Differentiable mappings on products with different degrees of differentiability in the two factors

We develop differential calculus of Cr,sCr,s-mappings on products of locally convex spaces and prove exponential laws for such mappings. As an application, we consider differential equations in Banach spaces depending on a parameter in a locally convex space. Under suitable assumptions, the associated flows are mappings of class Cr,sCr,s.