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Generic differentiability of order-bounded convex oparators

Published online by Cambridge University Press:  17 February 2009

Jonathan M. Borwein
Jonathan M. Borwein
Affiliation:
Dalhousie University, Canada
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We give sufficient conditions for order-bounded convex operators to be generically differentiable (Gâteaux or Fréchet). When the range space is a countably order-complete Banach lattice, these conditions are also necessary. In particular, every order-bounded convex operator from an Asplund space into such a lattice is generically Fréchet differentiable, if and only if the lattice has weakly-compact order intervals, if and only if the lattice has strongly-exposed order intervals. Applications are given which indicate how such results relate to optimization theory.

Type
Research Article
Information
The ANZIAM Journal , Volume 28 , Issue 1 , July 1986 , pp. 22 - 29
Copyright
Copyright © Australian Mathematical Society 1986

References

[1]Borwein, J. M., “Continuity and differentiability properties of convex operators”, Proc. London Math. Soc. 3 (1982), 420–444.CrossRefGoogle Scholar
[2]Borwein, J. M., “Subgradients of convex operators”, Math. Operationsforch. Statist. Ser. Optim. 15 (1984), 179–191.Google Scholar
[3]Davis, W. J., Ghossoub, N., and Lindenstrauss, J.,, “A lattice renorming theorem and applications to vector-valued processes”, Trans. Amer. Math. Soc. 90 (1981), 531–540.CrossRefGoogle Scholar
[4]Kirov, N. K., “Differentiability of convex mappings and generalized monotone mappings”, C. R. Acad. Bulgare Sci. 34 (1981), 1.Google Scholar
[5]Kirov, N. K., “Generalized Fréchet differentiability of convex operators”, Proc. Amer. Math. Soc. 94 (1985), 97–102.Google Scholar
[6]Lindenstrauss, J. and Tzafriri, L., Classical Banach spaces II. Function spaces (Springer-Verlag, Berlin, 1979).CrossRefGoogle Scholar
[7]Schaefer, H. H., Banach lattices and positive operators (Springer-Verlag, Berlin, 1974).CrossRefGoogle Scholar