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Smooth variational principles in Radon-Nikodým spaces

Published online by Cambridge University Press:  17 April 2009

Robert Deville  and
Abdelhakim Maaden
Robert Deville
Affiliation:
Université Bordeaux 1Faculté des SciencesLaboratoire de Mathématiques pures351 cours de la Libération33400 TalenceFrance
Abdelhakim Maaden
Affiliation:
Université Cadi-AyyadFaculté des Sciences et TechniquesDepartement de MathematiquesB.P. 523 Béni-MellalMaroc
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Abstract

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We prove that if f is a real valued lower semicontinuous function on a Banach space X, for which there exist a > 0 and b ∈ ℝ such that f(x) ≥ 2a∥x∥ + b, x ∈ X, and if X has the Radon-Nikody´m property, then for every Ε > 0 there exists a real function φ X such that φ is Fréchet differentiable, ∥φ∥∞ < Ε, ∥φ′∥∞ < Ε, φ′ is weakly continuous and f + φ attains a minimum on X. In addition, if we assume that the norm in X is β-smooth, we can take the function φ = g1 + g2 where g1 is radial and β-smooth, g2 is Fréchet differentiable, ∥g1∥∞ < Ε, ∥g2∥∞ < Ε, ∥g′1∥∞ < Ε, ∥g′1∥∞ < Ε, g′2 is weakly continuous and f + g1 + g2 attains a minimum on X.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 60 , Issue 1 , August 1999 , pp. 109 - 118
Copyright
Copyright © Australian Mathematical Society 1999

References

[1]Bollobás, B., ‘An extension to the theorem of Bishop and Phelps’, Bull. London Math. Soc. (2) (1970), 181–182.CrossRefGoogle Scholar
[2]Bonic, N. and Frampton, J., ‘Smooth functions on Banach manifolds’, J. Math. Mech. 15 (1966), 877–898.Google Scholar
[3]Borwein, J. 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
[4]Bourgin, R., Geometrical aspects of convex sets with the Radon-Nikodým property, Lecture notes in Math. 993 (Springer-Verlag, Berlin, Heidelberg, New York, 1983).CrossRefGoogle Scholar
[5]Deville, R., Godefroy, G. and Zizler, V., ‘A smooth variational principle with applications to Hamilton-Jacobi equations in infinite dimensions’, J. Func. Anal. 111 (1993), 197–212.CrossRefGoogle Scholar
[6]Deville, R., Godefroy, G. and Zizler, V., Smoothness and renormings in Banach spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics 64 (Longman Scientific and Technical, 1993).Google Scholar
[7]Deville, R., ‘Stability of subdifferentials of nonconvex functions in Banach spaces’, Set-Valued Anal. 2 (1994), 141–157.CrossRefGoogle Scholar
[8]Diestel, J. and Uhl, J., Vector measures (Amer. Math. Soc, Providence, RI, 1977).CrossRefGoogle Scholar
[9]Ekeland, I., ‘On the variational principle’, J. Math. Anal. Appl. 47 (1974), 324–353.CrossRefGoogle Scholar
[10]Haydon, R., ‘A counterexample to several questions about scattered compact spaces’, Bull. Lond. Math. Soc. 22 (1990), 261–268.CrossRefGoogle Scholar
[11]Phelps, R.R., Convex functions, monotone operators and differentiability, Lecture notes in Math. 1364 (Springer-Verlag, Berlin, Heidelberg, New York, 1991).Google Scholar
[12]Stegall, C., ‘Optimization of functions on certain subsets of Banach spaces’, Math. Ann. 236 (1978), 171–176.CrossRefGoogle Scholar