Existence and Algorithm of Solutions for a System of Generalized Mixed Implicit Equilibrium Problems in Banach Spaces

DING Xie-ping

doi:10.3879/j.issn.1000-0887.2010.09.001

Volume 31 Issue 9

Turn off MathJax

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2010 > 31(9): 1001-1015

DING Xie-ping. Existence and Algorithm of Solutions for a System of Generalized Mixed Implicit Equilibrium Problems in Banach Spaces[J]. Applied Mathematics and Mechanics, 2010, 31(9): 1001-1015. doi: 10.3879/j.issn.1000-0887.2010.09.001

Citation:

PDF( 468 KB)

Existence and Algorithm of Solutions for a System of Generalized Mixed Implicit Equilibrium Problems in Banach Spaces

doi: 10.3879/j.issn.1000-0887.2010.09.001

DING Xie-ping

College of Mathematics and Software Science, Sichuan Normal University, Chengdu 610068, P. R. China

Received Date: 1900-01-01
Rev Recd Date: 2010-08-17
Publish Date: 2010-09-15

Abstract

Abstract

A new system of generalized mixed implicity equilibrium problems was introduced and studied in Banach spaces.First,the notion of Yosida proximal mapping for generalized mixed implicity equilibrium problems was introduced.By using the notion,a system of generalized equation problems was considered and its equivalence with the system of generalized mixed implicity equilibrium problems was also proved. Next,by applying the system of generalized equation problems,an iterative algorithm to compute the approximate solutions of the system of generalized mixed implicity equilibrium problems was suggested and analyzed.The strong convergence of the iterative sequences generated by the algorithm was proved under quite mild conditions.The results are new and unify and generalize some recent results in this field.

FullText(HTML)

References(39)

