The following proposition is a main tool in our next proofs.Proposition 9 ��Assume that x is a mapping from I to a.e.??on??I.(23)Then DOT1L x?satisfying||x�B(t)||��c(t)(||x(t)||+||x(t)||p), ����0b2c(s)exp??((p?1)��(s))ds]1/(1?p),(24)on???????is bounded byM:=exp?(��(b2))[||x(0)||1?p+(1?p) the interval I1, where I1 = [0, T] when p (?��, 1) and I1 = [0, b1) for p (1, ��), where b1 is given as in Lemma 7.Proof ��Let v(t) = ||x(t)||. Since x is absolutely continuous on I, then the derivatives x�B(t) and v�B(t) exist a.e. on I and satisfyv�B(t)=?x�B(t),J(x(t))||x(t)||?,(25)where J is the normalized duality mapping (for the definition we refer to [25]). For such t, ��c(t)(v(t)+v(t)p).(26)Take??��c(t)(||x(t)||+||x(t)||p)??we havev�B(t)��||x�B(t)||||J(x(t))||||x(t)|| the functions h and k as in Lemma 7 satisfying h(t) = k(t) = c(t) > 0, for all t I.

Then by Lemmas 7 and 8 we get the conclusion of the proposition.In all what follows let b2 and M be as in Proposition 9. We recall from Deimling [2, Theorem 9, Page 117] the following existence result for u.s.c. set-valued mappings with values contained in a compact set.Theorem 10 ��Let be a Banach space, D a nonempty closed set, I = [0, T], and G : I �� satisfying the following:G is u.s.c. with closed convex values;G(t, x) c(t) on J �� D, for some convex compact set in and c C(I, +);G(t, x)��K(D; x) �� on I �� D.Then for every x0 D, there exists an absolutely continuous mapping x : I �� D such on??I.(27)We start now by?a.e.??on??I,x(0)=x0��D,x(t)��D,?thatx�B(t)��G(t,x(t)) proving the following proposition needed in the proof of the main result.

Proposition 11 ��Let D be a closed subset in and let F : D be an upper semicontinuous set-valued mapping with closed convex values and let r1, r2 > 0 be such that r1 < r2, and let �� : [0, +��)��[0,1] be a continuous function such that ��(s) = 1 for s �� r1 and ��(s) = 0 for s �� r2. Let G be a set-valued mapping ?(t,x)��I��D.(28)If F satisfies?defined on D as follows:G(t,x)=��(||x||)F(t,x) the nonlinear growth on I �� D; that is, F(t, x) c(t)(||x|| + ||x||p) on I �� D, for some c C(I, +), p with p �� 1, and is a convex compact set in , then G is upper semicontinuous on I �� D with closed convex values.Proof ��Clearly, G has closed convex values. Let 0 : = (r1 + r1p) 0.

Drug_discovery For any t I and any x D with ||x|| < r2, we have by the convexity of the following:G(t,x)=��(||x||)F(t,x)?��(||x||)c(t)(||x||+||x||p)??c(t)(r1+r1p)??c??0,(29)where c-:=max?t��I??c(t) and for any t I and any x D with x r2, we have G(t,x)=0?c-?0. Then G(I��D)?c-?0. Then, by Proposition 4, it is sufficient to prove that the graph of G is closed. To do that, we fix ((tn, xn), yn) gphG with ((tn,xn),yn)��((t-,x-),y-) and we have to prove that ((t-,x-),y-)��gph?G; that is y-��G(t-,x-). By definition of with??zn��F(tn,xn).