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Subharmonic solutions with prescribed minimal period of a discrete forced pendulum equation. (English) Zbl 1067.39022
Using variational methods the authors obtain sufficient conditions on the existence of sub-harmonic solutions with prescribed minimal period \(pT\) for the second order difference equation \[ x(n+1)-2x(n)+x(n-1)+A \sin(x(n))=f(n),\quad f(n+T)=f(n). \] An example is given. The idea of the paper is to find variational functionals to transfer the existence of sub-harmonic solutions into the existence of critical points of the corresponding functional.
This reviewer likes to add that the matrix \(B\) on page 579 in the paper is a circulant matrix some of its properties are well known [S. Barnett, Matrices: Methods and applications, Oxford University Press (1990; Zbl 0706.15001)].

39A11 Stability of difference equations (MSC2000)
39A12 Discrete version of topics in analysis
70J35 Forced motions in linear vibration theory
Full Text: DOI
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