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 ΩX
into ΩY
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.
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.