Hybrid Extragradient-Type Methods for Finding a Common Solution of an Equilibrium Problem and a Family of Strict Pseudo-Contraction Mappings

Affiliation(s)

Posts and Telecommunications Institute of Technology, Hanoi, Vietnam.

Hanoi University of Science, Hanoi, Vietnam.

Graduate student, Hanoi Institute of Mathematics, Hanoi, Vietnam.

Posts and Telecommunications Institute of Technology, Hanoi, Vietnam.

Hanoi University of Science, Hanoi, Vietnam.

Graduate student, Hanoi Institute of Mathematics, Hanoi, Vietnam.

ABSTRACT

This paper proposes a new hybrid variant of extragradient methods for finding a common solution of an equilibrium problem and a family of strict pseudo-contraction mappings. We present an algorithmic scheme that combine the idea of an extragradient method and a successive iteration method as a hybrid variant. Then, this algorithm is modified by projecting on a suitable convex set to get a better convergence property. The convergence of two these algorithms are investigated under certain assumptions.

This paper proposes a new hybrid variant of extragradient methods for finding a common solution of an equilibrium problem and a family of strict pseudo-contraction mappings. We present an algorithmic scheme that combine the idea of an extragradient method and a successive iteration method as a hybrid variant. Then, this algorithm is modified by projecting on a suitable convex set to get a better convergence property. The convergence of two these algorithms are investigated under certain assumptions.

Cite this paper

P. Anh, T. Quoc and D. Son, "Hybrid Extragradient-Type Methods for Finding a Common Solution of an Equilibrium Problem and a Family of Strict Pseudo-Contraction Mappings,"*Applied Mathematics*, Vol. 3 No. 10, 2012, pp. 1357-1367. doi: 10.4236/am.2012.330192.

P. Anh, T. Quoc and D. Son, "Hybrid Extragradient-Type Methods for Finding a Common Solution of an Equilibrium Problem and a Family of Strict Pseudo-Contraction Mappings,"

References

[1] E. Blum and W. Oettli, “From Optimization and Variational Inequality to Equilibrium Problems,” The Mathematics Student, Vol. 63, No. 1-4, 1994, pp. 127149.

[2] P. Daniele, F. Giannessi and A. Maugeri, “Equilibrium Problems and Variational Models,” Kluwer Academic Publisher, Dordrecht, 2003. doi:10.1007/978-1-4613-0239-1

[3] I. V. Konnov, “Combined Relaxation Methods for Variational Inequalities,” Springer-Verlag, Berlin, 2000.

[4] P. N. Anh and J. K. Kim, “The Interior Proximal Cutting Hyperplane Method for Multivalued Variational Inequalities,” Journal of Nonlinear and Convex Analysis, Vol. 11, No. 3, 2010, pp. 491-502.

[5] P. N. Anh, “A Logarithmic Quadratic Regularization Method for Solving Pseudo-Monotone Equilibrium Problems,” Acta Mathematica Vietnamica, Vol. 34, 2009, pp. 183-200.

[6] P. N. Anh, “An LQP Regularization Method for Equilibrium Problems on Polyhedral,” Vietnam Journal of Mathematics, Vol. 36, No. 2, 2008, pp. 209-228.

[7] G. Mastroeni, “Gap Function for Equilibrium Problems,” Journal of Global Optimization, Vol. 27, No. 4, 2004, pp. 411-426. doi:10.1023/A:1026050425030

[8] J. W. Peng, “Iterative Algorithms for Mixed Equilibrium Problems, Strict Pseudo Contractions and Monotone Mappings,” Journal of Optimization Theory and Applications, Vol. 144, No. 5, 2010, pp. 107-119. doi:10.1007/s10957-009-9585-5

[9] K. Goebel and W. A. Kirk, “Topics on Metric Fixed Point Theory,” Cambridge University Press, Cambridge, 1990. doi:10.1017/CBO9780511526152

[10] 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, 2006, pp. 191-201. doi:10.1007/s10957-005-7564-z

[11] 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, 2007, pp. 506-515. doi:10.1016/j.jmaa.2006.08.036

[12] 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, 2010, pp. 1293-1303.

