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Complementary to previous investigations in the case $\\int_0^T q(t) < 0$, we provide existence results for a suitable class of weights having (small) positive mean, when $g'(x) < 0$ at infinity. 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Complementary to previous investigations in the case $\\int_0^T q(t) < 0$, we provide existence results for a suitable class of weights having (small) positive mean, when $g'(x) < 0$ at infinity. 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