By Baldi P., Berti M.
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Extra resources for Forced vibrations of a nonhomogeneous string
34, for every j |λj (a) − λj (b)| = 1 1 |νj (α) − νj (β)| ≤ 2 α − β 2 c c 30 ∞ ≤K a−b H1 (68) and (17) follows by the mean value theorem because µΠV f (v(µ, w) + w) has bounded derivatives on bounded sets. Proof of (47). By the asymptotic formula (61) min |ωj+1 (µ, w) − ωj (µ, w)| ≥ j≥1 K 1 1 − > c j 2c if j > K/2c, uniformly in µ ∈ [µ1 , µ2 ], w ∈ BR . 34). Each mj > 0 because all the eigenvalues λj are simple. References  P. Acquistapace, Soluzioni periodiche di un’equazione iperbolica non lineare, Boll.
6, 214–228, 2000. -M. edu/users/jmfokam. J. McKenna, On solutions of a nonlinear wave question when the ratio of the period to the length of the intervals is irrational, Proc. Amer. Math. Soc. 93 (1985), no. 1, 59–64. I. N. Yungerman, Periodic solutions of a weakly nonlinear wave equation with an irrational relation of period to interval length, translation in Diff. Equations 24 (1988), no. 9, 1059–1065 (1989).  J. P¨oschel, E. , Orlando, Florida, 1987. 32  P. Rabinowitz, Periodic solutions of nonlinear hyperbolic partial differential equations, Comm.
30). Since p, ρ are positive and belong to H 3 , if a ∈ H 1 then q, α ∈ H 1 . π Integrating by parts | 0 cos(2jξ)(q + α) dξ| ≤ q + α H 1 /j and so νj = j 2 + 1 π π 0 (q + α) dξ + rj , |rj | ≤ C j for some C := C ( q + α H 1 ). Dividing by c2 and using the inverse Liouville change of variable we obtain the formula for the eigenvalues λj (a) of (65) √ π π Q ρ j2 a 1 1 C λj (a) = 2 + (67) √ dx + √ dx + rj (a), |rj (a)| ≤ c πc 0 p πc 0 ρp j for some C(ρ, p, a H 1 ) > 0. Formula (55) follows for a(x) = −µΠV f (v(µ, w) +w)(x) and some algebra.