 Solution

Orthogonality and $\Vert \tilde{\varphi }(x)\Vert =\Vert x\Vert$ giveso ${\psi }_{\epsilon }$ maps each boundary sphere to itself. AlsoOn compact $A$, this derivative is uniformly close to the identity; for sufficiently small $\epsilon$ its determinant is positive and it is locally invertible. The map is properly homotopic to the identity and has degree one, so the local diffeomorphism has one sheet and is a global diffeomorphism $A\to A$.