Un teorema di esistenza per equazioni contingenti in spazi di Banach

In this paper we study a contingent equation where the right hand side is a multifunction F, with nonempty, convex and compact values in a separable and reflexive Banach space X and where I is an interval [a, b] of the real line. For this problem we obtain, using a Carathéodory-Tonelli type procedure, an existence theorem for strongly absolutely continuous solutions when F is measurable in the variable t in I, satisfies property (Q) of L. Cesari [J. Optim. Theory Appl. 1 (1967), 87–112] in the variable x in X (equivalent in this setting to the upper semicontinuity in xX), and F(t, x) is contained in g(t)K for some nonnegative function g of class L1(I) and for some compact K of X. In this way some of the results noted in the literature [cf., e.g., C. Castaing, C. R. Acad. Sci. Paris S´ er. A-B 263 (1966), A63-A66; A. F. Filippov, Dokl. Akad. Nauk SSSR 151 (1963), 65–68; N. Kikuchi, Publ. Res. Inst. Math. Sci. Ser. A 3 (1967/68), 85–99] are extended to the infinite-dimensional case.”