Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-26T05:50:45.687Z Has data issue: false hasContentIssue false
  • English
  • Français

The generalised f-projection operator with an application

Published online by Cambridge University Press:  17 April 2009

Ke-Qing Wu  and
Nan-Jing Huang
Ke-Qing Wu
Affiliation:
Department of Mathematics, Sichuan University, Chengdu, Sichuan, 610064, Peoples Republic of China e-mail: [email protected]
Nan-Jing Huang
Affiliation:
Department of Mathematics, Sichuan University, Chengdu, Sichuan, 610064, Peoples Republic of China e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we introduce a new concept of generalised f-projection operator which extends the generalised projection operator πK : B* → K, where B is a reflexive Banach space with dual space B* and K is a nonempty, closed and convex subset of B. Some properties of the generalised f-projection operator are given. As an application, we study the existence of solution for a class of variational inequalities in Banach spaces.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 73 , Issue 2 , April 2006 , pp. 307 - 317
Copyright
Copyright © Australian Mathematical Society 2006

References

[1]Alber, Y., ‘Generalized projection operators in Banach spaces: properties and applications’, Funct. Differential Equations Israel 1 (1993), 1–21.Google Scholar
[2]Alber, Y., Metric and generalized projection operators in Banach spaces: properties and applications, Lecture Notes in Pure and Appl. Math. 178 (Dekker, New York, 1996).Google Scholar
[3]Alber, Y., Iusem, A. and Solodov, M., ‘Minimization of nonsmooth convex functionals in Banach spaces’, J. Convex Anal. 4 (1997), 235–254.Google Scholar
[4]Alber, Y., Burachik, R. and Iusem, A., ‘A proximal point method for nonsmooth convex optimization problems in Banach spaces’, Abstr. Appl. Anal. 2 (1997), 97–120.CrossRefGoogle Scholar
[5]Alber, Y. and Guerre-Delabriere, S., ‘On the projection methods for fixed point problem’, Analysis 21 (2001), 7–30.CrossRefGoogle Scholar
[6]Alber, Y. and Iusem, A., ‘Extension of subgradient techniques for nonsmooth optimization in Banach spaces’, Set-Valued Anal. 9 (2001), 315–335.CrossRefGoogle Scholar
[7]Alber, Y., ‘Proximal projection method for variational inequalities and Cesáro averaged approximations’, Comput. Math. Appl. 43 (2002), 1107–1124.CrossRefGoogle Scholar
[8]Alber, Y., ‘On average convergence of the iterative projection methods’, Taiwanese J. Math. 6 (2002), 323–341.Google Scholar
[9]Alber, Y. and Nashed, M., ‘Iterative-projection regularization of unstable variational inequalities’, Analysis 24 (2004), 19–39.CrossRefGoogle Scholar
[10]Baiocchi, C. and Capelo, A., Variational and quasi-variational inequalities, application to free boundary probles (Wiley, New York/London, 1984).Google Scholar
[11]Fan, K., ‘A generalization of Tychonoff's fixed point theorem’, Math. Ann. 142 (1961), 305–310.Google Scholar
[12]Giannessi, F. and Maugeri, A., Variational inequalities and network equilibrium problems (Plenum, New York, 1995).Google Scholar
[13]Isac, G., Complementarity problems, Lecture Notes in Math. 1528 (Springer-Verlag, Berlin, 1992).Google Scholar
[14]Isac, G., Sehgal, V.M. and Singh, S.P., ‘An alternate version of a variational inequality’, Indian J. Math. 41 (1999), 25–31.Google Scholar
[15]Kirk, W.K. and Sims, B., Handbook of metric fixed point theory (Kluwer Academic Publishers, Dordrecht, 2001).CrossRefGoogle Scholar
[16]Li, J., ‘On the existence of solutions of variational inequalities in Banach spaces’, J. Math. Anal. Appl. 295 (2004), 115–126.CrossRefGoogle Scholar
[17]Li, J., ‘The generalized projection operator on reflexive Banach spaces and its applications’, J. Math. Anal. Appl. 306 (2005), 55–71.CrossRefGoogle Scholar
[18]Takahashi, W., Nonlinear functional analysis (Yokohama Publishers, Yokohama, 2000).Google Scholar
[19]Vainberg, M.M., Variational methods and method of monotone operators (John Wiley & Sons, New York, 1973).Google Scholar
[20]Yuan, G.X.Z., KKM theory and applications in nonlinear analysis (Marcel-Dekker, New York, 1999).Google Scholar