## Abstract and Applied Analysis

### Hybrid Extragradient Method with Regularization for Convex Minimization, Generalized Mixed Equilibrium, Variational Inequality and Fixed Point Problems

#### Abstract

We introduce two iterative algorithms by the hybrid extragradient method with regularization for finding a common element of the set of solutions of the minimization problem for a convex and continuously Fréchet differentiable functional, the set of solutions of finite generalized mixed equilibrium problems, the set of solutions of finite variational inequalities for inverse strong monotone mappings and the set of fixed points of an asymptotically $\kappa$-strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. We prove some strong and weak convergence theorems for the proposed iterative algorithms under mild conditions.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 436069, 27 pages.

Dates
First available in Project Euclid: 6 October 2014

https://projecteuclid.org/euclid.aaa/1412606988

Digital Object Identifier
doi:10.1155/2014/436069

Mathematical Reviews number (MathSciNet)
MR3166613

Zentralblatt MATH identifier
07022387

#### Citation

Ceng, Lu-Chuan; Ho, Juei-Ling. Hybrid Extragradient Method with Regularization for Convex Minimization, Generalized Mixed Equilibrium, Variational Inequality and Fixed Point Problems. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 436069, 27 pages. doi:10.1155/2014/436069. https://projecteuclid.org/euclid.aaa/1412606988

#### References

• 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.
• G. Cai and S. 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.
• 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.
• J.-B. Baillon and G. Haddad, “Quelques propriétés des opérateurs angle-bornés et n-cycliquement monotones,” Israel Journal of Mathematics, vol. 26, no. 2, pp. 137–150, 1977.
• H.-K. Xu, “Averaged mappings and the gradient-projection algorithm,” Journal of Optimization Theory and Applications, vol. 150, no. 2, pp. 360–378, 2011.
• 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, 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.
• J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Paris, France, 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.
• 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.
• 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. Zeng and J.-C. Yao, “Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems,” Taiwanese Journal of Mathematics, vol. 10, no. 5, pp. 1293–1303, 2006.
• N. Nadezhkina and W. Takahashi, “Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz-continuous monotone mappings,” SIAM Journal on Optimization, vol. 16, no. 4, pp. 1230–1241, 2006.
• L.-C. Ceng and J.-C. Yao, “An extragradient-like approximation method for variational inequality problems and fixed point problems,” Applied Mathematics and Computation, vol. 190, no. 1, pp. 205–215, 2007.
• K. Goebel and W. A. Kirk, “A fixed point theorem for asymptotically nonexpansive mappings,” Proceedings of the American Mathematical Society, vol. 35, pp. 171–174, 1972.
• R. E. Bruck, T. Kuczumow, and S. Reich, “Convergence of iterates of asymptotically nonexpansive mappings in Banach spaces with the uniform Opial property,” Colloquium Mathematicum, vol. 65, no. 2, pp. 169–179, 1993.
• T.-H. Kim and H.-K. Xu, “Convergence of the modified Mann's iteration method for asymptotically strict pseudo-contractions,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 9, pp. 2828–2836, 2008.
• D. R. Sahu, H.-K. Xu, and J.-C. Yao, “Asymptotically strict pseudocontractive mappings in the intermediate sense,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 10, pp. 3502–3511, 2009.
• S. Huang, “Hybrid extragradient methods for asymptotically strict pseudo-contractions in the intermediate sense and variational inequality problems,” Optimization, vol. 60, no. 6, pp. 739–754, 2011.
• L.-C. Ceng and J.-C. Yao, “Strong convergence theorems for variational inequalities and fixed point problems of asymptotically strict pseudocontractive mappings in the intermediate sense,” Acta Applicandae Mathematicae, vol. 115, no. 2, pp. 167–191, 2011.
• Y. Yao, H. Zhou, and Y.-C. Liou, “Weak and strong convergence theorems for an asymptotically k-strict pseudo-contraction and a mixed equilibrium problem,” Journal of the Korean Mathematical Society, vol. 46, no. 3, pp. 561–576, 2009.
• 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, A. Petruşel, and J. C. Yao, “Iterative approaches to solving equilibrium problems and fixed point problems of infinitely many nonexpansive mappings,” Journal of Optimization Theory and Applications, vol. 143, no. 1, pp. 37–58, 2009.
• L.-C. Ceng and J.-C. Yao, “Approximate proximal methods in vector optimization,” European Journal of Operational Research, vol. 183, no. 1, pp. 1–19, 2007.
• L.-C. Ceng, Q. H. Ansari, and J.-C. Yao, “An extragradientčommentComment on ref. [34?]: This reference is a repetition of [29?]. Please check similar cases throughout. method for solving split feasibility and fixed point problems,” Computers & Mathematics with Applications, vol. 64, no. 4, pp. 633–642, 2012.
• 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.
• C. Byrne, “A unified treatment of some iterative algorithms in signal processing and image reconstruction,” Inverse Problems, vol. 20, no. 1, pp. 103–120, 2004.
• P. L. Combettes, “Solving monotone inclusions via compositions of nonexpansive averaged operators,” Optimization, vol. 53, no. 5-6, pp. 475–504, 2004.
• J. Górnicki, “Weak convergence theorems for asymptotically nonexpansive mappings in uniformly convex Banach spaces,” Commentationes Mathematicae Universitatis Carolinae, vol. 30, no. 2, pp. 249–252, 1989.
• G. Marino and H.-K. Xu, “Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 329, no. 1, pp. 336–346, 2007.
• H. K. Xu, “Existence and convergence for fixed points of mappings of asymptotically nonexpansive type,” Nonlinear Analysis: Theory, Methods & Applications, vol. 16, no. 12, pp. 1139–1146, 1991.
• M. O. Osilike, S. C. Aniagbosor, and B. G. Akuchu, “Fixed points of asymptotically demicontractive mappings in arbitrary Banach spaces,” Panamerican Mathematical Journal, vol. 12, no. 2, pp. 77–88, 2002.
• K.-K. Tan and H. K. Xu, “Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process,” Journal of Mathematical Analysis and Applications, vol. 178, no. 2, pp. 301–308, 1993.
• K. Geobel and W. A. Kirk, Topics on Metric Fixed-Point Theory, Cambridge University Press, Cambridge, UK, 1990.
• R. T. Rockafellar, “On the maximality of sums of nonlinear monotone operators,” Transactions of the American Mathematical Society, vol. 149, pp. 75–88, 1970.
• 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.
• C. Martinez-Yanes and H.-K. Xu, “Strong convergence of the CQ method for fixed point iteration processes,” Nonlinear Analysis: Theory, Methods & Applications, vol. 64, no. 11, pp. 2400–2411, 2006. \endinput