Existence and continuity of an implicit function in a neighborhood of an abnormal point

The article deals with the equation F(x,sigma) = 0 quad (x in U)tag1 where F: X times Sigma to Y is a twice differentiable map, X and Y are Banach spaces, Sigma a topological space, and U subset X a closed convex set. Under the assumption that F(x_*,sigma_*) = 0, the authors study the implicit function problem in the set U times {scr O} ({scr O} is a neighborhood of the point sigma_*). The main result is the following: If {rm ri}frac{partial F}{partial x}(x_*,sigma_*){scr U} ne emptyset ({scr U} = {rm cone}, (U - {x_*})) and for some h in X, frac{partial F}{partial x}(x_*,sigma_*){scr U} + frac{partial ^2F}{partial x^2}(x_*,sigma_*)bigg[h,{scr U} cap ker frac{partial F}{partial x}(x_*,sigma_*)bigg] = Y (the Avakov-Arutyunov property of 2-regularity with respect to the set U along h), then, in a neighborhood {scr O}_1 subset {scr O}, equation (1) defines an implicit function x(sigma):{scr O}_1 to U. Some properties of this implicit function are also described. This result is applied to the problem of local solvability for control systems.