[13] R. Chen, X. Shen and S. Cui, “Weak and Strong Convergence Theorems for Equilibrium Problems and Countable Strict Pseudocontractions Mappings in Hilbert Space,” Journal of Inequalities and Applications, Vol. 2010, 2010, Article ID: 474813. doi:10.1155/2010/474813

[14] P. N. Anh, “A Hybrid Extragradient Method Extended to Fixed Point Problems and Equilibrium Problems,” Optimization, 2011 (in press). doi:10.1080/02331934.2011.607497

[15] G. L. Acedo and H. K. Xu, “Iterative Methods for Strict Pseudo-Contractions in Hilbert Spaces,” Nonlinear Analysis, Vol. 67, No. 7, 2007, pp. 2258-2271. doi:10.1016/j.na.2006.08.036

[16] P. N. Anh and D. X. Son, “A New Iterative Scheme for Pseudomonotone Equilibrium Problems and a Finite Family of Pseudocontractions,” Journal of Applied Mathematics and Informatics, Vol. 29, No. 5-6, 2011, pp. 1179-1191.

[17] P. N. Anh, “Strong Convergence Theorems for Nonexpansive Mappings and Ky Fan Inequalities,” Journal of Optimization Theory and Applications, Vol. 154, No. 1, 2012, pp. 303-320. doi:10.1007/s10957-012-0005-x

[18] J. K. Kim, P. N. Anh and J. M. Nam, “Strong Convergence of an Extragradient Method for Equilibrium Problems and Fixed Point Problems,” Journal of the Korean Mathematical Society, Vol. 49, No. 1, 2012, pp. 187-200. doi:10.4134/JKMS.2012.49.1.187

[19] P. N. Anh and N. D. Hien, “The Extragradient-Armijo Method for Pseudomonotone Equilibrium Problems and Strict Pseudocontractions,” Fixed Point Theory and Applications, Vol. 2012, No. 1, 2012, pp. 1-16. doi:10.1186/1687-1812-2012-82

[20] L. C. Ceng, A. Petrusel, C. Lee and M. M. Wong, “Two Extragradient Approximation Methods for Variational Inequalities and Fixed Point Problems of Strict PseudoContractions,” Taiwanese Journal of Mathematics, Vol. 13, No. 2, 2009, pp. 607-632.

[21] S. Wang, Y. J. Cho and X. Qin, “A New Iterative Method for Solving Equilibrium Problems and Fixed Point Problems for Infinite Family of Nonexpansive Mappings,” Fixed Point Theory and Applications, Vol. 2010, 2010, p. 165098. doi:10.1155/2010/165098

[22] 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, 2010, pp. 2620-2630. doi:10.1016/j.cam.2009.11.008

[23] 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, No. 1, 2009, p. 632819.

[24] C. Marino-Yanes and H. K. Xu, “Strong Convergence of the CQ Method for Fixed Point Processes,” Nonlinear Analysis, Vol. 64, No. 11, 2006, pp. 2400-2411. doi:10.1016/j.na.2005.08.018

[25] S. Takahashi and M. Toyoda, “Weakly Convergence Theorems for Nonexpansive Mappings and Monotone Mappings,” Journal of Optimization Theory and Applications, Vol. 118, No. 2, 2003, pp. 417-428. doi:10.1023/A:1025407607560

[1] E. Blum and W. Oettli, “From Optimization and Variational Inequality to Equilibrium Problems,” The Mathematics Student, Vol. 63, No. 1-4, 1994, pp. 127149.

[2] P. Daniele, F. Giannessi and A. Maugeri, “Equilibrium Problems and Variational Models,” Kluwer Academic Publisher, Dordrecht, 2003. doi:10.1007/978-1-4613-0239-1

[3] I. V. Konnov, “Combined Relaxation Methods for Variational Inequalities,” Springer-Verlag, Berlin, 2000.

[4] P. N. Anh and J. K. Kim, “The Interior Proximal Cutting Hyperplane Method for Multivalued Variational Inequalities,” Journal of Nonlinear and Convex Analysis, Vol. 11, No. 3, 2010, pp. 491-502.

[5] P. N. Anh, “A Logarithmic Quadratic Regularization Method for Solving Pseudo-Monotone Equilibrium Problems,” Acta Mathematica Vietnamica, Vol. 34, 2009, pp. 183-200.

[6] P. N. Anh, “An LQP Regularization Method for Equilibrium Problems on Polyhedral,” Vietnam Journal of Mathematics, Vol. 36, No. 2, 2008, pp. 209-228.

