Abstract and Applied Analysis

A System of Generalized Mixed Equilibrium Problems, Maximal Monotone Operators, and Fixed Point Problems with Application to Optimization Problems

Pongsakorn Sunthrayuth and Poom Kumam

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We introduce a new iterative algorithm for finding a common element of the set of solutions of a system of generalized mixed equilibrium problems, zero set of the sum of a maximal monotone operators and inverse-strongly monotone mappings, and the set of common fixed points of an infinite family of nonexpansive mappings with infinite real number. Furthermore, we prove under some mild conditions that the proposed iterative algorithm converges strongly to a common element of the above four sets, which is a solution of the optimization problem related to a strongly positive bounded linear operator. The results presented in the paper improve and extend the recent ones announced by many others.

Article information

Source
Abstr. Appl. Anal. Volume 2012, Special Issue (2012), Article ID316276, 39 pages.

Dates
First available: 15 February 2012

Permanent link to this document
http://projecteuclid.org/euclid.aaa/1329337691

Digital Object Identifier
doi:10.1155/2012/316276

Mathematical Reviews number (MathSciNet)
MR2872300

Zentralblatt MATH identifier
1232.49037

Citation

Sunthrayuth, Pongsakorn; Kumam, Poom. A System of Generalized Mixed Equilibrium Problems, Maximal Monotone Operators, and Fixed Point Problems with Application to Optimization Problems. Abstract and Applied Analysis 2012, Special Issue (2012), 1--39. doi:10.1155/2012/316276. http://projecteuclid.org/euclid.aaa/1329337691.


