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I'm trying to solve the next problem: Let $c\in\mathbb{R}_{+}$and $u,v$ be continuous and postive functions from $\mathbb{R}_{+}$ to $\mathbb{R}$ such that for all $x\in\mathbb{R}_{+}$, $u\left(x\right)\leq c+\int_{0}^{x}u\left(t\right)v\left(t\right)dt$. Prove that $u\left(x\right)\leq c\cdot\exp\left(\int_{0}^{x}v\left(t\right)dt\right)$ for all $x\in\mathbb{R}_{+}$.

Using the first inequality, $u\left(x\right)\leq c+\int_{0}^{x}u\left(t\right)v\left(t\right)dt$, taking exponential in both sides, we have that $$ \exp u\left(x\right)\leq\exp\left(c+\int_{0}^{x}u\left(t\right)v\left(t\right)dt\right)=\exp c\cdot\exp\left(\int_{0}^{x}u\left(t\right)v\left(t\right)dt\right). $$

Using the fact that for all $x\in\mathbb{R}$ $x\leq\exp x$, we have that $u\left(x\right)\leq\exp c\cdot\exp\left(\int_{0}^{x}u\left(t\right)v\left(t\right)dt\right)$. Also using a kind of intermediate value theorem for integrals I have for the integral $\int_{0}^{x}u\left(t\right)v\left(t\right)dt$, that there exists $w\in\left[0,x\right]$ ($w$ could depend on $x$) such that $\int_{0}^{x}u\left(t\right)v\left(t\right)dt=u\left(w\right)\cdot\int_{0}^{x}v\left(t\right)dt$. I have tried that, but I couldn't obtain the inequality that we have to prove in the problem. Could you help me or give me some suggestions?

Thanks.

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