Computation of the tangent cone to the image of a mapping at a singular point

The paper concerns the calculus of the tangent cone to the set F(N) at the point F(p) for sufficiently small neighborhoods N of the singular point p, which means F'(p)(X)subset Y. The general inclusion F'(p)(X)subseteq Tsb{F(N)}(F(p)) is improved to the special equality F'(p)(X)+K=Tsb{F(N)}(F(p)) for a cone K provided that F'(p)(X)+L=Y for a complementing space L. The construction of K inside L depends on the behaviour of F"(p) towards the kernel of F'(p). Here X and Y are Banach spaces, the function Fcolon Xto Y is twice continuously Fréchet differentiable at the point p, F' and F" denote Fréchet differentials, and T stands for the tangent cone.