THREE-STEP ITERATIVE ALGORITHMS FOR FIXED POINT PROBLEMS AND VARIATIONAL INCLUSION PROBLEMS

- Journal title : Communications of the Korean Mathematical Society
- Volume 25, Issue 3, 2010, pp.419-426
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/CKMS.2010.25.3.419

Title & Authors

THREE-STEP ITERATIVE ALGORITHMS FOR FIXED POINT PROBLEMS AND VARIATIONAL INCLUSION PROBLEMS

Cho, Sun-Young; Hao, Yan;

Cho, Sun-Young; Hao, Yan;

Abstract

In this paper, a three-step iterative method is considered for finding a common element in the set of fixed points of a non-expansive mapping and in the set of solutions of a variational inclusion problem in a real Hilbert space.

Keywords

non-expansive mapping;fixed point;three-step iterative algorithm;resolvent operator;

Language

English

References

1.

S. Adly and W. Oettli, Solvability of generalized nonlinear symmetric variational inequalities,
J. Austral. Math. Soc. Ser. B 40 (1999), no. 3, 289–300.

2.

H. Brezis, Operateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Mathematics Studies, No. 5. Notas de Matematica (50). North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973.

3.

Y. J. Cho, S. M. Kang, and X. Qin, On systems of generalized nonlinear variational
inequalities in Banach spaces, Appl. Math. Comput. 206 (2008), no. 1, 214–220.

4.

Y. J. Cho and X. Qin, Generalized systems for relaxed cocoercive variational inequalities
and projection methods in Hilbert spaces, Math. Inequal. Appl. 12 (2009), no. 2, 365–375.

5.

Y. J. Cho and X. Qin, Systems of generalized nonlinear variational inequalities and its projection
methods, Nonlinear Anal. 69 (2008), no. 12, 4443–4451.

6.

Y. J. Cho, X. Qin, and J. I. Kang, Convergence theorems based on hybrid methods for
generalized equilibrium problems and fixed point problems, Nonlinear Anal. 71 (2009),
no. 9, 4203–4214.

7.

R. Glowinski and P. Le Tallec, Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics, SIAM Studies in Applied Mathematics, 9. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1989.

8.

A. Hamdi, A modified Bregman proximal scheme to minimize the difference of two
convex functions, Appl. Math. E-Notes 6 (2006), 132–140.

9.

S. Haubruge, V. H. Nguyen, and J. J. Strodiot, Convergence analysis and applications
of the Glowinski-Le Tallec splitting method for finding a zero of the sum of two maximal
monotone operators, J. Optim. Theory Appl. 97 (1998), no. 3, 645–673.

10.

A. Moudafi, On the difference of two maximal monotone operators: regularization and
algorithmic approaches, Appl. Math. Comput. 202 (2008), no. 2, 446–452.

11.

M. A. Noor, Three-step iterative algorithms for multivalued quasi variational inclusions,
J. Math. Anal. Appl. 255 (2001), no. 2, 589–604.

12.

M. A. Noor, General variational inequalities and nonexpansive mappings, J. Math. Anal.
Appl. 331 (2007), no. 2, 810–822.

13.

M. A. Noor and Z. Huang, Three-step methods for nonexpansive mappings and variational
inequalities, Appl. Math. Comput. 187 (2007), no. 2, 680–685.

14.

M. A. Noor and Z. Huang, Some resolvent iterative methods for variational inclusions and nonexpansive
mappings, Appl. Math. Comput. 194 (2007), no. 1, 267–275.

15.

M. A. Noor, K. I. Noor, A. Hamdi, and E. H. El-Shemas, On difference of two monotone
operators, Optim. Lett. 3 (2009), no. 3, 329–335.

16.

M. A. Noor, T. M. Rassias, and Z. Huang, Three-step iterations for nonlinear accretive
operator equations, J. Math. Anal. Appl. 274 (2002), no. 1, 59–68.

17.

X. Qin, Y. Su, and M. Shang, Approximating common fixed points of asymptotically
nonexpansive mappings by composite algorithm in Banach spaces, Cent. Eur. J. Math.
5 (2007), no. 2, 345–357.

18.

S. Reich, Constructive Techniques for Accretive and Monotone Operators, Applied nonlinear analysis (Proc. Third Internat. Conf., Univ. Texas, Arlington, Tex., 1978), pp. 335–345, Academic Press, New York-London, 1979.

19.

M. Shang, Y. Su, and X. Qin, Three-step iterations for nonexpansive mappings and
inverse-strongly monotone mappings, J. Syst. Sci. Complex. 22 (2009), no. 2, 333–344.