e.?on??I.(35)Assume that p �� 1.?so we get||x�B(t)||��k?(||x(t)||+||x(t)||p), Using Proposition 9, we get ||x(t)|| �� M a.e. on I1 and so by the definition of G and �� we get G(t, x(t)) promotion info = ��(||x(t)||)F(t, x(t)) = F(t, x(t)) that is, x is a solution of (31) on I1.5. Nonconvex Sweeping Process with Perturbations Having Nonlinear Growth ConditionsOur purpose, in this section, is to use the techniques developed previously to extend some existing results, in separable Hilbert spaces, of nonconvex sweeping processes with perturbations from the case of perturbation with linear growth to the case of perturbation with nonlinear growth. For this end let be a separable Hilbert space, let I : = [0, T] (T > 0), and let C : I be a set-valued mapping satisfying the following Lipschitz condition for any y and any t, t�� I:|dC(t)(y)?dC(t��)(y)|��L|t?t��|.

(36)We start with the following existence result which is a consequence of Theorem 4.1 in [19].Theorem 13 ��Let be a separable Hilbert space and let r (0, ��]. Assume that C(t) is r-prox-regular for every t I and that the assumption (36) holds. Let F : I �� �� be a set-valued mapping with convex compact values in such that F is u.s.c. on I �� . Assume that F(t, x) 1 �� for all (t, x) I �� , for some compact set 1 in . Then, for any x0 C(0), the sweeping process (SPP) with the perturbation F has at least one Lipschitz continuous solution; that is, there exists an Lipschitz continuous mapping x : I �� such ?t��I,x(0)=x0,(37)and?a.e.??t��I,x(t)��C(t),?that?x�B(t)��NC(C(t);x(t))+F(t,x(t)) ||x�B(t)||��L(2��+1), a.e. on I.

Using the techniques from the previous section and Theorem 13 we prove our main result in this section.Theorem 14 ��Let r (0, +��]. Assume that C(t) is r-prox-regular for every t I and that the assumption (36) holds. Let F : I �� �� be a set-valued mapping with convex compact values. Assume also that F has nonlinear growth; that is, there exist a positive continuous function c : I �� (0, ��), a convex compact set , and k > ?(t,x)��I��?.(38)Assume?0 such thatF(t,x)?c(t)(||x||+||x||p)??c(t)(||x||+||x||p)k?, further that the following conditions on the constants L, k, c-, p, and T are satisfied:p��1,c?:=max?t��I??c(t)<18TLk(1+4p?1(TL+||x0||)p?1).(39)Then for any u0 C(0), there exists a Lipschitz continuous mapping u : I �� satisfying the following sweeping process with a ?t��I,x(0)=x0.

(40)Proof?a.e.??t��I,x(t)��C(t),?perturbation:?x�B(t)��NC(C(t);x(t))+F(t,x(t)) Carfilzomib ��Assume without loss of generality that 0 . By our assumptions on the constants L, c-, and T we have after simple computations4p?1(LT+||x0||)p?1<18c?LTk?1.(41)So we can find some positive number �� > 0 such that4p?1(LT+||x0||)p?1<��<18c?LTk?1.(42)Let �� = ��1/(p?1). Then��>2c?(��p?1+1)??4(LT+||x0||),<14LTk.(43)Define then the function �� : [0, +��)��[0,1] to be a continuous function such that ��(s) = 1 for s �� ��/2 and ��(s) = 0 for s �� �� and define the set-valued mapping G on as on??I��?.