广义均衡问题的算法及强收敛性——基于Wiener-Hopf方程技巧

Abstract

Abstract:

In this paper, we use generalized Wiener-Hopf equation techniques to construct an iterative scheme for generalized mixed equilibrium problems, an infinite family of non-expansive mappings and variational inequality problems. Furthermore, two strong convergence theorems are also established based on different monotonicity in Hilbert spaces. Finally, we utilize our results to study several optimization problems. The methods and results presented in this paper are different from the ones announced by many others and they extend the results of Shi and Noor.

Key words: Generalized mixed equilibrium problems, Operator equation techniques, Algorithms, Strong convergence, Optimization

CLC Number:

Wang Yuehu, Zhang Congjun. Strong Convergence Based on Generalized Wiener-Hopf Equation Techniques for Solving Generalized Equilibrium Problems[J].Acta mathematica scientia,Series A, 2015, 35(4): 695-709.

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References

[1] Saewan S, Kumam P. A modified hybrid projection method for solving generalized mixed equilibrium problems and fixed point problems in Banach spaces. Comput Math Appl, 2011, 62: 1723-1735
[2] Zhang S. Generalized mixed equilibrium problem in Banach spaces. Appl Math Mech, 2009, 30: 1105-1112
[3] Yao Y, Cho Y J, Liou Y. Algorithms of common solutions for variational inclusions, mixed equilibrium problems and fixed point problems. Eur J Oper Res, 2011, 212: 242-250
[4] Kim J K, Cho S Y, Qin X. Some results on generalized equilibrium problems involving strictly pseudocontractive mappings. Acta Math Sci, 2011, 31B(5): 985-996
[5] Stampacchia G. Formes bilineaires coercitives sur les ensembles convexes. Académie des Sciences de Paris, 1964, 258: 4413-4416
[6] Blum E, Oettli W. From optimization and variational inequalities to equilibrium problems. Math Student, 1994, 63: 123-145
[7] Noor M A. Some predictor-corrector algorithms for multivalued variational inequalities. Journal of Optimization Theory and Applications, 2001, 108(3): 659-671
[8] Noor M A. On a class of nonconvex equilibrium problems. Appl Math Comput, 2004, 157: 653-666
[9] Noor M A, Themistocles M. On general hemiequilibrium problems. J Math Anal Appl, 2006, 324: 1417-1428
[10] Noor M A. Multivalued general equilibrium problems. J Math Anal Appl, 2003, 283: 140-149
[11] Kim J K, Anh P N, Hyun H G. A proximal point-type algorithm for pseudomonotone equilibrium problems. Bull Korean Math Soc, 2012, 49: 749-759
[12] Burachik R, Kassay G. On a generalized proximal point method for solving equilibrium problems in Banach spaces. Nonlinear Anal, 2012, 75: 6456-6464
[13] Tada A, Takahashi W. Weak and strong convergence theorems for nonexpansive mappings and equilibrium problems. J Optim Theory Appl, 2007, 133: 359-370
[14] Changa S, Joseph Leeb H W, Chanb C K. A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization. Nonlinear Analysis: Theory, Methods & Applications, 2009, 70: 3307-3319
[15] Noor M A. Some iterative schemes for solving extended general quasi variational inequalitie. Appl Math Inf Sci, 2013, 7(3): 917-925
[16] Takahashi S, Takahashi W. Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. J Math Anal Appl, 2007, 331: 506-515
[17] Kumam P. A hybrid approximation method for equilibrium and fixed point problems for a monotone mapping and a nonexpansive. Nonlinear Anal, Hybrid Syst, 2008, 18: 1245-1255
[18] Plubtieng S, Punpaeng R. A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces. J Math Anal Appl, 2007, 336(1): 455-469
[19] Shehu Y. Iterative approximation for common solutions of equilibrium problems, variational inequality and fixed point problems. Mathematical and Computer Modelling, 2013, 57: 1489-1503
[20] Shi P. Equivalence of variational inequalities with Wiener-Hopf equations. Proc Amer Math Soc, 1991, 111: 339-346
[21] Verma R U. Generalized variational inequalities involving multivalued relaxed monotone operators. Appl Math Lett, 1997, 10: 107-109
[22] Robinson S M. Normal maps induced by linear transformations. Math Oper Res, 1992, 17(3): 691-714
[23] Noor M A, Huang Z. Wiener-Hopf equation technique for variational inequalities and nonexpansive mappings. Applied Mathematics and Computation, 2007, 191: 504-510
[24] Al-Shemas E. General nonconvex Wiener-Hopf equations and general nonconvex variational inequalities. J Math Sci Adv Appl, 2013, 19(1): 1-11
[25] Shimoji K, Takahashi W. Strong convergence to common fixed points of infinite nonexpansive mappings and applications. Taiwanese J Math, 2001, 5: 387-404
[26] Marino G, Xu H K. A general iterative method for non-expansive mapping in Hilbert spaces. J Math Anal Appl, 2006, 318: 43-52
[27] Yao Y, Kang S M, Liou Y C. Algorithms for approximating minimization problems in Hilbert spaces. J Comput Anal Appl, 2011, 235: 3515-3526

Strong Convergence Based on Generalized Wiener-Hopf Equation Techniques for Solving Generalized Equilibrium Problems

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