[7] G. Mastroeni, “Gap Function for Equilibrium Problems,” Journal of Global Optimization, Vol. 27, No. 4, 2004, pp. 411-426. doi:10.1023/A:1026050425030

[8] J. W. Peng, “Iterative Algorithms for Mixed Equilibrium Problems, Strict Pseudo Contractions and Monotone Mappings,” Journal of Optimization Theory and Applications, Vol. 144, No. 5, 2010, pp. 107-119. doi:10.1007/s10957-009-9585-5

[9] K. Goebel and W. A. Kirk, “Topics on Metric Fixed Point Theory,” Cambridge University Press, Cambridge, 1990. doi:10.1017/CBO9780511526152

[10] 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, 2006, pp. 191-201. doi:10.1007/s10957-005-7564-z

[11] 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, 2007, pp. 506-515. doi:10.1016/j.jmaa.2006.08.036

[12] 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, 2010, pp. 1293-1303.

[13] R. Chen, X. Shen and S. Cui, “Weak and Strong Convergence Theorems for Equilibrium Problems and Countable Strict Pseudocontractions Mappings in Hilbert Space,” Journal of Inequalities and Applications, Vol. 2010, 2010, Article ID: 474813. doi:10.1155/2010/474813

[14] P. N. Anh, “A Hybrid Extragradient Method Extended to Fixed Point Problems and Equilibrium Problems,” Optimization, 2011 (in press). doi:10.1080/02331934.2011.607497

[15] G. L. Acedo and H. K. Xu, “Iterative Methods for Strict Pseudo-Contractions in Hilbert Spaces,” Nonlinear Analysis, Vol. 67, No. 7, 2007, pp. 2258-2271. doi:10.1016/j.na.2006.08.036

[16] P. N. Anh and D. X. Son, “A New Iterative Scheme for Pseudomonotone Equilibrium Problems and a Finite Family of Pseudocontractions,” Journal of Applied Mathematics and Informatics, Vol. 29, No. 5-6, 2011, pp. 1179-1191.

[17] P. N. Anh, “Strong Convergence Theorems for Nonexpansive Mappings and Ky Fan Inequalities,” Journal of Optimization Theory and Applications, Vol. 154, No. 1, 2012, pp. 303-320. doi:10.1007/s10957-012-0005-x

[18] J. K. Kim, P. N. Anh and J. M. Nam, “Strong Convergence of an Extragradient Method for Equilibrium Problems and Fixed Point Problems,” Journal of the Korean Mathematical Society, Vol. 49, No. 1, 2012, pp. 187-200. doi:10.4134/JKMS.2012.49.1.187

[19] P. N. Anh and N. D. Hien, “The Extragradient-Armijo Method for Pseudomonotone Equilibrium Problems and Strict Pseudocontractions,” Fixed Point Theory and Applications, Vol. 2012, No. 1, 2012, pp. 1-16. doi:10.1186/1687-1812-2012-82

[20] L. C. Ceng, A. Petrusel, C. Lee and M. M. Wong, “Two Extragradient Approximation Methods for Variational Inequalities and Fixed Point Problems of Strict PseudoContractions,” Taiwanese Journal of Mathematics, Vol. 13, No. 2, 2009, pp. 607-632.

[21] S. Wang, Y. J. Cho and X. Qin, “A New Iterative Method for Solving Equilibrium Problems and Fixed Point Problems for Infinite Family of Nonexpansive Mappings,” Fixed Point Theory and Applications, Vol. 2010, 2010, p. 165098. doi:10.1155/2010/165098

[22] 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, 2010, pp. 2620-2630. doi:10.1016/j.cam.2009.11.008

[23] 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, No. 1, 2009, p. 632819.

[24] C. Marino-Yanes and H. K. Xu, “Strong Convergence of the CQ Method for Fixed Point Processes,” Nonlinear Analysis, Vol. 64, No. 11, 2006, pp. 2400-2411. doi:10.1016/j.na.2005.08.018

[25] S. Takahashi and M. Toyoda, “Weakly Convergence Theorems for Nonexpansive Mappings and Monotone Mappings,” Journal of Optimization Theory and Applications, Vol. 118, No. 2, 2003, pp. 417-428. doi:10.1023/A:1025407607560