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Generic Gateaux differentiability via smooth perturbations

Published online by Cambridge University Press:  17 April 2009

Pando Gr Georgiev  and
Nadia P. Zlateva
Pando Gr Georgiev
Affiliation:
Department of Mathematics and InformaticsUniversity of Sofia5 James Bourchier Blvd.1126 SofiaBulgaria
Nadia P. Zlateva
Affiliation:
Department of Mathematics and InformaticsUniversity of Sofia5 James Bourchier Blvd.1126 SofiaBulgaria
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Abstract

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We prove that in a Banach space with a Lipschitz uniformly Gateaux smooth bump function, every continuous function which is directionally differentiable on a dense Gδ subset of the space, is Gateaux differentiable on a dense Gδ subset of the space. Applications of this result are given.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 56 , Issue 3 , December 1997 , pp. 421 - 428
Copyright
Copyright © Australian Mathematical Society 1997

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.CrossRefGoogle Scholar
[2]Deville, R., Godefroy, G. and Zizler, V., ‘A smooth variational principle with applications to Hamilton-Jacobi equations in infinite dimensions’, J. Funct. Anal. 111 (1993), 197–212.CrossRefGoogle Scholar
[3]Ekeland, I., ‘Nonconvex minimization problems’, Bull. Amer. Math. Soc. 1 (1979), 443–474.CrossRefGoogle Scholar
[4]Ekeland, I. and Lebourg, G., ‘Generic Frechet differentiability and perturbed optimization problems in Banach spaces’, Trans. Amer. Math. Soc. 224 (1976), 193–216.Google Scholar
[5]Fabian, M. and Preiss, D., ‘On intermediate differentiability of Lipschitz functions on certain spaces’, Proc. Amer. Math. Soc. 113 (1991), 733–740.CrossRefGoogle Scholar
[6]Georgiev, P.G., ‘The smooth variational principle and generic differentiability’, Bull. Austral. Math. Soc. 43 (1991), 169–175.Google Scholar
[7]Kuratowski, K., Topology I (Academic Press, New York and London, 1966).Google Scholar
[8]Zajicek, L., ‘A generalization of an Ekeland-Lebourg theorem and the differentiability of distance functions’, in Proc. 11-th Winter School on Abstract Analysis, Suppl. Rend. Circolo Mat. di Palermo, Ser. II 3, 1984.Google Scholar
[9]Zhivkov, N.V., ‘Generic Gateaux differentiability of directionally differentiable mappings’, Rev. Roumaine Math. Pures Appl. 32 (1987), 179–188.Google Scholar