Paraconvex analysis on -manifolds

Abstract

Let be real Banach spaces. Let f(·) be a real-valued locally uniformly approximate convex function defined on an open subset . Let be an open subset. Let σ (·) be a differentiable mapping of Ω_X into Ω_Y such that the differentials of are locally uniformly continuous function of x. Then f(σ (·)) is also a locally uniformly approximate convex function. Therefore the function f(σ (·)) is Fréchet differentiable on a dense G _δ-set, provided X is an Asplund space, and Gateaux differentiable on a dense G _δ-set, provided X is separable. As a consequence, we obtain that a locally uniformly approximate convex function defined on a -manifold is Fréchet differentiable on a dense G _δ-set, provided is an Asplund space, and Gateaux differentiable on a dense G _δ-set, provided E is separable.

†Dedicated to Professor Diethard Pallaschke on the occasion of his 65th birthday.

Keywords:

• Fréchet differentiability
• Gateaux differentiability
• Paraconvex functions
• C ^1,u _E -manifolds

Keywords:

• Mathematics Subject Classifications 2000: 58C20
• 46G05
• 26E15

Acknowledgment

The author would like to express his thanks to Professor Janusz Grabowski, whose valuable remarks permitted to improve the redaction of the article.

Notes

†Dedicated to Professor Diethard Pallaschke on the occasion of his 65th birthday.

¹ We shall say briefly differentials, since under assumptions of continuity each Gateaux differential is also a Fréchet differential.