# If $u\left(x\right)\leq c+\int_{0}^{x}u\left(t\right)v\left(t\right)dt$ then $u\left(x\right)\leq c\exp\left(\int_{0}^{x}v\left(t\right)dt\right).$

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.