For an m-dimensional differential inclusion of the form ie A(t)x(t) + F[t,x(t)], with A and F T-periodic in t, we prove the existence of a nonconstant periodic solution. Our hypotheses require m to be odd, and require F to have different growth behavior for |i| small and |i| large (often the case in applications). The idea is to guarantee that the topological degree associated with the system has different values on two distinct neighborhoods of the origin. We prove the existence of a

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... t T-periodic solution to a time-dependent differential inclusion of the form a.e. in [0,T], T > 0 given. We assume the following: x upper-semicontinuous multivalued map with nonempty, compact, convex values such that 0 G F(t,0) a.e. in [0, T], and satisfying \F(t,x)\d= max{\y\: y G F(t,x)} < a(t)\x\ + ß(t) a.e. in [0,T] for some pair (a,ß) from L1([0,T],R+). (H2) If 0 G A(t)c + F(t, c) a.e. in [0, T] for some fixed c G Rn, then c = 0. (H3) We define i(t,x) = inf{(x,A(f)x + y): y G F(t,x)},s(t,x) = sup{(x, A(t)x + y): y G F(t, x)}, where (a, b) denotes the inner product of a and b in Rm. Assume / liminf^iSdOO, Jo M-oo |x|z and suppose that there exists an r > 0 such that s(t, x) < 0 for all 0 < |x| < r a.e. in[0,T]. In the following, the maps A(-) and F(-,x) will be considered extended from the interval [0, T] to the real line R by T-periodicity. (L~)m is the Banach space of functions in L1^,T],Rm), extended to R by T-periodicity. In the same way ÍACt)™ and (Gr)m are the Banach spaces of T-periodic functions which are absolutely continuous and continuous endowed with the usual norms ||x||>ic and HxHoo respectively.