On the existence theorem for differential equations

PROOF. We suppose without loss of generality that xo is the origin of E and that U is an open ball with center xo. Take Uo to be the open ball whose center is xo and whose radius is half the radius of U. Let I denote the closed interval -1, 1CR. For p an integer >0 let CP(I, E) denote the Banach space of CP maps from I to E (with the CP topology), Cg(I, E) be the (closed) subspace of CP(I, E) consisting of all yyECP(I, E) with y(0) = 0, and C:(I, UO) the set of all yE CI(I, E) such that zy(I)C Uo. Note that Cg(I, Uo) is open in the Banach space CO(I, E). D denotes the differentiation operator (see 4 or 5) and Di denotes partial differentiation with respect to the jth variable. Let F: RX UoXCo(I, Uo) ->CO(I, E) be the map defined by