 450011, China
e-mail: qxlxajh@163.com

Department of Mathematics, Gyeongsang National University, Chinju 660-701, Korea
e-mail: ooly61@yahoo.co.kr

Department of Mathematics Education and RINS, Gyeongsang National University, Chinju 660-701, Korea
e-mail: yjcho@gnu.ac.kr


Abstract.   In this paper, we introduce an iterative scheme for a general variational inequality. Strong convergence theorems of common solutions of two variational inequalities are established in a uniformly convex and 2-uniformly smooth Banach space. As applications, we, still in Banach spaces, consider the convex feasibility problem.

2000 Mathematics Subject Classification.   47H05, 47H09, 47J25.

Key words and phrases.   Iterative algorithm, variational inequality, inverse-strongly accretive mapping, sunny nonexpansive retraction.


Full text (PDF) (access from subscribing institutions only)

DOI: 10.3336/gm.46.1.17


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), Art. ID 35390.
    MathSciNet     CrossRef

  2. F. E. Browder and W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl. 20 (1967), 197-228.
    MathSciNet     CrossRef

  3. F. E. Browder, Fixed point theorems for noncompact mappings in Hilbert spaces, Proc. Nat. Acad. Sci. U.S.A. 53 (1965), 1272-1276.
    MathSciNet     CrossRef

  4. R. E. Bruck, Properties of fixed-point sets of nonexpansive mappings in Banach spaces, Tras. Amer. Math. Soc. 179 (1973), 251-262.
    MathSciNet     CrossRef

  5. Y. J. Cho, Y. Yao and H. Zhou, Strong convergence of an iterative algorithm for accretive operators in Banach spaces, J. Comput. Appl. Anal. 10 (2008), 113-125.
    MathSciNet    

  6. L. C. Ceng, C. Y. Wang and J. C. Yao, Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities, Math. Meth. Oper. Res. 67 (2008), 375-390.
    MathSciNet     CrossRef

  7. P. L. Combettes, The convex feasibility problem, in: Image recovery, Advances in Imaging and Electron Physics, P. Hawkes, Ed., vol. 95, pp. 155-270, Academic Press, Orlando, Fla, USA, 1996.

  8. Y. Censor and S. A. Zenios, Parallel Optimization. Theory, Algorithms, and Applications, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, NY, USA, 1997.
    MathSciNet    

  9. Y. Hao, Strong convergence of an iterative method for inverse strongly accretive operators, J. Inequal. Appl. 2008, Art. ID 420989.
    MathSciNet     CrossRef

  10. H. Iiduka and W. Takahashi, Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings, Nonlinear Anal. 61 (2005), 341-350.
    MathSciNet     CrossRef

  11. S. Kitahara and W. Takahashi, Image recovery by convex combinations of sunny nonexpansive retractions, Topol. Meth. Nonlinear Anal. 2 (1993), 333-342.
    MathSciNet    

  12. T. Kotzer, N. Cohen and J. Shamir, Images to ration by a novel method of parallel projection onto constraint sets, Opt. Lett. 20 (1995), 1172-1174.
    CrossRef

  13. X. Qin and Y. Su, Approximation of a zero point of accretive operator in Banach spaces, J. Math. Anal. Appl. 329 (2007), 415-424.
    MathSciNet     CrossRef

  14. X. Qin, Y. Su and M. Shang, Strong convergence of the composite Halpern iteration, J. Math. Anal. Appl. 339 (2008), 996-1002.
    MathSciNet     CrossRef

  15. S. Reich, Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal. Appl. 75 (1980), 287-292.
    MathSciNet     CrossRef

  16. S. Reich, Asymptotic behavior of contractions in Banach spaces, J. Math. Anal. Appl. 44 (1973), 57-70.
    MathSciNet     CrossRef

  17. T. Suzuki, Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochne integrals, J. Math. Anal. Appl. 305 (2005), 227-239.
    MathSciNet     CrossRef

  18. M. I. Sezan and H. Stark, Application of convex projection theory to image recovery in tomograph and related areas, in Image Recovery: Theory and Application, H. Stark, Ed., pp. 155-270 Academic Press, Orlando, Fla, USA, 1987.

  19. W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, 2000.
    MathSciNet    

  20. H. K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal. 16 (1991), 1127-1138.
    MathSciNet     CrossRef

  21. H. K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc. 66 (2002), 240-256.
    MathSciNet     CrossRef

  22. Y. Yao and J. C. Yao, On modified iterative method for nonexpansive mappings and monotone mappings, Appl. Math. Comput. 186 (2007), 1551-1558.
    MathSciNet     CrossRef

Glasnik Matematicki Home Page