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I am having a problem with the following exercise.

Let $f$, $g$ be continuous non-negative functions on $[a,b]$, and let $C$ a positive constant.

Suppose that: $f(x) \leq C+ \int_{a}^x f(t)g(t)dt$,

for all $x \in [a,b]. $ Show that:

$$f(x) \leq C\exp\left(\int_{a}^x g(t)dt\right).$$

Thank you in advance

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2 Answers 2

up vote 1 down vote accepted

Let $h(x):=\int_a^xf(t)g(t)dt$. Then $g(x)(C+h(x))\geq h'(x)$. So $h'(x)-g(x)h(x)\leq Cg(x)$. Now multiplying by $\exp\left(-\int_0^xg(t)dt\right)$, we get $$\frac d{dx}\left(h(x)\exp\left(-\int_0^xg(t)dt\right)\right)\leq Cg(x)\exp\left(-\int_0^xg(t)dt\right),$$ and integrating $$h(x)\exp\left(-\int_0^xg(t)dt\right)\leq C\int_0^xg(u)\exp\left(-\int_0^ug(t)dt\right)du.$$ The RHS is $C-C\exp\left(-\int_0^xg(t)dt\right)$, so $$h(x)\leq C\exp\left(\int_0^xg(t)dt\right)-C.$$ As $h(x)\geq f(x)-C$, we get the wanted result.

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Can you detail a bit more. I do not see how to use this.. –  user43418 Oct 31 '12 at 11:05
    
$h'(x)=f(x)g(x)$ and $f(x)\le C+h(x)$. –  Berci Oct 31 '12 at 11:08
    
Why do I multiply by $\exp\left(-\int_0^xg(t)dt\right)$ ? –  user43418 Oct 31 '12 at 14:02
    
The LHS will be the derivative to a product. –  Davide Giraudo Oct 31 '12 at 14:02
    
what is does LHS stand for? –  user43418 Oct 31 '12 at 14:05

you can use the idea of wansik inequality in ODE

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I can't use that –  user43418 Oct 31 '12 at 11:01
    
I can only use properties regarding differentiation and integrals –  user43418 Oct 31 '12 at 11:01
    
If you need to answer a question in a certain way, you should say so when you ask the question, by providing context and a description of what attempts you have made. It is frustrating to answer a question only to have the goalposts move. –  Douglas S. Stones Oct 31 '12 at 11:08
    
ok sorry about that. I am still new to the website –  user43418 Oct 31 '12 at 11:13
    
That being said, what is the "wansik inequality"? –  Douglas S. Stones Oct 31 '12 at 11:18

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