By Calvin H. Wilcox
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Extra info for Asymptotic solutions of differential equations and their applications
In [13, Sect. 1], we assumed that the map t → T0 (t)p0 (0) ∈ Z is H¨olderian of order a0 , 0 < a0 ≤ 1 and, that the linear map t → eB0t ∈ L(Z, X) as well as the linearized operator t → Π0 (t, 0) ∈ L(Z, X) are H¨olderian of order a, 0 < a ≤ 1 (see [13, Assumption (A3) in Sect. 1]). K. Hale and G. Raugel to the domain D(B∗0 ) [13, Assumption (A4) in Sect. 1], but this assumption could be replaced by a much weaker one. If the H¨older exponent a satisfies the condition a > 1/2 [13, Assumption (A5)(a) in Sect.
1, we have never used the fact that X is finite-dimensional. So the above proof is valid in the general case, where X is any Banach space, B0 is a linear continuous mapping from X into X and f0 (x) + h(ε , x) is a C1 -perturbation in (ε , x). 1], Hale and Verduyn Lunel have used the method of Sect. 1 to prove that regular perturbations of RFDE’s have still a periodic solution pε of period ωε , close to ω0 , if the unperturbed equation has a non-degenerate periodic solution p0 (t) of period ω0 .
Let x(t) be a periodic solution of (3) of period ω . For any ω ∈ R, ω > 0, the function x∗ (t) = x( ωω0 t) is periodic of period ω0 if and only if x(t) is periodic of period ω . As a consequence, if we use this rescaling of time, then x∗ (t) satisfies the equation xt∗ (t) = ω ω B0 x∗ (t) + ( f0 (x∗ (t)) + h(ε , x∗ (t))). ω0 ω0 The objective is to find an ω0 -periodic solution of (144) for x∗ = p0 + z, then z satisfies the equation ω ω0 (144) − 1 and ε small. K. Hale and G. Raugel and G(ε , ω ,t, z) = ω [ f0 (p0 + z) − f0(p0 ) − D f0 (p0 )z + h(ε , p0 + z)] ω0 ω + ( − 1) A0 z + B0 p0 + f0 (p0 ) .
Asymptotic solutions of differential equations and their applications by Calvin H. Wilcox