By Berti M., Bolle P.

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93) An elementary calculus, using that maxk≥1 k α exp{−(σi − σ)k} ≤ C(α)/(σi − σ)α , gives hi σ,s+ 2(τ −1) β hi ≤ C(τ ) σi ,s (σi − σ) 2(τ −1) β . 1-d), provided δ0 is small enough, also v2 (δ, v1 , w) a σ,s+ 2(τ −1) β σi ,s σ,s+ = (∂u g) δ, x, v1 + w + v2 (δ, v1 , w) 2(τ −1) β 2(τ −1) β σ,s+ ≤ C(τ )2µ . ≤ C and therefore 2(τ −1) β ≤ C. This bound is a consequence of the analyticity assumption (H) on the nonlinearity f , the Banach algebra property of Xσ,s+2(τ −1)/β , and can be obtained as in (22).

Paleari, Normal form and exponential stability for some nonlinear string equations, Z. Angew. Math. Phys, 52, 6, 1033-1052, 2001. [4] P. Baldi, M. Berti, Periodic solutions of nonlinear wave equations for asymptotically full measure set of frequencies, to appear on Rend. Mat. Acc. Lincei. [5] M. Berti, P. Bolle, Periodic solutions of nonlinear wave equations with general nonlinearities, Comm. Math. Phys. Vol. 243, 2, 315-328, 2003. [6] M. Berti, P. Bolle, Multiplicity of periodic solutions of nonlinear wave equations, Nonlinear Analysis, 56/7, 1011-1046, 2004.

As a result, v 1 = ΠV1 v is a non-degenerate solution of (98). 2 By assumption, v is a non-degenerate solution of equation (11). 1, v 1 = ΠV1 v ∈ B(R, V1 ) is a non degenerate solution of (98). Since the map (δ, v1 ) → −∆v1 − ΠV1 G(δ, v1 ) is in C ∞ ([0, δ0 ) × V1 ; V1 ), by the Implicit Function Theorem, there is a C ∞ path δ → v1 (δ) ∈ B(2R, V1 ) such that v1 (δ) is a solution of (97) and v1 (0) = v 1 . 1, the function u(δ) := δ v1 (δ) + v2 δ, v1 (δ), w(δ, v1 (δ)) + w(δ, v1 (δ)) ∈ Xσ/2,s (103) is a solution of equation (3) if δ belongs to the Cantor-like set C := δ ∈ [0, δ0 ) | (δ, v1 (δ)) ∈ B∞ .