LINEAR-EQUATIONS DEPENDING DIFFERENTIABLY ON A PARAMETER

We investigate operator functions T(x) in Banach spaces, depending differentiably (meaning of class Cμ or C∞) on a parameter x and enjoying a certain "regularity property". If f is a given differentiable function such that the equation T(x)e=f(x) is solvable for each x then the existence of a function e is proved which belongs to the same differentiability class as f and T, solving the equation T(x)e(x)≡f(x) identically in x. As an application we extend a result of J. Leiterer [9] and give a comprehensive answer to a question posed by J.L. Taylor in [15] concerning the exactness of certain cochain complexes. © 1990 Birkhäuser Verlag.