Abstract
In the present paper, we propose a simpler explicit iterative algorithm for finding a solution for variational inequalities over the set of common fixed points of a finite family of nonexpansive mappings on Hilbert spaces. A strong convergence theorem is proved under fewer restrictions imposed on the mappings and parameters. An extension and numerical result are also given to illustrate the effectiveness and superiority of the proposed algorithm.
Similar content being viewed by others
References
Hartman, P., Stampacchia, G.: On some nonlinear elliptic differential functional equations. Acta. Math. 115, 271–310 (1996)
Browder, F.E.: Nonlinear monotone operators and convex sets in Banach spaces. Bull. Am. Math. Soc. 71, 780–785 (1965)
Browder, F.E.: Nonexpansive nonlinear operators in a Banach space. Proc. Natl. Acad. Sci. USA 54, 1041–1044 (1965)
Browder, F.E.: Existence and approximation of solutions of nonlinear variational inequalities. Proc. Natl. Acad. Sci. USA 56, 1080–1086 (1966)
Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971)
Lions, J.L., Stampaccchia, G.: Variational inequalities. Commun. Pure Appl. Math. 20, 493–519 (1967)
Duvaut, D., Lions, J.L.: Inequalities in Mechanics and Physics. Springer, Berlin (1976)
Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Academic Press, New York (1980)
Hlavacek, I., Haslinger, J., Necas, J., Lovicek, J.: Solution of Variational Inequalities in Mechanics. Springer, New York (1982)
Glowinski, R.: Numerical Methods for Nonlinear Variational Problems. Springer, New York (1984)
Panagiotopoulos, P.D.: Inequality Problems in Mechanics and Applications. Birkhäuser, Boston (1985)
Zeidler, E.: Nonlinear Functional Analysis and Its Applications. Springer, New York (1985)
Yamada, Y.: The hybrid steepest-descent method for variational inequalities problems over the intersection of the fixed point sets of nonexpansive mappings. In: Butnariu, D., Censor, Y., Reich, S. (eds.) Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, pp. 473–504. North-Holland, Amsterdam (2001)
Xu, H.K., Kim, T.H.: Convergence of hybrid steepest-descent methods for variational inequalities. J. Optim. Theory Appl. 119, 185–201 (2003)
Liu, X., Cui, Y.: The common minimal-norm fixed point of a finite family of nonexpansive mappings. Nonlinear Anal. 73, 76–83 (2010)
Buong, D., Duong, L.T.: An explicit iterative algorithm for a class of variational inequalities in Hilbert spaces. J. Optim. Theory Appl. 151, 513–524 (2011)
Byrne, C.: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl. 20, 103–120 (2004)
Xu, H.K.: A variable Krasnosel’skii–Mann algorithm and the multiple-set split feasibility problem. Inverse Probl. 22, 2021–2034 (2006)
Xu, H.K.: Iterative methods for the split feasibility problem in infinite dimensional Hilbert spaces. Inverse Probl. 26, 105018 (2010)
Xu, H.K.: Averaged mappings and the gradient-projection algorithm. J. Optim. Theory Appl. 150, 360–378 (2011)
López, G., Martin, V., Xu, H.K.: Iterative algorithm for the multiple-sets split feasibility problem. In: Biomedical Math, pp. 243–279 (2009)
Goebel, K., Kirk, W.A.: Topics in Metric Fixed Point Theory. Cambridge Studies in Advanced Math., vol. 28. Cambridge University Press, Cambridge (1990)
Suzuki, T.: Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces. Proc. Am. Math. Soc. 135, 99–106 (2007)
Xu, H.K.: An iterative approach to quadratic optimization. J. Optim. Theory Appl. 116, 659–678 (2003)
Zhou, H.: Convergence theorems of fixed points for k-strict pseudo-contractions in Hilbert spaces. Nonlinear Anal. 69, 456–462 (2008)
Gerchberg, R.W.: Super-restoration through error energy reduction. Opt. Acta 21, 709–720 (1974)
Papoulis, A.: A new algorithm in spectral analysis and band-limited extrapolation. IEEE Trans. Circuits Syst. 22, 735–742 (1975)
Piana, M., Bertero, M.: Projected Landweber method and preconditioning. Inverse Probl. 13, 441–463 (1997)
Acknowledgements
The authors would like to thank the editors and the referees for their valuable comments and suggestions which improved the original submission of this paper. This work is supported by the National Natural Science Foundation of China (11071053).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zhou, H., Wang, P. A Simpler Explicit Iterative Algorithm for a Class of Variational Inequalities in Hilbert Spaces. J Optim Theory Appl 161, 716–727 (2014). https://doi.org/10.1007/s10957-013-0470-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-013-0470-x