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?



Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.