Export citation

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 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.
  • 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.
  • E. Blum and W. Oettli, “From optimization and variational inequalities to equilibrium problems,” The Mathematics Student, vol. 63, no. 1–4, pp. 123–145, 1994.
  • C. Jaiboon and P. Kumam, “A general iterative method for addressing mixed equilibrium problems and optimization problems,” Nonlinear Analysis. Theory, Methods & Applications, vol. 73, no. 5, pp. 1180–1202, 2010.
  • L.-C. Ceng, A. R. Khan, Q. H. Ansari, and J.-C. Yao, “Viscosity approximation methods for strongly positive and monotone operators,” Fixed Point Theory, vol. 10, no. 1, pp. 35–72, 2009.
  • 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.
  • H.-Y. Hu and L.-C. Ceng, “A general system of generalized nonlinear mixed composite-type equilibria in Hilbert spaces,” Taiwanese Journal of Mathematics, vol. 15, no. 3, pp. 927–959, 2011.
  • 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.
  • 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.
  • M. A. Noor, “General variational inequalities and nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 331, no. 2, pp. 810–822, 2007.
  • P. Kumam and P. Katchang, “A viscosity of extragradient approximation method for finding equilibrium problems, variational inequalities and fixed point problems for nonexpansive mappings,” Nonlinear Analysis. Hybrid Systems, vol. 3, no. 4, pp. 475–486, 2009.
  • S. Plubtieng and P. Kumam, “Weak convergence theorem for monotone mappings and a countable family of nonexpansive mappings,” Journal of Computational and Applied Mathematics, vol. 224, no. 2, pp. 614–621, 2009.
  • X. Qin, Y. J. Cho, and S. M. Kang, “Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces,” Journal of Computational and Applied Mathematics, vol. 225, no. 1, pp. 20–30, 2009.
  • S. Saewan and P. Kumam, “A hybrid iterative scheme for a maximal monotone operator and two countable families of relatively quasi-nonexpansive mappings for generalized mixed equilibrium and variational inequality problems,” Abstract and Applied Analysis, vol. 2010, Article ID 123027, 31 pages, 2010.
  • S. Saewan and P. Kumam, “Modified hybrid block iterative algorithm for convex feasibility problems and generalized equilibrium problems for uniformly quasi-ø-asymptotically nonexpansive mappings,” Abstract and Applied Analysis, vol. 2010, Article ID 357120, 22 pages, 2010.
  • S. Saewan and P. Kumam, “A new modified block iterative algorithm for uniformly quasi-ø-asymptotically nonexpansive mappings and a system of generalized mixed equilibrium problems,” Fixed Point Theory and Applications, vol. 2011, 35 pages, 2011.
  • S. Saewan and P. Kumam, “A modified hybrid projection method for solving generalized mixed equilibrium problems and fixed point problems in Banach spaces,” Computers and Mathematics with Applications, vol. 62, pp. 1723–1735, 2011.
  • S. Saewan and P. Kumam, “Strong convergence theorems for countable families of uniformly quasi-ø-asymptotically nonexpansive mappings and a system of generalized mixed equilibrium problems,” Abstract and Applied Analysis, vol. 2011, Article ID 701675, 2011.
  • S. Saewan and P. Kumam, “The shrinking projection method for solving generalized equilibrium problem and common fixed points for asymptotically quasi-ø-nonexpansive mappings,” Fixed Point Theory and Applications, vol. 2011, 9 pages, 2011.
  • A. Tada and W. Takahashi, “Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem,” Journal of Optimization Theory and Applications, vol. 133, no. 3, pp. 359–370, 2007.
  • W. Takahashi and M. Toyoda, “Weak convergence theorems for nonexpansive mappings and monotone mappings,” Journal of Optimization Theory and Applications, vol. 118, no. 2, pp. 417–428, 2003.
  • P. Kumam, W. Kumam, and P. Junlouchai, “Generalized systems of variational inequalities and projection methods for inverse-strongly monotone mappings,” Discrete Dynamics in Nature and Society, vol. 2011, Article ID 976505, 2011.
  • P. Kumam, “A relaxed extragradient approximation method of two inverse-strongly monotone mappings for a general system of variational inequalities, fixed point and equilibrium problems,” Bulletin of the Iranian Mathematical Society, vol. 36, no. 1, pp. 227–250, 2010.
  • P. Kumam and C. Jaiboon, “Approximation of common solutions to system of mixed equilibrium problems, variational inequality problem, and strict pseudo-contractive mappings,” Fixed Point Theory and Applications, vol. 2011, Article ID 347204, 30 pages, 2011.
  • P. Katchang and P. Kumam, “Convergence of iterative algorithm for finding common solution of fixed points and general system of variational inequalities for two accretive operators,” Thai Journal of Mathematics, vol. 9, no. 2, pp. 343–360, 2011.
  • P. Katchang and P. Kumam, “A general iterative method of fixed points for mixed equilibrium problems and variational inclusion problems,” Journal of Inequalities and Applications, vol. 2010, Article ID 370197, 25 pages, 2010.
  • P. Katchang, T. Jitpeera, and P. Kumam, “Strong convergence theorems for solving generalized mixed equilibrium problems and general system of variational inequalities by the hybrid method,” Nonlinear Analysis. Hybrid Systems, vol. 4, no. 4, pp. 838–852, 2010.
  • N. Petrot, R. Wangkeeree, and P. Kumam, “A viscosity approximation method of common solutions for quasi variational inclusion and fixed point problems,” Fixed Point Theory, vol. 12, no. 1, pp. 165–178, 2011.
  • S. Saewan and P. Kumam, “A hybrid iterative scheme for a maximal monotone operator and two countable families of relatively quasi-nonexpansive mappings for generalized mixed equilibrium and variational inequality problems,” Abstract and Applied Analysis, vol. 2010, Article ID 123027, 31 pages, 2010.
  • Y. Yao and J.-C. Yao, “On modified iterative method for nonexpansive mappings and monotone mappings,” Applied Mathematics and Computation, vol. 186, no. 2, pp. 1551–1558, 2007.
