We study the positive subharmonic solutions to the second order nonlinear ordinary differential equation
$\begin{equation*}u'' + q(t) g(u) = 0,\end{equation*}$
where $g(u)$ has superlinear growth both at zero and at infinity, and $q(t)$ is a $T$ -periodic sign-changing weight. Under the sharp mean value condition $\int_{0}^{T}{q\left( t \right)dt<0}$ , combining Mawhin's coincidence degree theory with the Poincaré-Birkhoff fixed point theorem, we prove that there exist positive subharmonic solutions of order $k$ for any large integer $k$ . Moreover, when the negative part of $q(t)$ is sufficiently large, using a topological approach still based on coincidence degree theory, we obtain the existence of positive subharmonics of order $k$ for any integer $k≥2$ .
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Figure 1.
The figure shows an example of multiple positive solutions for the
Figure 2.
Using a numerical simulation we have studied the
Figure 3.
The figure shows the aperiodic necklaces made by arranging
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