By C Truesdell
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According to the discussion Persistence of Periodic Orbits for Perturbed Dissipative Dynamical Systems 45 made in Sect. 1, we set Zε = Xεs1 , Z ≡ Z0 = X0s1 , where 0 < s1 < 1/2 in the case ∗ n = 1 and where s1 < inf(1/2, 1−2α ) in the case n = 2. 5. There, we gave an estimate of Jε (ω )w − J0 w Cω (X) when w belongs to Cω0 (Z). Here, since 0 Jε and J0 act on different spaces, we will need to replace this estimate by an estimate of Jε (ω )W − J0 MW Cω (Xε ) for W in Cω0 (Zε ), where M is an appropriate 0 continuous linear operator from Xε into Xε .
12. There exists a positive constant C1 such that, for any s, 0 < s ≤ 1, for any U ∈ Xεs , we have eBε t U − eB0t MU Xε ≤ C1 ε s/2 eC1t U Xεs , (141) and also, for 0 ≤ σ < 1/2 and 0 ≤ σ ≤ s ≤ 1, eBε t U − eB0t MU Xεσ ≤ C1 ε (s−σ )/2 eC1t U Xεs . 9 and thereby obtain the existence and uniqueness of a periodic orbit Γε = {Pε (t) |t ∈ [0, ωε )} of Tε (t), close to the image of P0 (t), of minimal period ωε close to ω0 . 1 consists in arguments combining the Fredholm alternative with a Lyapunov-Schmidt procedure.
In the case n = 3, α ∗ = 1, we simply choose s2 satisfying 0 < s2 < 1/2. The regularity results of [12] imply that (p0t (t), p0tt (t), q0t (t), q0tt (t)) is continuous from R into Z. Since (p0 (t), q0 (t)) belongs to C0 (R, (H 2 (Ω))2 ), gi (p0 , q0 ) belongs to C0 (R, H 1 (Ω)) and thus Hypothesis (H2) is satisfied. In the cases n = 1, n = 2 or n = 3 with α ∗ < 1, DU fε (U) is a linear continuous mapping from X into (H s1 +1 (Ω) × H s1 (Ω))2 , for any U ∈ X, and thus Hypothesis (H3) is automatically satisfied.