# American Institute of Mathematical Sciences

March  2022, 12(1): 63-78. doi: 10.3934/naco.2021051

## Weak convergence theorems for symmetric generalized hybrid mappings and equilibrium problems

 1 Department of Applied Mathematics, Pukyong National University, Busan, 48513, Korea 2 Department of Scientific Fundamentals, Vietnam Trade Union University, Hanoi, Vietnam 3 Department of Mathematics, Faculty of Information Technology, Le Quy Don Technical University, Hanoi, Vietnam

*Corresponding author

Received  March 2020 Revised  October 2021 Published  March 2022 Early access  November 2021

Fund Project: This work was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korean Government (NRF-2019R1A2C1008672)

In this paper, we introduce three new iterative methods for finding a common point of the set of fixed points of a symmetric generalized hybrid mapping and the set of solutions of an equilibrium problem in a real Hilbert space. Each method can be considered as an combination of Ishikawa's process with the proximal point algorithm, the extragradient algorithm with or without linesearch. Under certain conditions on parameters, the iteration sequences generated by the proposed methods are proved to be weakly convergent to a solution of the problem. These results extend the previous results given in the literature. A numerical example is also provided to illustrate the proposed algorithms.

Citation: Do Sang Kim, Nguyen Ngoc Hai, Bui Van Dinh. Weak convergence theorems for symmetric generalized hybrid mappings and equilibrium problems. Numerical Algebra, Control and Optimization, 2022, 12 (1) : 63-78. doi: 10.3934/naco.2021051
##### References:
 [1] P. N. Anh, A hybrid extragradient method extended to fixed point problems and equilibrium problems, Optim., 62 (2013), 271-283.  doi: 10.1080/02331934.2011.607497. [2] G. Bigi, M. Castellani, M. Pappalardo and M. Passacantando, Existence and solution methods for equilibria, Eur. J. Oper. Res., 227 (2013), 1-11.  doi: 10.1016/j.ejor.2012.11.037. [3] E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student, 63 (1994), 127-149. [4] P. L. Combettes and A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal., 6 (2005), 117-136. [5] J. Contreras, M. Klusch and J. B. Krawczyk, Numerical solution to Nash-Cournot equilibria in coupled constraint electricity markets, IEEE Trans. Power Syst., 19 (2004), 195-206. [6] B. V. Dinh and D. S. Kim, Extragradient algorithms for equilibrium problems and symmetric generalized hybrid mappings, Optim. Lett., 11 (2016), 537-553.  doi: 10.1007/s11590-016-1025-5. [7] B. V. Dinh and L. D. Muu, A projection algorithm for solving pseudomonotone equilibrium problems and it's application to a class of bilevel equilibria, Optim., 64 (2015), 559-575.  doi: 10.1080/02331934.2013.773329. [8] B. V. Dinh, P. G. Hung and L. D. Muu, Bilevel optimization as a regularization approach to pseudomonotone equilibrium problems, Numer. Funct. Anal. Optim., 35 (2014), 539-563.  doi: 10.1080/01630563.2013.813857. [9] B. V. Dinh and L. D. Muu, On penalty and gap function methods for bilevel equilibrium problems, J. Appl. Math., (2011) DOI: 10.1155/2011/646452. doi: 10.1155/2011/646452. [10] F. Facchinei and J. S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer, New York, 2003. [11] A. Genel and J. Lindenstrass, An example concerning fixed points, Isarel J. Math., 22 (1975), 81-86.  doi: 10.1007/BF02757276. [12] D. V. Hieu, L. D. Muu and P. K. Anh, Parallel hybrid extragradient methods for pseudomonotone equilibrium problems and nonexpansive mappings, Numer. Algor., 73 (2016), 197-217.  doi: 10.1007/s11075-015-0092-5. [13] M. Hojo, T. Suzuki and W. Takahashi, Fixed point theorems and convergence theorems for generalized hybrid non-self mappings in Hilbert spaces, J. Nonlinear Convex Anal., 14 (2013), 363-376. [14] H. Iiduka, Fixed point optimization algorithm and its application to power control in CDMA data networks, Math. Program., Ser. A, 133 (2012), 227-242.  