Which of the following statements are ??

a. Let $\phi$ be a non-negative and continuously differentiable function on $(0,\infty)$ such that $\phi'(x)\le\phi(x)$ for all $x$ $\in (0,\infty)$. Then

$$lim_{x\to \infty}\phi(x)=0$$

b. Let $\phi$ be a non-negative function continuous on $[0,\infty)$ and differentiable on $(0,\infty)$ such that $\phi(0)=0$ and such that $\phi'(x)\le\phi(x)$ for all $x$ $\in (0,\infty)$. Then $\phi=0$.

c. Let $\phi$ be a non-negative function continuous on $[0,\infty)$ and such that $$\phi(x) \le \int_{0}^{x}\phi(t) dt$$ for all $x \in [0,\infty)$. Then $\phi=0$.


Hint(Grownwall's Inequality)

If $$f(t)\leq K+\int_a^tf(s)g(s)ds,$$ then $$f(t)\leq K\exp(\int_a^tg(s)ds)$$

And $K\geq 0$; $f,g$ is nonnegative continuous in $[a,b]$.

  • 2
    $\begingroup$ so (b) and (c) are right?? $\endgroup$ – Kayoken Jan 26 '14 at 5:01
  • $\begingroup$ (a) and (b) are right. $\endgroup$ – gaoxinge Jan 26 '14 at 5:06
  • $\begingroup$ (c) is lack of $\phi(0)=0$ $\endgroup$ – gaoxinge Jan 26 '14 at 5:06
  • 2
    $\begingroup$ how can (a) be right?? suppose $\phi(x)=e^x$?? $\endgroup$ – Kayoken Jan 26 '14 at 5:08
  • 2
    $\begingroup$ in (c) $\phi(0)=0$... $\endgroup$ – Kayoken Jan 26 '14 at 5:10

All of them are wrong.

For statement 1, consider $\phi(x) = e^x$. For statement 2, consider $\phi(x) = e^x - 1$. For statement 3, consider $\phi(x) = e^x$.

  • $\begingroup$ Your examples for statement 2 and statement 3 are wrong. $\endgroup$ – Error 404 Jan 19 '17 at 15:08

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