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Search: MSC category 58C20 ( Differentiation theory (Gateaux, Frechet, etc.) [See also 26Exx, 46G05] )

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1. CJM 2005 (vol 57 pp. 961)

Borwein, Jonathan M.; Wang, Xianfu
 Cone-Monotone Functions: Differentiability and Continuity We provide a porosity-based approach to the differentiability and continuity of real-valued functions on separable Banach spaces, when the function is monotone with respect to an ordering induced by a convex cone $K$ with non-empty interior. We also show that the set of nowhere $K$-monotone functions has a $\sigma$-porous complement in the space of continuous functions endowed with the uniform metric. Keywords:Cone-monotone functions, Aronszajn null set, directionally porous, sets, GÃ¢teaux differentiability, separable spaceCategories:26B05, 58C20

2. CJM 1999 (vol 51 pp. 250)

Combari, C.; Poliquin, R.; Thibault, L.
 Convergence of Subdifferentials of Convexly Composite Functions In this paper we establish conditions that guarantee, in the setting of a general Banach space, the Painlev\'e-Kuratowski convergence of the graphs of the subdifferentials of convexly composite functions. We also provide applications to the convergence of multipliers of families of constrained optimization problems and to the generalized second-order derivability of convexly composite functions. Keywords:epi-convergence, Mosco convergence, PainlevÃ©-Kuratowski convergence, primal-lower-nice functions, constraint qualification, slice convergence, graph convergence of subdifferentials, convexly composite functionsCategories:49A52, 58C06, 58C20, 90C30

3. CJM 1999 (vol 51 pp. 26)

Fabian, Marián; Mordukhovich, Boris S.
 Separable Reduction and Supporting Properties of FrÃ©chet-Like Normals in Banach Spaces We develop a method of separable reduction for Fr\'{e}chet-like normals and $\epsilon$-normals to arbitrary sets in general Banach spaces. This method allows us to reduce certain problems involving such normals in nonseparable spaces to the separable case. It is particularly helpful in Asplund spaces where every separable subspace admits a Fr\'{e}chet smooth renorm. As an applicaton of the separable reduction method in Asplund spaces, we provide a new direct proof of a nonconvex extension of the celebrated Bishop-Phelps density theorem. Moreover, in this way we establish new characterizations of Asplund spaces in terms of $\epsilon$-normals. Keywords:nonsmooth analysis, Banach spaces, separable reduction, FrÃ©chet-like normals and subdifferentials, supporting properties, Asplund spacesCategories:49J52, 58C20, 46B20
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