doi: 10.1007/s10107-010-0427-x. [15] S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc., 40 (1974), 147-150.  doi: 10.2307/2039245. [16] S. Itoh and W. Takahashi, The common fixed point theory of single-valued mappings and multi-valued mappings, Pacific J. Math., 79 (1978), 493-508. [17] T. Kawasaki and W. Takahashi, Existence and mean approximation of fixed points of generalized hybrid mappings in Hilbert spaces, J. Nonlinear Convex Anal., 14 (2013), 71-87. [18] F. Kohsaka and W. Takahashi, Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces, Arch. Math., 91 (2008), 166-177.  doi: 10.1007/s00013-008-2545-8. [19] I. V. Konnov, Combined Relaxation Methods for Variational Inequalities, Lecture Notes in Economics and Mathematical System, 495 (2001), Springer, Berlin. doi: 10.1007/978-3-642-56886-2. [20] G. M. Korpelevich, The extragradient method for finding saddle points and other problems, Matekon, 12 (1976), 747-756. [21] W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc., 4 (1953), 506-510.  doi: 10.2307/2032162. [22] F. Moradlou and Al izadeh, Strong convergence theorem by a new iterative method for equilibrium problems and symmetric generalized hybrid mappings, Mediterr. J. Math., 13 (2016), 379-390.  doi: 10.1007/s00009-014-0462-6. [23] A. Moudafi, Proximal point algorithm extended to equilibrium problems, J. Nat. Geom., 15 (1999), 91-100. [24] L. D. Muu and W. Oettli, Convergence of an adaptive penalty scheme for finding constrained equilibria, Nonlinear Anal.: TMA., 18 (1992), 1159-1166.  doi: 10.1016/0362-546X(92)90159-C. [25] L. D. Muu and T. D. Quoc, Regularization algorithms for solving monotone Ky Fan inequalities with application to a Nash-Cournot equilibrium model, J. Optim. Theory Appl., 142 (2009), 185-204.  doi: 10.1007/s10957-009-9529-0. [26] R. T. Rockafellar, Convex Analysis, Princeton University Press, 1970. [27] J. Schu, Weak and strong convergence to fixed points of asymptotically noexpansive mappings, Bulletin of the Australian Math. Soc., 43 (1991), 153-159.  doi: 10.1017/S0004972700028884. [28] A. Tada and W. Takahashi, Weak and strong convergence theorem for nonexpansive mapping and equilibrium problem, J. Optim. Theory Appl., 133 (2007), 359-370.  doi: 10.1007/s10957-007-9187-z. [29] W. Takahashi and M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 118 (2003), 417-428.  doi: 10.1023/A:1025407607560. [30] W. Takahashi, N. C. Wong and J. C. Yao, Fixed point theorems for new generalized hybrid mappings in Hilbert spaces and applications, Taiwanese J. Math., 17 (2013), 1597-1611.  doi: 10.11650/tjm.17.2013.2921. [31] W. Takahashi, Fixed point theorems for new nonlinear mappings in a Hilbert space, J. Nonlinear Convex Anal., 11 (2010), 79-88. [32] D. Q. Tran, L. M. Dung and V. H. Nguyen, Extragradient algorithms extended to equilibrium problems, Optim., 57 (2008), 749-776.  doi: 10.1080/02331930601122876. [33] N. N. Tam, J. C. Yao and N. D. Yen, Solution methods for pseudomonotone variational inequalities, J. Optim. Theory Appl., 38 (2008), 253-273.  doi: 10.1007/s10957-008-9376-4. [34] P. T. Vuong, J. J. Strodiot and V. H. Nguyen, Extragradient methods and linesearch algorithms for solving Ky Fan inequalities and fixed point problems, J. Optim. Theory Appl., 155 (2013), 605-627.  doi: 10.1007/s10957-012-0085-7. [35] H. K. Xu, A variable Krasnosel'ski$\mathop {\rm{i}}\limits^ \vee$-Mann algorithm and the multiple-set split feasibility problem, Inverse Problems, 22 (2006), 2021-2034.  doi: 10.1088/0266-5611/22/6/007. [36] C. M. Yanes and H. K. Xu, Strong convergence of the $C Q$ method for fixed point iteration processes, Nonlinear Anal. TMA., 64 (2006), 2400-2411.  doi: 10.1016/j.na.2005.08.018. [37] E. Zeidler, Nonlinear Functional Analysis and Its Applications I, Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4612-4838-5.

