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A Generalized Variational Principle

Published online by Cambridge University Press:  20 November 2018

Philip D. Loewen  and
Xianfu Wang
Philip D. Loewen
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2. email: loew@math.ubc.ca
Xianfu Wang
Affiliation:
Department of Mathematics & Statistics, Okanagan University College, 3333 College Way, Kelowna, BC, V1V 1V7. email: xwang@cecm.sfu.ca
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Abstract

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We prove a strong variant of the Borwein-Preiss variational principle, and show that on Asplund spaces, Stegall's variational principle follows from it via a generalized Smulyan test. Applications are discussed.

Keywords

49J52Keywords: variational principlestrong minimizergeneralized Smulyan testAsplund spacedimple pointporosity
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 53 , Issue 6 , 01 December 2001 , pp. 1174 - 1193
Copyright
Copyright © Canadian Mathematical Society 2001

References

[1] Borwein, J. M. and Preiss, D., A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions. Trans. Amer. Math. Soc. 303(1987), 517–527.Google Scholar
[2] Clarke, F. H., Ledyaev, Yu. S., Stern, R. J. and Wolenski, P. R., Nonsmooth Analysis and Control Theory. Graduate Texts in Math. 178, Springer-Verlag, New York, 1998.Google Scholar
[3] Collier, J. B., The dual of a space with the Radon-Nikodym property. Pacific J. Math. 64(1976), 103–106.Google Scholar
[4] Deville, R., Godefroy, G. and Zizler, V., Smoothness and Renorming in Banach Spaces. Pitman Monographs Surveys Pure Appl. Math. Vol. 64, Longman, 1993.Google Scholar
[5] Deville, R. and Revalski, J. P., Porosity of ill-posed problems. Proc. Amer.Math. Soc. 128(1999), 1117–1124.Google Scholar
[6] Ekeland, I., On the variational principle. J. Math. Anal. Appl. 47(1974), 324–353.Google Scholar
[7] Fabian, M. and Zizler, V., An elementary approach to some questions in higher order smoothness in Banach spaces. Extracta Math. 14(1999), 295–327.Google Scholar
[8] Fabian, M. and Mordukhovich, B. S., Nonsmooth characterizations of Asplund spaces and smooth variational principles. Set-Valued Anal. 6(1998), 381–406.Google Scholar
[9] Giles, J. R., Convex analysis with application in differentiation of convex functions. Research Notes in Mathematics 58, Pitman, 1982.Google Scholar
[10] Georgiev, P. G., The strong Ekeland variational principle, the strong drop theorem and applications. J. Math. Anal. Appl. 131(1988), 1–21.Google Scholar
[11] Ioffe, A. D., Approximate subdifferentials and applications II. Mathematika 33(1986), 111–128.Google Scholar
[12] Li, Yongxin and Shi, Shuzhong, A generalization of Ekeland's ε-variational principle and its Borwein-Preiss smooth variant. J. Math. Anal. Appl. 246(2000), 308–319.Google Scholar
[13] Phelps, R. R., Convex functions, monotone operators and differentiability. 2nd edition, Lecture Notes in Mathematics 1364, Springer-Verlag, Berlin, 1993.Google Scholar
[14] Preiss, D. and Zajicek, L., Fréchet differentiation of convex functions in a Banach space with a separable dual. Proc. Amer. Math. Soc. 91(1984), 202–204.Google Scholar
[15] Stegall, C., Optimization of functions on certain subsets of Banach spaces. Math. Ann. 236(1978), 171–176.Google Scholar
[16] Sullivan, F., A characterization of complete metric spaces. Proc. Amer. Math. Soc. 83(1981), 345–346.Google Scholar