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Under the sharp mean value condition $\\int_{0}^{T} q(t) ~\\!dt < 0$, combining Mawhin's coincidence degree theory with the Poincar\\'e-Birkhoff fixed point theorem, we prove that there exist positive subharmonic solutions of order $k$ for any large integer $k$. 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Under the sharp mean value condition $\\int_{0}^{T} q(t) ~\\!dt < 0$, combining Mawhin's coincidence degree theory with the Poincar\\'e-Birkhoff fixed point theorem, we prove that there exist positive subharmonic solutions of order $k$ for any large integer $k$. 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