On a problem in the theory of nonlinear Fredholm operators

We prove that for any Fredholm operator $A(x)$ of class $C^1$ and zero index in a Hilbert space, in a neighborhood of any star compactum $T$ lying in the domain of a we can define a completely continuous and continuously differentiable operator $C$, so that the linear operator $A'(x)+C'(x)$ has a bounded inverse for all $x\in T$.