References

[1]	Blum E, Oettli W. From optimization and variational inequalities to equilibrium problems[J]. Math Student, 1994,63(1): 123-145.
[2]	Moudafi A, Théra M. Proximal and dynamical approaches to equilibrium problems[C]Lecture Notes in Economics and Mathematical Systems.477. Berlin: Springer,1999: 187-201.
[3]	Moudafi A. Mixed equilibrium problems: sensitivity analysis and algorithmic aspects[J]. Comput Math Appl, 2002, 44(8/9): 1099-1108. doi: 10.1016/S0898-1221(02)00218-3
[4]	Giannessi F, Maugeri A, Pardalos M. Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models[M]. Dordrecht: Kluwer Academic, 2001.
[5]	Giannessi F, Maugeri A. Variational Inequalities and Network Equilibrium Problems[M]. New York: Plenum, 1995.
[6]	Noor M A. Multivalued general equilibrium problems[J].J Math Anal Appl, 2003, 283(1): 140-149. doi: 10.1016/S0022-247X(03)00251-8
[7]	Noor M A. Auxiliary principle technique for equilibrium problem[J].J Optim Theory Appl, 2004, 122(2): 371-386. doi: 10.1023/B:JOTA.0000042526.24671.b2
[8]	Noor M A. Generalized mixed quasi-equilibrium problems with trifunction[J].Appl Math Lett, 2005, 18(5): 695-700. doi: 10.1016/j.aml.2004.04.015
[9]	Ding X P. Iterative algorithm of solutions for generalized mixed implicit equilibrium-like problems[J]. Appl Math Comput, 2005, 162(2): 799-809. doi: 10.1016/j.amc.2003.12.127
[10]	丁协平, 林炎诚, 姚任之. 求解广义混合隐拟平衡问题的预测修正算法[J]. 应用数学和力学, 2006, 27(9): 1009-1016.
[11]	Xia F Q, Ding X P. Predictor-corrector algorithms for solving generalized mixed implicit quasi-equilibrium problems[J]. Appl Comput Math, 2007, 188(1): 173-179. doi: 10.1016/j.amc.2006.09.095
[12]	Tada A, Takahashi W. Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem[J]. J Optim Theory Appl, 2007, 133(3): 359-370. doi: 10.1007/s10957-007-9187-z
[13]	Ceng L C, Yao J C. A hybrid iterative scheme for mixed equilibrium problems and fixed point problems[J]. J Comput Appl Math, 2008, 214(1):186-201. doi: 10.1016/j.cam.2007.02.022
[14]	Bigi G, Castellani M, Kassay G. A dual view of equilibrium problems[J].J Math Anal Appl, 2008,342(1): 17-26. doi: 10.1016/j.jmaa.2007.11.034
[15]	Colao V, Marino G, Xu H-K. An iterative method for finding common solutions of equilibrium and fixed point problems[J]. J Math Anal Appl, 2008, 344(2): 340-352. doi: 10.1016/j.jmaa.2008.02.041
[16]	Plubtieng S, Punpaeng R. A new iterative method for equilibrium problems and fixed point problems of nonexpansive mappings and monotone mappings[J]. Appl Math Comput, 2008, 197(2): 548-558. doi: 10.1016/j.amc.2007.07.075
[17]	Chang S S, Lee H W J, Chan C K. A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization[J]. Nonlinear Anal, 2009, 70(9): 3307-3319. doi: 10.1016/j.na.2008.04.035
[18]	Ceng L C, Al-Homidan S, Ansari Q H, Yao J C. An iterative scheme for equilibrium problems and fixed point problems of strict pseudo-contraction mappings[J]. J Comput Appl Math, 2009, 223(2): 967-974. doi: 10.1016/j.cam.2008.03.032
[19]	Qin X L, Cho Y J, Kang S M. Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces[J]. J Comput Appl Math, 2009, 225(1): 20-30. doi: 10.1016/j.cam.2008.06.011
[20]	Jung J S. Strong convergence of composite iterative methods for equilibrium problems and fixed point problems[J]. Appl Math Comput, 2009, 213(2): 489-505.
[21]	Colao V, Acedo G L, Marino G. An implicit method for finding common solutions of variational inequalities and system of equilibrium problems and fixed points of infinite family of nonexpansive mappings[J]. Nonlinear Anal, 2009, 71(7/8): 2708-2715. doi: 10.1016/j.na.2009.01.115
[22]	Huang N A, Lan H Y, Cho Y J. Sensitivity analysis for nonlinear generalized mixed implicit equilibrium problems with non-monotone set-valued mappings[J]. J Comput Appl Math, 2006, 196(2): 608-618. doi: 10.1016/j.cam.2005.10.014
[23]	Kazmi K R, Khan F A. Existence and iterative approximation of solutions of generalized mixed equilibrium problems[J]. Comput Math Appl, 2008, 56(5):1314-1321. doi: 10.1016/j.camwa.2007.11.051
[24]	Ding X P, Ho J L. New iterative algorithm for solving a system of generalized mixed implicit equilibrium problems in Banach spaces[J]. Taiwanese J Math, this paper is available online at
[25]	张石生. 变分不等式和相补问题理论及应用[J]. 上海:上海科学技术文献出版社, 1991.
[26]	Noor M A. Auxiliary principle for generalized mixed variational-like inequalities[J]. J Math Anal Appl,1997, 215(1): 78-85.
[27]	Ding X P. On generalized mixed variational-like inequalities[J]. J Sichuan Normal Univ, 1999,22(5): 494-503.
[28]	Ding X P, Tarafdar E. Existence and uniqueness of solutions for a general nonlinear variational inequality[J]. Appl Math Lett, 1995, 8(1): 31-36.
[29]	Ding X P. Generalized quasi-variational-like inclusions with nonconvex functions[J]. Appl Math Comput, 2001, 122(3): 267-282. doi: 10.1016/S0096-3003(00)00027-8
[30]	Ansari Q H, Yao J C. Iterative schemes for solving mixed variational-like inequalities[J]. J Optim Theory Appl, 2001, 108(3): 527-541. doi: 10.1023/A:1017531323904
[31]	Huang N J, Deng C X. Auxiliary principle and iterative algorithms for generalized set-valued strongly nonlinear mixed variational-like inequalities[J]. J Math Anal Appl, 2001, 256(2): 345-359. doi: 10.1006/jmaa.2000.6988
[32]	Ding X P. Existence and algorithm of solutions for nonlinear mixed quasi-variational inequalities in Banach spaces[J]. J Comput Appl Math, 2003, 157(2): 419-434. doi: 10.1016/S0377-0427(03)00421-7
[33]	Ding X P. Existence and algorithm of solutions for mixed variational-like inequalities in Banach spaces[J]. J Optim Theory Appl, 2005, 127(2): 285-302. doi: 10.1007/s10957-005-6540-y
[34]	Ding X P, Yao J C. Existence and algorithm of solutions for mixed quasi-variational-like inclusions in Banach spaces[J]. Comput Math Appl, 2005, 49(5/6): 857-869. doi: 10.1016/j.camwa.2004.05.013
[35]	Ding X P, Yao J C, Zeng L C. Existence and algorithm of solutions for generalized strongly nonlinear mixed variational-like inequalities in Banach spaces[J]. Comput Math Appl, 2008, 55(4): 669-679. doi: 10.1016/j.camwa.2007.06.004
[36]	Zeng L C, Guu S M, Yao J C. Three-step iterative algorithms for solving the system of generalized mixed quasi-variational-like inclusions[J]. J Comput Math Appl, 2007, 53(10): 1572-1581. doi: 10.1016/j.camwa.2006.05.024
[37]	Zeng L C, Schaible S, Yao J C. Iterative algorithm for generalized set-valued strongly nonlinear mixed variational-like inequalities[J]. J Optim Theory Appl, 2005,124(3): 725-738. doi: 10.1007/s10957-004-1182-z
[38]	Zeng L C, Guu S M, Yao J C. Iterative algorithm for completely generalized set-valued strongly nonlinear mixed variational-like inequalities[J]. J Comput Math Appl, 2005,50(5/6): 935-945. doi: 10.1016/j.camwa.2004.12.017
[39]	Nadler S B. Multivalued contraction mapping[J]. Pacific J Math, 1969, 30: 475-488.