- Title
- Duality inequalities and sandwiched functions
- Creator
- Borwein, J. M.; Fitzpatrick, S. P.
- Relation
- Nonlinear Analysis: Theory, Methods & Applications Vol. 46, Issue 3, p. 365-380
- Publisher Link
- http://dx.doi.org/10.1016/S0362-546X(00)00129-2
- Publisher
- Pergamon
- Resource Type
- journal article
- Date
- 2001
- Description
- Motivated by Clarke and Ledyaev's [3] striking multi-directional mean-value theorem and its elegant reformulation by Lewis and Ralph [9] as a non-smooth sandwich theorem, our intentions in this paper are two-fold. Firstly, in Section 2 we provide a self-contained proof of a general convex/Lipschitz inequality from which both results follow. In Section 3 we exploit the underlying technique to obtain some more refined – and perhaps surprising – inequalities. These results rely on the Brouwer/Schauder fixed-point theorem. Secondly, in Section 4, we use variational methods to provide an affirmative answer to several open questions posed by Lewis and Lucchetti [8] on the existence of common subgradients for a finite family of Lipschitz functions. Section 5 provides various infinite dimensional and non-compact extensions of our results and poses several open questions. The rest of this section is dedicated to three preparatory results.
- Subject
- nonconvex separation; sandwich theorem; mean value inequalities; Fenchel duality; Schauder fixed point theorem; Ekeland variational principle
- Identifier
- http://hdl.handle.net/1959.13/940705
- Identifier
- uon:13074
- Identifier
- ISSN:0362-546X
- Language
- eng
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