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##### References:
 [1] P. N. Anh, A hybrid extragradient method extended to fixed point problems and equilibrium problems, Optim., 62 (2013), 271-283.  doi: 10.1080/02331934.2011.607497. [2] G. Bigi, M. Castellani, M. Pappalardo and M. Passacantando, Existence and solution methods for equilibria, Eur. J. Oper. Res., 227 (2013), 1-11.  doi: 10.1016/j.ejor.2012.11.037. [3] E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student, 63 (1994), 127-149. [4] P. L. Combettes and A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal., 6 (2005), 117-136. [5] J. Contreras, M. Klusch and J. B. Krawczyk, Numerical solution to Nash-Cournot equilibria in coupled constraint electricity markets, IEEE Trans. Power Syst., 19 (2004), 195-206. [6] B. V. Dinh and D. S. Kim, Extragradient algorithms for equilibrium problems and symmetric generalized hybrid mappings, Optim. Lett., 11 (2016), 537-553.  doi: 10.1007/s11590-016-1025-5. [7] B. V. Dinh and L. D. Muu, A projection algorithm for solving pseudomonotone equilibrium problems and it's application to a class of bilevel equilibria, Optim., 64 (2015), 559-575.  doi: 10.1080/02331934.2013.773329. [8] B. V. Dinh, P. G. Hung and L. D. Muu, Bilevel optimization as a regularization approach to pseudomonotone equilibrium problems, Numer. Funct. Anal. Optim., 35 (2014), 539-563.  doi: 10.1080/01630563.2013.813857. [9] B. V. Dinh and L. D. Muu, On penalty and gap function methods for bilevel equilibrium problems, J. Appl. Math., (2011) DOI: 10.1155/2011/646452. doi: 10.1155/2011/646452. [10] F. Facchinei and J. S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer, New York, 2003. [11] A. Genel and J. Lindenstrass, An example concerning fixed points, Isarel J. Math., 22 (1975), 81-86.  doi: 10.1007/BF02757276. [12] D. V. Hieu, L. D. Muu and P. K. Anh, Parallel hybrid extragradient methods for pseudomonotone equilibrium problems and nonexpansive mappings, Numer. Algor., 73 (2016), 197-217.  doi: 10.1007/s11075-015-0092-5. [13] M. Hojo, T. Suzuki and W. Takahashi, Fixed point theorems and convergence theorems for generalized hybrid non-self mappings in Hilbert spaces, J. Nonlinear Convex Anal., 14 (2013), 363-376. [14] H. Iiduka, Fixed point optimization algorithm and its application to power control in CDMA data networks, Math. Program., Ser. A, 133 (2012), 227-242.  doi: 10.1007/s10107-010-0427-x. [15] S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc., 40 (1974), 147-150.  doi: 10.2307/2039245. [16] S. Itoh and W. Takahashi, The common fixed point theory of single-valued mappings and multi-valued mappings, Pacific J. Math., 79 (1978), 493-508. [17] T. Kawasaki and W. Takahashi, Existence and mean approximation of fixed points of generalized hybrid mappings in Hilbert spaces, J. Nonlinear Convex Anal., 14 (2013), 71-87. [18] F. Kohsaka and W. Takahashi, Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces, Arch. Math., 91 (2008), 166-177.  doi: 10.1007/s00013-008-2545-8. [19] I. V. Konnov, Combined Relaxation Methods for Variational Inequalities, Lecture Notes in Economics and Mathematical System, 495 (2001), Springer, Berlin. doi: 10.1007/978-3-642-56886-2. [20] G. M. Korpelevich, The extragradient method for finding saddle points and other problems, Matekon, 12 (1976), 747-756. [21] W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc., 4 (1953), 506-510.  doi: 10.2307/2032162. [22] F. Moradlou and Al izadeh, Strong convergence theorem by a new iterative method for equilibrium problems and symmetric generalized hybrid mappings, Mediterr. J. Math., 13 (2016), 379-390.  doi: 10.1007/s00009-014-0462-6. [23] A. Moudafi, Proximal point algorithm extended to equilibrium problems, J. Nat. Geom., 15 (1999), 91-100. [24] L. D. Muu and W. Oettli, Convergence of an adaptive penalty scheme for finding constrained equilibria, Nonlinear Anal.: TMA., 18 (1992), 1159-1166.  