The tangential condition will also appear in other theorems in the sequel (it may be slightly changed) with an exception of our theorem, where we use GDQs instead of the contingent derivative.
Our idea was to use for the first time in viability theory another tool of differentiation: GDQs. One of the advantages of GDQs over the contingent derivative is that, as Example 2.6 shows, SGDQ contains all important directions while the contingent derivative has, besides these directions, some superfluous elements.
The main difference between Theorem 3.6 and others is, as it was mentioned before, that we use GDQs theory instead of the contingent derivative to formulate the tangential condition for problem (2.1).
Indeed, in (3.4) we intersect F with, possibly smaller than the contingent derivative, the closed union of minimal GDQs ofK.
As a generalized derivative we choose the generalized differential quotient (GDQ), introduced recently by Sussmann [11;12].
We say that [LAMBDA] is a generalized differential quotient (GDQ) of F at ([bar.x], [bar.y]) in the direction S, and write [lambda] [member of] GDQ(F, [bar.x], [bar.y], S) if for every positive real number [delta] there exist U;G such that
A minimal GDQ of F at ([bar.x], [bar.y]) in the direction S is a minimal element of the set GDQ(F; [bar.x]; [bar.y]; S) (minimal in the sense of inclusions of sets).
As we can have more than one minimal GDQ, we introduce the following concept of SGDQ.
multifunction with nonempty closed values such that for all (t; y) [member of] GrK, where t [member of] [0; a), K is GDQ differentiable at (t; y) in the direction of [R.sub.+] and for every [epsilon] > 0 there exists [T.sub.[epsilon]] [subset.bar] T such that [lambda](T\[T.sub.[epsilon]]) < [epsilon] and the map (t; y) [??] SGDQ(K; t; y;[R.sub.+]) is u.s.c.