Strong convergence with a modified iterative projection method for hierarchical fixed point problems and variational inequalities

Yıl 2016, Cilt: 4 Sayı: 2, 193 - 202, 01.03.2016

İbrahim Karahan Murat Özdemir

Öz

Let  be a nonempty closed convex subset of a real
Hilbert space . Let  be a sequence of nearly nonexpansive mappings
such that . Let  be a -Lipschitzian mapping and  be a -Lipschitzian and -strongly monotone operator. This
paper deals with a modified iterative projection method for approximating a
solution of the hierarchical fixed point problem. It is shown that under
certain approximate assumptions on the operators and parameters, the modified
iterative sequence  converges strongly to  which is also the unique solution of the
following variational inequality: 

                                        

 As a special case, this projection method can
be used to find the minimum norm solution of above variational inequality;
namely, the unique solution  to the quadratic minimization problem: . The results here improve and
extend some recent corresponding results of other authors.

Anahtar Kelimeler

Variational inequality, hierarchical fixed point, nearly nonexpansivemappings

Kaynakça

• R. P. Agarwal, D. O’Regan, and D. R. Sahu, Iteratıve Constructıon of Fıxed Poınts of Nearly Asymptotıcally Nonexpansıve Mappıngs, Journal of Nonlinear and Convex Analysis, Vol.8, No.1, 61-79, 2007.
• R. P. Agarwal, D. O’Regan, and D. R. Sahu, Fixed Point Theory for Lipschitzian-Type Mappings with Applications, Topological Fixed Point Theory and Its Applications, Springer, New York, NY, USA, 2009.
• F. Cianciaruso, G. Marino, L. Muglia, Y. Yao, On a two-steps algorithm for hierarchical fixed point problems and variational inequalities. J. Inequal. Appl., 1-13, 2009.
• M. Tian, A general iterative algorithmfor nonexpansivemappings in Hilbert spaces, Nonlinear Analysis, Theory,Methods and Applications, vol. 73, no. 3, 689–694, 2010.
• Y. Yao, Y.J. Cho, Y.C. Liou, Iterative algorithms for hierarchical fixed points problems and variational inequalities. Math. Comput.Model. 52 (9-10), 1697-1705, 2010.
• G. Gu, S. Wang, Y.J. Cho, Strong convergence algorithms for hierarchical fixed points problems and variational inequalities. J. Appl.Math. 2011, 1-17, 2011
• Y. Yao, R. Chen, Regularized algorithms for hierarchical fixed-point problems, Nonlinear Analysis, 74, 6826–6834, 2011.
• M, Tian and L.H.Huang, Iterativemethods for constrained convexminimization probleminHilbert spaces, Fixed Point Theory and Applications, 2013:105, 2013.
• I. Yamada, The hybrid steepest-descent method for variational inequality problems over the intersection of the fixed point sets of nonexpansivemappings. In: Butnariu,D, Censor, Y, Reich, S (eds.) Inherently Parallel Algorithms and Optimization and Their Applications, pp. 473-504. North-Holland, Amsterdam, 2001.
• A.Moudafi, Krasnoselski-Mann iteration for hierarchical fixed-point problems. Inverse Probl. 23 (4), 1635-1640, 2007.
• P.E. Mainge, A, Moudafi, Strong convergence of an iterativemethod for hierarchical fixed-point problems. Pac. J. Optim. 3 (3), 529-538, 2007.
• Y. Yao and Y. C. Liou, Weak and strong convergence of Krasnoselski–Mann iteration for hierarchical fixed point problems, Inverse Problems 24, 015015, 8pp, 2008,
• H.K. Xu, Vıscosıtymethod for hıerarchıcal fıxed poınt approach to varıatıonal ınequalıtıes, Taıwanese Journal ofMathematıcs, Vol. 14, No. 2, 463-478, 2010.
• G. Marino and H.K. Xu, Explicit hierarchical fixed point approach to variational inequalities. J. Optim. Theory Appl. 149 (1), 61-78, 2011.
• A. Bnouhachem and M.A. Noor, An iterative method for approximating the common solutions of a variational inequality, a mixed equilibriumproblemand a hierarchical fixed point problem, Journal of Inequalities and Applications, 2013:490, 2013.
• G. Marino and H.K. Xu, A general iterative method for nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl., 318, 43-52, 2006.
• L.C. Ceng, Q.H. Ansari and J.C. Yao, Some iterative methods for finding fixed points and for solving constrained convex minimization problems. Nonlinear Analysis, 74, 5286-5302, 2011.
• N.C.Wong,D.R. Sahu and J.C. Yao, A generalized hybrid steepest-descentmethod for variational inequalities in Banach spaces, Fixed Point Theory and Applications, vol. 2011, Article ID 754702, 28 pages, 2011.
• D.R. Sahu, S.M. Kang and V. Sagar, Approximation of Common Fixed Points of a Sequence of Nearly NonexpansiveMappings and Solutions of Variational Inequality Problems, Journal of AppliedMathematics, Article ID 902437, 12 pages, 2012.
• Y.Wang andW. Xu, Strong convergence of a modified iterative algorithmfor hierarchical fixed point problems and variational inequalities, Fixed Point Theory and Applications, 2013:121, 2013.
• K. Goebel, W. A. Kirk, Topics onMetric Fixed-Point Theory, Cambridge University Press, Cambridge, England, 1990.
• H.K. Xu and T.H.Kim, Convergence of hybrid steepest-descent methods for variational inequalities,” Journal of Optimization Theory and Applications, vol. 119, no. 1, 185–201, 2003.
• N.C.Wong,D.R. Sahu and J.C. Yao, Solving variational inequalities involving nonexpansive typemappings, Nonlinear Analysis, Theory,Methods and Applications, vol. 69, no. 12, 4732-4753, 2008.
• Hu, H. Y., & Ceng, L. C. (2015). A hybrid iterativemethod for a common solution of variational inequalities, generalized mixed equilibriumproblems, and hierarchical fixed point problems. Journal of Inequalities and Applications, 2015(1), 1-29.
• Bnouhachem, A. (2014). A modified projection method for a common solution of a system of variational inequalities, a split equilibriumproblemand a hierarchical fixed-point problem. Fixed Point Theory and Applications, 2014(1), 1-25.
• Karahan, I., & Ozdemir, M. (2014). Convergence Theorems for Hierarchical Fixed Point Problems and Variational Inequalities. Journal of AppliedMathematics, 2014.
• Karahan, I., Secer, A., Ozdemir, M., & Bayram, M. (2015). The common solution for a generalized equilibrium problem, a variational inequality problemand a hierarchical fixed point problem. Journal of Inequalities and Applications, 2015(1), 1-25.
• Bnouhachem, A., Al-Homidan, S., & Ansari, Q. H. (2014). An iterativemethod for common solutions of equilibrium problems and hierarchical fixed point problems. Fixed Point Theory and Applications, 2014(1), 194.