Extending the Kantorovich theory for solving equations

Let X,  Y be Banach spaces, D ⊂ X be convex, F:D ⊂ X → Y be a Fréchet differentiable operator. We shall determine a solution x ^* of the equation 6.1 F ( x ) = 0 , $$ F(x) = 0 , $$ https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315153469/276b2768-c97f-4f2f-95fc-28fbbbbbe91f/content/math6_1.tif"/>