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A new explicit iterative algorithm for solving split variational inclusion and fixed point problem for the infinite family of nonexpansive operators

Cuijie Zhang (zhang_cui_jie***at***126.com)
Zhihui Xu (zhihui_xu007***at***163.com)

Abstract: In this paper, we introduce a new explicit iterative algorithm for finding a solution of split variational inclusion problem over the common fixed points set of a infinite family of nonexpansive mappings in Hilbert spaces. To reach this goal, the iterative algorithms which combine Tian's method with some fixed point technically proving methods are utilized for solving the problem. Under suitable assumptions, we prove that the sequence generated by the iterative algorithm converges strongly to the unique solution of the considered problem. Our result improves and extends the corresponding results announced by many others.

Keywords: Split variational inclusion problem, Strong convergence, Variational inequality, Common fixed point, resolvent operator.

Category 1: Complementarity and Variational Inequalities

Citation: 1.Moudafi, A., Split Monotone Variatioal Inclusions, J. Optim. Theory Appl., 150,275-283, 2011. 2.Moudafi, A.,Viscosity approximation methods for fixed points problem, J. Math. Anal. Appl., 241,46-55,2000. 3.Nimana, N., Ptrot, N.,Viscosity approximation methods for split variational inclusion and fixed point problems in Hilbert spaces, In: Proceedings fo the international multiconference of engineers and computer scientists 2014 Vol $\amalg$. 4.Aoyama, K., Kimura, Y., Takahashi, W., Toyoda, M.: Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space. Nonlinear Anal. 67, 2350-2360,2007. 5.Marino, G., Xu, H.K., A general iterative method for nonexpansive mappings in Hilbert spaces, Journal of Mathematical Analysis and Applications, 318,43-52,2006. 6.Yamada, I., The hybrid steepest descent for the variational inequality problems over the intersection of fixed points sets of nonexpansive mapping,in: D. Butnariu, Y. Censor, S. Reich (Eds.), Inherently Parallel Algorithms in Feasibility and Optimization and Their Application, Elservier, New York, 473-504,2001. 7.Tian, M.,A general iterative algorithm for nonexpansive mappings in Hilbert spaces,73,689-694,2010. 8.Crombez, G., A hierarchical presentation of operators with fixed points on Hilbert spaces. Numer. Funct. Anal. Optim.,27,259-277,2006. 9.Crombez, G., A geometrical look at iterative methods for operators with fixed points,Numer. Funct. Anal. Optim.,26,157-175,2005. 10. Xu, H.K., Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl.,298,279-291,2004. 11.Goebel, K., Kirk, W. A.,Topics in Metric Fixed-Point Theory,Cambridge University Press, Cambridge, England, 1990. 12.Kazmi, K. R.,Rizvi, S. H., An iterative method for split variational inclusion problem and fixed point problem for a nonexpansive mapping,Optim Lett.DOI 10.1007/s11590-013-0629-2. 13.Lopez, G., Martin, V. Xu, H. K., Iterative algorithm for the multiple-sets split feasibility problem, Biomedical Math,243-279, 2009. 14. Xu, H.K., An iterative approch to quadratic optimization,J. Optim. Theory Appl., 116,659-678,2003. 15. Marino, G, Xu, H.K., A general iterative method for nonexpansive mappings in Hilbert spaces,J. Math. Anal. Appl.,318,43-52,2006. 16.Yamada, I.,The hybrid steepest descent for the variational inequality problems over the intersection of fixed points sets of nonexpansive mappings, In: Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications. Studies in Computatinal Mathematics, 8, 473-504,2001.

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Entry Submitted: 09/04/2015
Entry Accepted: 09/04/2015
Entry Last Modified: 09/05/2015

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