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A new system of variational inclusions with (H, η)-monotone operators

Published online by Cambridge University Press:  17 April 2009

Jianwen Peng  and
Jianrong Huang
Jianwen Peng
Affiliation:
College of Mathematics and Computer Science, Chongqing Normal University, Chongqing 400047, Peoples Republic of China and School of Management, Fudan University, Shanghai 200433, Peoples Republic of China e-mail: [email protected]
Jianrong Huang
Affiliation:
College of Mathematics and Computer Science, Chongqing Normal University, Chongqing 400047, Peoples Republic of China
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In this paper, We introduce and study a new system of variational inclusions involving(H, η)-monotone operators in Hilbert spaces. By using the resolvent operator method associated with (H, η)-monotone operators, we prove the existence and uniqueness of solutions and the convergence of some new three-step iterative algorithms for this system of variational inclusions and its special cases. The results in this paper extends and improves some results in the literature.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 74 , Issue 2 , October 2006 , pp. 301 - 319
Copyright
Copyright © Australian Mathematical Society 2006

References

[1]Agawal, R.P., Huang, N. J. and Tan, M.Y., ‘Sensitivity analysis for a new system of generalized nonlinear mixed quasi-variational inclusions’, Appl. Math. Lett. 17 (2004), 345–352.CrossRefGoogle Scholar
[2]Allevi, E., Gnudi, A. and Konnov, I.V., ‘Generalized vector variational inequalies over product sets’, Nonlinear Anal. 47 (2001), 573–582.CrossRefGoogle Scholar
[3]Ansari, Q.H., Schaible, S. and Yao, J.C., ‘Systems of vector equilibrium problems and its applications’, J. Optim. Theory Appl. 107 (2000), 547–557.CrossRefGoogle Scholar
[4]Ansari, Q.H. and Yao, J.C., ‘A fixed point theorem and its applications to a system of variational inequalities’, Bull. Austral. Math. Soc. 59 (1999), 433–442.CrossRefGoogle Scholar
[5]Bianchi, M., Pseudo P-monotone operators and variational inequalities, Istituto di econometria e Matematica per le decisioni economiche, (Report 6) (Universita Cattolica del Sacro Cuore, Milan, Italy, 1993).Google Scholar
[6]Chen, G.Y. and Craven, B.D., ‘Approximate dual and approximate vetor variational inequality for multiobjective optimization’, J. Austral. Math. Soc. Ser. A 47 (1989), 418–423.CrossRefGoogle Scholar
[7]Cho, Y. J., Fang, Y.P. and Huang, N.J., ‘Algorithms for systems of nonlinear variational variational inequalities’, J. Korean Math. Soc. 41 (2004), 489–499.CrossRefGoogle Scholar
[8]Cohen, G. and Chaplais, F., ‘Nested monotony for variational inequalities over a product of spaces and convergence of iterative algorithms’, J. Optim. Theory Appl. 59 (1988), 360–390.CrossRefGoogle Scholar
[9]Fang, Y.P. and Huang, N.J., ‘H-monotone operators and system of variational inclusions’, Comm. Appl. Nonlinear Anal. 11 (2004), 93–101.Google Scholar
[10]Fang, Y.P. and Huang, N.J., ‘H-Monotone operator resolvent operator technique for quasi-variational inclusions’, Appl. Math. Comput. 145 (2003), 795–803.Google Scholar
[11]Fang, Y.P., Huang, N.J. and Thompson, H.B., ‘A new system of variational inclusions with (H, η)-monotone operators in Hilbert spaces’, Comput. Math. Appl. 49 (2005), 365–374.CrossRefGoogle Scholar
[12]Huang, N.J. and Fang, Y.P., ‘A new class of general variational inclusions invoving maximal ηmonotone mappings’, Publ. Math. Debrecen 62 (2003), 83–98.CrossRefGoogle Scholar
[13]Kassay, G. and Kolumbán, J., ‘System of multi-valued variational variational inequalities’, Publ. Math. Debrecen 54 (1999), 267–279.CrossRefGoogle Scholar
[14]Kassay, G., Kolumbán, J. and Páles, Z., ‘Factorization of Minty and Stampacchia variational inequality system’, European J Oper Res. 143 (2002), 377–389.CrossRefGoogle Scholar
[15]Kazmi, K.R. and Bhat, M.I., ‘Iterative algorithm for a system of nonlinear variational-like inclusions’, Comput. Math. Appl. 48 (2004), 1929–1935.CrossRefGoogle Scholar
[16]Kim, J.K. and Kim, D.S., ‘A new system of generalized nonlinear mixed variational inequalities in Hilbert spaces’, J Convex Anal. 11 (2004), 235–243.Google Scholar
[17]Pang, J.S., ‘Asymmetric variational inequality problems over product sets: Applications and iterative methods’, Math. Programming 31 (1985), 206–219.CrossRefGoogle Scholar
[18]Peng, J.W., ‘System of generalized set-valued quasi-variational-like inequalities’, Bull. Austral. Math. Soc. 68 (2003), 501–515.CrossRefGoogle Scholar
[19]Peng, J.W. and Yang, X.M., ‘On existence of a solution for the system of generalized vector quasi-equilibrium problems with upper semicontinuous set-valued maps’, Internat. J Math. Math. Sci. 15 (2005), 2409–2420.CrossRefGoogle Scholar
[20]Verma, R.U., ‘On a new system of nonlinear variational inequalities and associated iterative algorithms’, Math. Sci. Res. Hot-Line 3 (1999), 65–68.Google Scholar
[21]Verma, R.U., ‘Projection methods, algorithms and a new system of nonlinear variational inequalities’, Comput. Math. Appl. 41 (2001), 1025–1031.CrossRefGoogle Scholar
[22]Verma, R.U., ‘Iterative algorithms and a new system of nonlinear quasivariational inequalities’, Adv. Nonlinear Var Inequal. 4 (2001), 117–124.Google Scholar
[23]Verma, R.U., ‘Generalized system for relaxed cocoercive variational inequalities and problems and projection methods’, J Optim. Theory and Appl. 121 (2004), 203–210.CrossRefGoogle Scholar
[24]Verma, R.U., ‘General convergence analysis for two-step projection methods and application to variational problems’, Appl. Math. Lett. 18 (2005), 1286–1292.CrossRefGoogle Scholar
[25]Verma, R.U., ‘A-monotononicity and applications to nonlinear variational inclusion problems’, J Appl. Math. Stoch. Anal. 17 (2004), 193–195.CrossRefGoogle Scholar