## Abstract and Applied Analysis

### Hybrid Algorithms for Solving Variational Inequalities, Variational Inclusions, Mixed Equilibria, and Fixed Point Problems

#### Abstract

We present a hybrid iterative algorithm for finding a common element of the set of solutions of a finite family of generalized mixed equilibrium problems, the set of solutions of a finite family of variational inequalities for inverse strong monotone mappings, the set of fixed points of an infinite family of nonexpansive mappings, and the set of solutions of a variational inclusion in a real Hilbert space. Furthermore, we prove that the proposed hybrid iterative algorithm has strong convergence under some mild conditions imposed on algorithm parameters. Here, our hybrid algorithm is based on Korpelevič’s extragradient method, hybrid steepest-descent method, and viscosity approximation method.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 208717, 22 pages.

Dates
First available in Project Euclid: 6 October 2014

https://projecteuclid.org/euclid.aaa/1412607614

Digital Object Identifier
doi:10.1155/2014/208717

Mathematical Reviews number (MathSciNet)
MR3176723

Zentralblatt MATH identifier
07021931

#### Citation

Ceng, Lu-Chuan; Petrusel, Adrian; Wong, Mu-Ming; Yao, Jen-Chih. Hybrid Algorithms for Solving Variational Inequalities, Variational Inclusions, Mixed Equilibria, and Fixed Point Problems. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 208717, 22 pages. doi:10.1155/2014/208717. https://projecteuclid.org/euclid.aaa/1412607614

