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φ-regular functions in Asplund spaces

Ngai H. V., Thera M.

Control and Cybernetics

2007

Vol. 36, no 3

755-774

We introduce in the context of Asplund spaces, a new class of (φ-regular functions. This new concept generalizes the one of prox-regularity introduced by Poliquin & Rockafellar (2000) in Rn and extended to Banach spaces by Bernard & Thibault (2004). In particular, the class of φ-regular functions includes all lower semi-continuous convex functions, all lower-C2 functions, and convexly C1,0-composite functions as well. Geometrical and subdifferential characterizations for this new class of functions are investigated.

Critical points for vector-valued functions

Lucchetti R., Revalski J., Thera M.

Control and Cybernetics

2002

Vol. 31, no 3

545-555

This paper contains a mountain pass theorem for continuous mappings, defined on a complete metric space and taking values in a real Banach space, ordered by a closed convex cone. We use the concept of critical point introduced by Degiovanni, Lucchetti and Ribarska, and we furnish a variant of their result, allowing for a localization both of the critical point and of the critical value.