On abnormal problems with a nonclosed image

The authors consider Banach spaces X and Y, a differentiable map Fcolon X to Y, and points x^* in X and y^* in Y with y^*=F(x^*). For y in Y belonging to a neighbourhood of y^* they consider the equation F(x)=y, where x (belonging to a neighbourhood of x^*) is unknown. Then they study the question of under which assumptions there exists a solution x(y) of the equation F(x) = y such that |x(y) - x^*| le{rm const} |y - y^*|^alpha is fulfilled for some alpha > 0. par The authors also discuss applications to optimization problems in general spaces using Lagrangian technique.