Un equazione differenziale del primo ordine in spazi di Hilbert le cui soluzioni verificano una condizione di periodicita' generalizzata

Let H and Q be Hilbert spaces, H* the dual of H, H subset-of Q subset-of H*, H dense in Q. Let i: H --> Q continuous, J is-an-element-of L(H*, H) the operator defined by the relation (Ju,v)H = [u,v] for-all u is-an-element-of H*, for-all v is-an-element-of H. Let [alpha, beta] be an interval of R, A(t) is-an-element-of L(H, H), B is-an-element-of L(Q, Q). We prove the existence of a unique solution u is-an-element-of L2(alpha, beta, H) and C0([alpha, beta], Q) and H1/2(alpha, beta, Q) of the problem: A(t)u(t) + (Ju(t))' = Jf(t) on [alpha, beta], where f is-an-element-of L2(alpha, beta, H*), u(alpha) = Bu(beta). (If B = Id(Q), this it is the classic periodic problem). If i is compact, if parallel-to Bu parallel-to H less-than-or-equal-to parallel-to u parallel-to H if f is-an-element-of L2(alpha, beta, Q) and A(t) = kI (k > 0) we prove that u is-an-element-of H1/2 (alpha, beta, H) and H1 (alpha, beta, Q) and C0 ([alpha, beta], H). If A(t) = A(t + T), for-all t is-an-element-of R, we study the analogous problem in R.