Fixed point theorems in the Arnol'd model about instability of the action-variables in phase-space

We consider the hamiltonian $H=1/2(I_1^2+I_2^2)+\varepsilon(\cos\varphi_1-1) (1+\mu(\sin\varphi_2+\cos t))$ $I\in\mathbb{R}^2$ ("Arnol'd model about diffusion"); by means of fixed point theorems, the existence of the stable and unstable manifolds (whiskers) of invariant, "a priori unstable tori", for any vector-frequency $(\omega,1)\in\mathbb{R}^2$ is proven. Our aim is to provide detailed proofs which are missing in Arnol'd's paper, namely prove the content of the Assertion B pag.583 of [A]. Our proofs are based on technical tools suggested by Arnol'd i.e. the contraction mapping method together with the "conical metric" (see the footnote ** of pag. 583 of [A]).