doi: 10.1016/0362-546X(92)90159-C. [25] L. D. Muu and T. D. Quoc, Regularization algorithms for solving monotone Ky Fan inequalities with application to a Nash-Cournot equilibrium model, J. Optim. Theory Appl., 142 (2009), 185-204.  doi: 10.1007/s10957-009-9529-0. [26] R. T. Rockafellar, Convex Analysis, Princeton University Press, 1970. [27] J. Schu, Weak and strong convergence to fixed points of asymptotically noexpansive mappings, Bulletin of the Australian Math. Soc., 43 (1991), 153-159.  doi: 10.1017/S0004972700028884. [28] A. Tada and W. Takahashi, Weak and strong convergence theorem for nonexpansive mapping and equilibrium problem, J. Optim. Theory Appl., 133 (2007), 359-370.  doi: 10.1007/s10957-007-9187-z. [29] W. Takahashi and M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 118 (2003), 417-428.  doi: 10.1023/A:1025407607560. [30] W. Takahashi, N. C. Wong and J. C. Yao, Fixed point theorems for new generalized hybrid mappings in Hilbert spaces and applications, Taiwanese J. Math., 17 (2013), 1597-1611.  doi: 10.11650/tjm.17.2013.2921. [31] W. Takahashi, Fixed point theorems for new nonlinear mappings in a Hilbert space, J. Nonlinear Convex Anal., 11 (2010), 79-88. [32] D. Q. Tran, L. M. Dung and V. H. Nguyen, Extragradient algorithms extended to equilibrium problems, Optim., 57 (2008), 749-776.  doi: 10.1080/02331930601122876. [33] N. N. Tam, J. C. Yao and N. D. Yen, Solution methods for pseudomonotone variational inequalities, J. Optim. Theory Appl., 38 (2008), 253-273.  doi: 10.1007/s10957-008-9376-4. [34] P. T. Vuong, J. J. Strodiot and V. H. Nguyen, Extragradient methods and linesearch algorithms for solving Ky Fan inequalities and fixed point problems, J. Optim. Theory Appl., 155 (2013), 605-627.  doi: 10.1007/s10957-012-0085-7. [35] H. K. Xu, A variable Krasnosel'ski$\mathop {\rm{i}}\limits^ \vee$-Mann algorithm and the multiple-set split feasibility problem, Inverse Problems, 22 (2006), 2021-2034.  doi: 10.1088/0266-5611/22/6/007. [36] C. M. Yanes and H. K. Xu, Strong convergence of the $C Q$ method for fixed point iteration processes, Nonlinear Anal. TMA., 64 (2006), 2400-2411.  doi: 10.1016/j.na.2005.08.018. [37] E. Zeidler, Nonlinear Functional Analysis and Its Applications I, Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4612-4838-5.
Results computed with Algorithm 2
 N.P Size (n) Average Times Average Iterations 10 5 2.3766 152 10 10 4.2141 223 10 20 6.7813 457 10 30 10.5266 515 10 50 17.4891 567 10 100 29.2406 674
 N.P Size (n) Average Times Average Iterations 10 5 2.3766 152 10 10 4.2141 223 10 20 6.7813 457 10 30 10.5266 515 10 50 17.4891 567 10 100 29.2406 674
Results computed with Algorithm 3
 N.P Size (n) Average Times Average Iterations 10 5 2.4656 99 10 10 4.1422 132 10 20 6.6375 164 10 30 8.0672 170 10 50 11.8828 192 10 100 21.4953 210
 N.P Size (n) Average Times Average Iterations 10 5 2.4656 99 10 10 4.1422 132 10 20 6.6375 164 10 30 8.0672 170 10 50 11.8828 192 10 100 21.4953 210
Results computed with Algorithm 1 in [6]
 N.P Size (n) Average Times Average Iterations 10 5 23.2484 826 10 10 34.7438 1445 10 20 87.1016 2346 10 30 157.5781 2715 10 50 255.4578 3839
 N.P Size (n) Average Times Average Iterations 10 5 23.2484 826 10 10 34.7438 1445 10 20 87.1016 2346 10 30 157.5781 2715 10 50 255.4578 3839
Results computed with Algorithm 2 in [6]
 N.P Size (n) Average Times Average Iterations 10 5 38.5938 904 10 10 106.3172 2242 10 20 163.1266 3050 10 30 250.9313 3001 10 50 359.1094 3592
 N.P Size (n) Average Times Average Iterations 10 5 38.5938 904 10 10 106.3172 2242 10 20 163.1266 3050 10 30 250.9313 3001 10 50 359.1094 3592

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