Abstract and Applied Analysis

A New Iterative Method for Equilibrium Problems and Fixed Point Problems

Abdul Latif and Mohammad Eslamian

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Abstract

Introducing a new iterative method, we study the existence of a common element of the set of solutions of equilibrium problems for a family of monotone, Lipschitz-type continuous mappings and the sets of fixed points of two nonexpansive semigroups in a real Hilbert space. We establish strong convergence theorems of the new iterative method for the solution of the variational inequality problem which is the optimality condition for the minimization problem. Our results improve and generalize the corresponding recent results of Anh (2012), Cianciaruso et al. (2010), and many others.

Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 178053, 9 pages.

Dates
First available in Project Euclid: 26 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393442174

Digital Object Identifier
doi:10.1155/2013/178053

Mathematical Reviews number (MathSciNet)
MR3147862

Zentralblatt MATH identifier
1364.47031

Citation

Latif, Abdul; Eslamian, Mohammad. A New Iterative Method for Equilibrium Problems and Fixed Point Problems. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 178053, 9 pages. doi:10.1155/2013/178053. https://projecteuclid.org/euclid.aaa/1393442174


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References

  • K. Fan, “A minimax inequality and applications,” in Inequality III, pp. 103–113, Academic Press, New York, NY, USA, 1972.
  • P. N. Anh, “A hybrid extragradient method extended to fixed point problems and equilibrium problems,” Optimization, vol. 62, no. 2, pp. 271–283, 2013.
  • 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.
  • 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.
  • G. Stampacchia, “Formes bilinéaires coercitives sur les ensembles convexes,” Comptes rendus de l'Académie des Sciences, vol. 258, pp. 4413–4416, 1964.
  • P. Daniele, F. Giannessi, and A. Maugeri, Equilibrium Problems and Variational Models, vol. 68, Kluwer Academic, Norwell, Mass, USA, 2003.
  • J.-W. Peng and J.-C. Yao, “Some new extragradient-like methods for generalized equilibrium problems, fixed point problems and variational inequality problems,” Optimization Methods & Software, vol. 25, no. 4–6, pp. 677–698, 2010.
  • D. R. Sahu, N.-C. Wong, and J.-C. Yao, “Strong convergence theorems for semigroups of asymptotically nonexpansive mappings in Banach spaces,” Abstract and Applied Analysis, vol. 2013, Article ID 202095, 8 pages, 2013.
  • S. Takahashi and W. Takahashi, “Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 331, no. 1, pp. 506–515, 2007.
  • S. Wang and B. Guo, “New iterative scheme with nonexpansive mappings for equilibrium problems and variational inequality problems in Hilbert spaces,” Journal of Computational and Applied Mathematics, vol. 233, no. 10, pp. 2620–2630, 2010.
  • Y. Yao, Y.-C. Liou, and Y.-J. Wu, “An extragradient method for mixed equilibrium problems and fixed point problems,” Fixed Point Theory and Applications, vol. 2009, Article ID 632819, 15 pages, 2009.
  • F. E. Browder, “Nonexpansive nonlinear operators in a Banach space,” Proceedings of the National Academy of Sciences of the United States of America, vol. 54, pp. 1041–1044, 1965.
  • R. Chen and Y. Song, “Convergence to common fixed point of nonexpansive semigroups,” Journal of Computational and Applied Mathematics, vol. 200, no. 2, pp. 566–575, 2007.
  • A. Aleyner and Y. Censor, “Best approximation to common fixed points of a semigroup of nonexpansive operators,” Journal of Nonlinear and Convex Analysis, vol. 6, no. 1, pp. 137–151, 2005.
  • H. K. Xu, “An iterative approach to quadratic optimization,” Journal of Optimization Theory and Applications, vol. 116, no. 3, pp. 659–678, 2003.
  • G. Marino and H.-K. Xu, “A general iterative method for nonexpansive mappings in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 318, no. 1, pp. 43–52, 2006.
  • A. Moudafi, “Viscosity approximation methods for fixed-points problems,” Journal of Mathematical Analysis and Applications, vol. 241, no. 1, pp. 46–55, 2000.
  • F. Cianciaruso, G. Marino, and L. Muglia, “Iterative methods for equilibrium and fixed point problems for nonexpansive semigroups in Hilbert spaces,” Journal of Optimization Theory and Applications, vol. 146, no. 2, pp. 491–509, 2010.
  • S. Li, L. Li, and Y. Su, “General iterative methods for a one-parameter nonexpansive semigroup in Hilbert space,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 9, pp. 3065–3071, 2009.
  • D. R. Sahu, N. C. Wong, and J. C. Yao, “A unified hybrid iterative method for solving variational inequalities involving generalized pseudocontractive mappings,” SIAM Journal on Control and Optimization, vol. 50, no. 4, pp. 2335–2354, 2012.
  • N. Shioji and W. Takahashi, “Strong convergence theorems for asymptotically nonexpansive semigroups in Hilbert spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 34, no. 1, pp. 87–99, 1998.
  • P. N. Anh, “Strong convergence theorems for nonexpansive mappings and Ky Fan inequalities,” Journal of Optimization Theory and Applications, vol. 154, no. 1, pp. 303–320, 2012.
  • L.-C. Ceng, Q. H. Ansari, and J.-C. Yao, “Hybrid proximal-type and hybrid shrinking projection algorithms for equilibrium problems, maximal monotone operators, and relatively nonexpansive mappings,” Numerical Functional Analysis and Optimization, vol. 31, no. 7-9, pp. 763–797, 2010.
  • L.-C. Ceng, Q. H. Ansari, and J.-C. Yao, “Viscosity approximation methods for generalized equilibrium problems and fixed point problems,” Journal of Global Optimization, vol. 43, no. 4, pp. 487–502, 2009.
  • 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, 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. 2012, article 92, 2012.
  • M. Eslamian, “Convergence theorems for nonspreading mappings and nonexpansive multivalued mappings and equilibrium problems,” Optimization Letters, vol. 7, no. 3, pp. 547–557, 2013.
  • M. Eslamian, “Hybrid method for equilibrium problems and fixed point problems of finite families of nonexpansive semigroups,” RACSAM, vol. 107, pp. 299–307, 2013.
  • M. Eslamian and A. Abkar, “One-step iterative process for a finite family of multivalued mappings,” Mathematical and Computer Modelling, vol. 54, no. 1-2, pp. 105–111, 2011.
  • P.-E. Maingé, “Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization,” Set-Valued Analysis, vol. 16, no. 7-8, pp. 899–912, 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.
  • J.-B. Baillon, “Un théorème de type ergodique pour les contractions non linéaires dans un espace de Hilbert,” Comptes Rendus de l'Académie des Sciences, vol. 280, no. 22, pp. A1511–A1514, 1975.
  • W. Kaczor, T. Kuczumow, and S. Reich, “A mean ergodic theorem for nonlinear semigroups which are asymptotically nonexpansive in the intermediate sense,” Journal of Mathematical Analysis and Applications, vol. 246, no. 1, pp. 1–27, 2000. \endinput