Eusebio Corbacho Rosas and Vaja Tarieladze Automatic continuity of directional derivatives

Abstract: Let f be a mapping defined on an open set G of a Banach space E with values in a Banach space F and x_{0} ∈ G. Definition. The mapping f is said to be directionally differentiable x_{0} ∈ G if there is a mapping T_{x_{0},f} : E → F such that

T_{x_{0},f}(u) = lim_{t→0} \frac{f(x_{0} + th) − f(x_{0})}{t} ∀h ∈ E.  (1)

If f is directionally differentiable at x_{0} and its directional derivative T_{x_{0},f} : E → F is a continuous linear operator, then f is said to be Gâteaux differentiable at x_{0}. We consider two natural questions: Question 1. Assume that the given continuous f is directionally differentiable at x_{0} and its directional derivative T_{x_{0},f} : E → F is an additive mapping. Is then f Gâteaux differentiable at x_{0}? Question 2. Is there a wide class Φ of mappings from G to F such that if a given continuous f ∈ Φ is the directionally differentiable at x_{0}, then f Gâteaux differentiable at x_{0}?

In the given talk we will discuss the following two results:

Theorem 1. Question 1 has the affirmative answer.

Theorem 2. If G is convex, F = R and Φ consists of convex functionals, then Question 2 has the affirmative answer as well.

Theorem 1 seems not to be noticed earlier in the literature. Theorem 2 is known [G, p. 127, Theorem 7 ]. This talk is based on [BCPT], where more general versions of Theorem 1 and Theorem 2 are contained.

REFERENCES

[BCPT] Banakh, T., Corbacho, E., Plichko, A., Tarieladze V., Automatic continuity and linearity of Gâteaux derivatives, In preparation.

[G] Giles, J.R. Convex analysis with applications in differentation of convex functions. Pitman, London, 1982, 278 p.

AUTHORS

Eusebio Corbacho Rosas

Departamento de Matematica Aplicada, E. T. S. E. de Telecomunicacion, Universidad de Vigo, Vigo, Spain. E-mail: corbacho@uvigo.es

Vaja Tarieladze Muskhelishvili Institute of Computational Mathematics, Georgian Academy of Sciences, Tbilisi–93, Georgia. E-mails: tar@gw.acnet.ge visit01@dma.uvigo.es