Let S be a locally convex space.We use the product integration to show that Cauchy initial value problem: u'(x)=Au(x),u)o)=p,has a unique solution,where A is a function from S into itself and u is a continuously differentiable function from [0,∞) into S . The conditions required on A arethat A is dissipative, and there exists an open subset C of S such that A is continuous on C and the range of (I-εA) contains C as εis sufficiently small.
