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Playing Around Resonance

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Playing Around Resonance

Part of the book series: Birkhäuser Advanced Texts Basler Lehrbücher ((BAT))

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Abstract

In this chapter, we will consider again the periodic problem

$$\displaystyle{ (P)\quad \left \{\begin{array}{l} x^{{\prime\prime}} + g(t,x) = 0\,, \\ x(0) = x(T)\,,\;\;x^{{\prime}}(0) = x^{{\prime}}(T)\,, \end{array} \right. }$$

where \(g: [0,T] \times \mathbb{R} \rightarrow \mathbb{R}\) is a continuous function, and we will concentrate in finding sufficient conditions for the existence of a solution in the case when

$$\displaystyle{ g(t,x) =\lambda _{N}x + h(t,x)\,, }$$

where \(\lambda _{N} = \left (\frac{2\pi N} {T} \right )^{2}\) is an eigenvalue of the differential operator, and h is a bounded function.

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Notes

  1. 1.

    The notation o(ɛ) used here has the following meaning: for some function R(ɛ; θ0 ),

    $$\displaystyle{ R(\varepsilon;\theta _{0}) = o(\varepsilon )\quad \Longleftrightarrow\quad \lim _{\varepsilon \rightarrow 0^{+}} \frac{1} {\varepsilon } R(\varepsilon;\theta _{0}) = 0\,,\quad \mbox{ uniformly in }\theta _{0} \in [0,\tau ]. }$$

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Authors and Affiliations

  1. Dipartimento di Matematica e Geosci, Università degli Studi di Trieste, Trieste, Italy

    Alessandro Fonda

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  1. Alessandro Fonda
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Fonda, A. (2016). Playing Around Resonance. In: Playing Around Resonance. Birkhäuser Advanced Texts Basler Lehrbücher. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-47090-0_6

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