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I want to derive a Gronwall-type inequality from the inequality below. Here all the functions are nonnegative, continuous and if you need some assumptions you may use that. $$ f^2(t) \leqslant g^2(t) + \int_0^t (f(s) +c) f(s) ds \;\;\;\; (t \in [0,T]) $$ So please help!

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up vote 2 down vote accepted

If $c >0$ then by the Young's inequality, $$ \begin{eqnarray*} f^2 (t) &\leqslant& g^2 (t) + \int_0^t (f^2 (s) + \frac{c^2 + f^2 (s)}{2} ) ds \\ & \leqslant & g^2 (t) + \frac{c^2}{2}T + \int_0^t \frac{3}{2} f^2 (s) ds \end{eqnarray*}$$ So you can apply the Gronwall's inequality now.

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