  • S. Saewan, P. Kumam, and K. Wattanawitoon, “Convergence theorem based on a new hybrid projection method for finding a common solution of generalized equilibrium and variational inequality problems in Banach spaces,” Abstract and Applied Analysis, vol. 2010, Article ID 734126, 25 pages, 2010.
  • J. Zhao and S. He, “A new iterative method for equilibrium problems and fixed point problems of infinitely nonexpansive mappings and monotone mappings,” Applied Mathematics and Computation, vol. 215, no. 2, pp. 670–680, 2009.
  • 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.
  • Y. Yao and N. Shahzad, “New methods with perturbations for non-expansive mappings in Hilbert spaces,” Fixed Point Theory and Applications, vol. 2011, no. 79, 2011.
  • Y. Yao and N. Shahzad, “Strong convergence of a proximal point algorithm with general errors,” Optimization Letters, vol. 4, pp. 635–641, 2010.
  • Y. Yao, Y.-C. Liou, and C.-P. Chen, “Algorithms construction for nonexpansive mappings and inversestrongly monotone mappings,” Taiwanese Journal of Mathematics, vol. 15, pp. 1979–1998, 2011.
  • Y. Yao, R. Chen, and Y.-C. Liou, “A unified implicit algorithm for solving the triple-hierarchical constrained optimization problem,” Mathematical & Computer Modelling. In press.
  • P. Sunthrayuth and P. Kumam, “A general iterative algorithm for the solution of variational inequalities for a nonexpansive semigroup in Banach spaces,” Journal of Nonlinear Analysis and Optimization, vol. 1, no. 1, pp. 139–150, 2010.
  • T. Jitpeera and P. Kumam, “A general iterative algorithm for generalized mixed equilibrium problems and variational inclusions approach to variational inequalities,” International Journal of Mathematics and Mathematical Sciences, vol. 2011, Article ID 619813, 25 pages, 2011.
  • T. Jitpeera and P. Kumam, “An extragradient type method for a system of equilibrium problems, variational inequality problems and fixed points of finitely many nonexpansive mappings,” Journal of Nonlinear Analysis and Optimization, vol. 1, no. 1, pp. 71–91, 2010.
  • T. Jitpeera and P. Kumam, “A composite iterative method for generalized mixed equilibrium problems and variational inequality problems,” Journal of Computational Analysis and Applications, vol. 13, no. 2, pp. 345–361, 2011.
  • T. Jitpeera and P. Kumam, “A new hybrid algorithm for a system of mixed equilibrium problems, fixed point problems for nonexpansive semigroup, and variational inclusion problem,” Fixed Point Theory and Applications, Article ID 217407, 27 pages, 2011.
  • W. Chantarangsi, C. Jaiboon, and P. Kumam, “A viscosity hybrid steepest descent method for generalized mixed equilibrium problems and variational inequalities for relaxed cocoercive mapping in Hilbert spaces,” Abstract and Applied Analysis, vol. 2010, Article ID 390972, 39 pages, 2010.
  • L. Yu and M. Liang, “Convergence of iterative sequences for fixed point and variational inclusion problems,” Fixed Point Theory and Applications, vol. 2011, Article ID 368137, 15 pages, 2011.
  • H. H. Bauschke and J. M. Borwein, “On projection algorithms for solving convex feasibility problems,” SIAM Review, vol. 38, no. 3, pp. 367–426, 1996.
  • P. L. Combettes, “Hilbertian convex feasibility problem: convergence of projection methods,” Applied Mathematics and Optimization, vol. 35, no. 3, pp. 311–330, 1997.
  • F. Deutsch and I. Yamada, “Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings,” Numerical Functional Analysis and Optimization, vol. 19, no. 1-2, pp. 33–56, 1998.
  • H.-K. Xu, “Iterative algorithms for nonlinear operators,” Journal of the London Mathematical Society, vol. 66, no. 1, pp. 240–256, 2002.
  • T. Jitpeera, U. Witthayarat, and P. Kumam, “Hybrid algorithms of common solutions of generalized mixed equilibrium problems and the common variational inequality problems with applications,” Fixed Point Theory and Applications, vol. 2011, Article ID 971479, 28 pages, 2011.
  • T. Jitpeera and P. Kumam, “Hybrid algorithms for minimization problems over the solutions of generalized mixed equilibrium and variational inclusion problems,” Mathematical Problems in Engineering, vol. 2011, Article ID 648617, 26 pages, 2011.
  • R. Wangkeeree, N. Petrot, P. Kumam, and C. Jaiboon, “Convergence theorem for mixed equilibrium problems and variational inequality problems for relaxed cocoercive mappings,” Journal of Computational Analysis and Applications, vol. 13, no. 3, pp. 425–449, 2011.
  • A. Kangtunyakarn, “A new iterative algorithm for the set of fixed-point problems of nonexpansive mappings and the set of equilibrium problem and variational inequality problem,” Abstract and Applied Analysis, vol. 2011, Article ID 562689, 24 pages, 2011.
  • 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.
  • W. Takahashi, Nonlinear Functional analysis, Yokohama Publishers, Yokohama, Japan, 2000.
  • K. Aoyama, Y. Kimura, W. Takahashi, and M. Toyoda, “On a strongly nonexpansive sequence in Hilbert spaces,” Journal of Nonlinear and Convex Analysis, vol. 8, no. 3, pp. 471–489, 2007.
  • S. Takahashi, W. Takahashi, and M. Toyoda, “Strong convergence theorems for maximal monotone operators with nonlinear mappings in Hilbert spaces,” Journal of Optimization Theory and Applications, vol. 147, no. 1, pp. 27–41, 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. In press.
  • 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.
  • J. T. Oden, Qualitative Methods on Nonlinear Mechanics, Prentice Hall, Englewood Cliffs, NJ, USA, 1986.
  • 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.
  • J. S. Jung, “Iterative algorithms with some control conditions for quadratic optimizations,” Panamerican Mathematical Journal, vol. 16, no. 4, pp. 13–25, 2006.
  • Z. Opial, “Weak convergence of the sequence of successivečommentComment on ref. [25?]: Please update the information of these references [25,62,65?], if possible. approximations for nonexpansive mappings,” Bulletin of the American Mathematical Society, vol. 73, pp. 591–597, 1967.
  • L. Yu and M. Liang, “Convergence of iterative sequences for fixed point and variational inclusion problems,” Fixed Point Theory and Applications, Article ID 368137, 15 pages, 2011.
  • R. T. Rockafellar, “On the maximal monotonicity of subdifferential mappings,” Pacific Journal of Mathematics, vol. 33, pp. 209–216, 1970.