#### References

• F. E. Browder and W. V. Petryshyn, “Construction of fixed points of nonlinear mappings in Hilbert space,” Journal of Mathematical Analysis and Applications, vol. 20, pp. 197–228, 1967.
• J.-W. Peng and J.-C. Yao, “A new hybrid-extragradient method for generalized mixed equilibrium problems, fixed point problems and variational inequality problems,” Taiwanese Journal of Mathematics, vol. 12, no. 6, pp. 1401–1432, 2008.
• L. C. Ceng, H.-Y. Hu, and M. M. Wong, “Strong and weak convergence theorems for generalized mixed equilibrium problem with perturbation and fixed pointed problem of infinitely many nonexpansive mappings,” Taiwanese Journal of Mathematics, vol. 15, no. 3, pp. 1341–1367, 2011.
• Y. Yao, Y. J. Cho, and Y.-C. Liou, “Algorithms of common solutions for variational inclusions, mixed equilibrium problems and fixed point problems,” European Journal of Operational Research, vol. 212, no. 2, pp. 242–250, 2011.
• L.-C. Ceng and J.-C. Yao, “A relaxed extragradient-like method for a generalized mixed equilibrium problem, a general system of generalized equilibria and a fixed point problem,” Nonlinear Analysis. Theory, Methods & Applications, vol. 72, no. 3-4, pp. 1922–1937, 2010.
• L.-C. Ceng, Q. H. Ansari, and S. Schaible, “Hybrid extragradient-like methods for generalized mixed equilibrium problems, systems of generalized equilibrium problems and optimization problems,” Journal of Global Optimization, vol. 53, no. 1, pp. 69–96, 2012.
• L.-C. Ceng, S.-M. Guu, and J.-C. Yao, “Hybrid iterative method for finding common solutions of generalized mixed equilibrium and fixed point problems,” Fixed Point Theory and Applications, vol. 92, p. 19, 2012.
• G. Cai and S. Q. Bu, “Strong and weak convergence theorems for general mixed equilibrium problems and variational inequality problems and fixed point problems in Hilbert spaces,” Journal of Computational and Applied Mathematics, vol. 247, pp. 34–52, 2013.
• S. Takahashi and W. Takahashi, “Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space,” Nonlinear Analysis. Theory, Methods & Applications, vol. 69, no. 3, pp. 1025–1033, 2008.
• L.-C. Ceng and J.-C. Yao, “A hybrid iterative scheme for mixed equilibrium problems and fixed point problems,” Journal of Computational and Applied Mathematics, vol. 214, no. 1, pp. 186–201, 2008.
• R. T. Rockafellar, “Monotone operators and the proximal point algorithm,” SIAM Journal on Control and Optimization, vol. 14, no. 5, pp. 877–898, 1976.
• N.-J. Huang, “A new completely general class of variational inclusions with noncompact valued mappings,” Computers & Mathematics with Applications, vol. 35, no. 10, pp. 9–14, 1998.
• L.-C. Zeng, S.-M. Guu, and J.-C. Yao, “Characterization of $H$-monotone operators with applications to variational inclusions,” Computers & Mathematics with Applications, vol. 50, no. 3-4, pp. 329–337, 2005.
• Y.-P. Fang and N.-J. Huang, “$H$-accretive operators and resolvent operator technique for solving variational inclusions in Banach spaces,” Applied Mathematics Letters, vol. 17, no. 6, pp. 647–653, 2004.
• S.-S. Zhang, J. H. W. Lee, and C. K. Chan, “Algorithms of common solutions to quasi variational inclusion and fixed point problems,” Applied Mathematics and Mechanics, vol. 29, no. 5, pp. 571–581, 2008.
• J.-W. Peng, Y. Wang, D. S. Shyu, and J.-C. Yao, “Common solutions of an iterative scheme for variational inclusions, equilibrium problems, and fixed point problems,” Journal of Inequalities and Applications, vol. 2008, Article ID 720371, 15 pages, 2008.
• L.-C. Ceng, Q. H. Ansari, M. M. Wong, and J.-C. Yao, “Mann type hybrid extragradient method for variational inequalities, variational inclusions and fixed point problems,” Fixed Point Theory, vol. 13, no. 2, pp. 403–422, 2012.
• G. M. Korpelevič, “An extragradient method for finding saddle points and for other problems,” Èkonomika i Matematicheskie Metody, vol. 12, no. 4, pp. 747–756, 1976.
• F. Facchinei and J.-S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems. Vol. I, Springer, New York, NY, USA, 2003.
• L.-C. Ceng, S.-M. Guu, and J.-C. Yao, “Finding common solutions of a variational inequality, a general system of variational inequalities, and a fixed-point problem via a hybrid extragradient method,” Fixed Point Theory and Applications, vol. 2011, Article ID 626159, 22 pages, 2011.
• N. Nadezhkina and W. Takahashi, “Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings,” Journal of Optimization Theory and Applications, vol. 128, no. 1, pp. 191–201, 2006.
• L.-C. Ceng, Q. H. Ansari, and J.-C. Yao, “An extragradient method for solving split feasibility and fixed point problems,” Computers & Mathematics with Applications, vol. 64, no. 4, pp. 633–642, 2012.
• L. C. Ceng, M. Teboulle, and J. C. Yao, “Weak convergence of an iterative method for pseudomonotone variational inequalities and fixed-point problems,” Journal of Optimization Theory and Applications, vol. 146, no. 1, pp. 19–31, 2010.
• L.-C. Ceng, Q. H. Ansari, and J.-C. Yao, “Relaxed extragradient methods for finding minimum-norm solutions of the split feasibility problem,” Nonlinear Analysis. Theory, Methods & Applications, vol. 75, no. 4, pp. 2116–2125, 2012.
• L.-C. Ceng, Q. H. Ansari, and J.-C. Yao, “Relaxed extragradient iterative methods for variational inequalities,” Applied Mathematics and Computation, vol. 218, no. 3, pp. 1112–1123, 2011.
• L.-C. Ceng, Q. H. Ansari, N.-C. Wong, and J.-C. Yao, “An extragradient-like approximation method for variational inequalities and fixed point problems,” Fixed Point Theory and Applications, vol. 2011, article 22, 2011.
• L.-C. Ceng, N. Hadjisavvas, and N.-C. Wong, “Strong convergence theorem by a hybrid extragradient-like approximation method for variational inequalities and fixed point problems,” Journal of Global Optimization, vol. 46, no. 4, pp. 635–646, 2010.
• L.-C. Ceng, Q. H. Ansari, M. M. Wong, and J.-C. Yao, “Mann type hybrid extragradient method for variational inequalities, variational inclusions and fixed point problems,” Fixed Point Theory, vol. 13, no. 2, pp. 403–422, 2012.
• J. L. Lions, Quelques MéthoDes De Résolution Des Problémes Aux Limites Non Linéaires, Dunod, Paris, 1969.
• R. Glowinski, Numerical Methods for Nonlinear Variational Problems, Springer, New York, NY, USA, 1984.
• W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, Japan, 2000.
• J. T. Oden, Quantitative Methods on Nonlinear Mechanics, Prentice-Hall, Englewood Cliffs, NJ, USA, 1986.
• E. Zeidler, Nonlinear Functional Analysis and Its Applications, Springer, New York, NY, USA, 1985.
• J. G. O'Hara, P. Pillay, and H.-K. Xu, “Iterative approaches to convex feasibility problems in Banach spaces,” Nonlinear Analysis. Theory, Methods & Applications, vol. 64, no. 9, pp. 2022–2042, 2006.
• L.-C. Ceng and J.-C. Yao, “Relaxed viscosity approximation methods for fixed point problems and variational inequality problems,” Nonlinear Analysis. Theory, Methods & Applications, vol. 69, no. 10, pp. 3299–3309, 2008.
• S. Atsushiba and W. Takahashi, “Strong convergence theorems for a finite family of nonexpansive mappings and applications,” Indian Journal of Mathematics, vol. 41, no. 3, pp. 435–453, 1999.
• V. Colao, G. Marino, and H.-K. Xu, “An iterative method for finding common solutions of equilibrium and fixed point problems,” Journal of Mathematical Analysis and Applications, vol. 344, no. 1, pp. 340–352, 2008.
• I. Yamada, “The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings,” in Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, D. Butnariu, Y. Censor, and S. Reich, Eds., vol. 8 of Studies in Computational Mathematics, pp. 473–504, North-Holland, Amsterdam, The Netherlands, 2001.
• K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge University Press, Cambridge, UK, 1990.
• A. Moudafi and M. Théra, “Proximal and dynamical approaches to equilibrium problems,” in Ill-posed Variational Problems and Regularization Techniques, vol. 477 of Lecture Notes in Economics and Mathematical Systems, pp. 187–201, Springer, Berlin, Germany, 1999.
• T. Suzuki, “Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals,” Journal of Mathematical Analysis and Applications, vol. 305, no. 1, pp. 227–239, 2005.
• K. Shimoji and W. Takahashi, “Strong convergence to common fixed points of infinite nonexpansive mappings and applications,” Taiwanese Journal of Mathematics, vol. 5, no. 2, pp. 387–404, 2001.
• Y. Yao, M. A. Noor, S. Zainab, and Y.-C. Liou, “Mixed equilibrium problems and optimization problems,” Journal of Mathematical Analysis and Applications, vol. 354, no. 1, pp. 319–329, 2009.
• 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, pp. 185–201, 2003. \endinput