Let $f:[0,2\pi]\times \mathbb{R} \rightarrow \mathbb{R}$ a differential function satisfying : $\displaystyle k^2\leq \liminf_{|x|\rightarrow \infty} \frac{f(t,x)}{x}\leq \limsup_{|x|\rightarrow \infty}\frac{f(t,x)}{x} \leq (k+1)^2$
Let $(x_n)\subset H^1([0,2\pi],\mathbb{R})=\lbrace x\in L^2([0,2\pi],\mathbb{R}),x'\in L^2([0,2\pi],\mathbb{R}),x(0)=x(2\pi)\rbrace$
such that $\|x_n\|\rightarrow \infty$ when $n \rightarrow \infty$
Why : the sequence $(\displaystyle\frac{f(t,x_n)-k^2 x_n}{\|x_n\|})$ is bounded ?
This is an answer of a professor and I don't understand the line 6.
How to use :
$$\displaystyle k^2\leq \liminf_{|x|\rightarrow \infty} \frac{f(t,x)}{x}\leq \limsup_{|x|\rightarrow \infty}\frac{f(t,x)}{x} \leq (k+1)^2$$
to find : $\displaystyle \| f(t,x_n)\|_{L^2}^2 =\int_0^{2\pi} f(t,x_n)^2 dt \leq \max\lbrace (k+1)^4\int_0^{2\pi} |x_n|^2 dt , 2\pi M\rbrace$
where : $\displaystyle M=\sup_{(t,x)\in [0,2\pi]\times[-a,a]}|f(t,x)|$
Please help me ,
Thank you .
After the details you've provided, the situation is quite different, but anyways the method is almost the same. The first trick is to notice that bounds on $\liminf$ and $\limsup$ tell you, that for any $\delta>0$ there exists $a(\delta)$ big enough such that $$ |x|\geq a(\delta)\implies |f(t,x)|\leq((k+1)^2+\delta)|x|. $$ Let us denote $a:=a(1)$. Then $$ \|f(\cdot,x_n(\cdot))\|_2^2 = \int |f(t,x_n(t))|^2\mathrm dt = \int\limits_{\{t:|x_n|\leq a\}}|f(t,x_n(t))|^2\mathrm dt + \int\limits_{\{t:|x_n|\leq a\}}|f(t,x_n(t))|^2\mathrm dt $$ $$ \leq \int \sup_{t\in [0,2\pi],|x|\leq a}|f(t,x)|^2\mathrm dt + \int ((k+1)^2+\delta)^2|x_n(t)|^2\mathrm dt $$ $$ \leq 2\pi\sup_{t\in [0,2\pi],|x|\leq a}|f(t,x)|^2 + ((k+1)^2+\delta)^2\|x_n\|_2^2. $$ where we always integrate over $t\in[0,2\pi]$ plus other possible constraints. This is a bit different from the bound in your notes, but still similar and works for showing that the sequence is bounded as well. Please tell me, whether it something is still unclear to you.
Another point is that you shall not rely precisely upon the notes of your professor, that often may have just the general idea rather than the perfectly precise sequence of statements, especially when it comes to the asymptotic results.