 2022-03-27

How do I get an estimate for this nonlocal ODE?
Consider the following nonlocal ODE on $\left[1,\mathrm{\infty }\right)$:

$f\left(1\right)=\alpha$
$\underset{r\to \mathrm{\infty }}{lim}f\left(r\right)=0$
where l is a positive integer and $\alpha$ is a real number.
Define the following norm $‖·‖$
$‖f{‖}^{2}:={\int }_{1}^{\infty }{r}^{2}f\text{'}\left(r{\right)}^{2}dr+l\left(l+1\right){\int }_{1}^{\infty }f\left(r{\right)}^{2}dr$
I want to prove the estimate:
$‖f‖\le C\sqrt{l\left(l+1\right)}|\alpha |$
for some constant C independent of $\alpha$, l and f. But I am stuck.
Here is what I tried. Multiply both sides by f and integrate by parts to get:

where I used Cauchy-Schwartz in the before last line. I am not sure how to continue and how to get rid of the f'(1) term.
Any help is appreciated. Jonathon Hanson

Step 1
Here is how we deal with the f'(1) term:

$\begin{array}{rl}{f}^{\prime }\left(1\right)& =-{\int }_{1}^{\mathrm{\infty }}{f}^{″}\left(r\right)dr\\ & =-{\int }_{1}^{\mathrm{\infty }}\left[-\frac{f\left(1\right)+{f}^{\prime }\left(1\right)}{{r}^{4}}+\frac{l\left(l+1\right)}{{r}^{2}}f\left(r\right)-\frac{1}{r}{f}^{\prime }\left(r\right)\right]dr\end{array}$

Step 2
Since ${\int }_{1}^{\mathrm{\infty }}\frac{1}{{r}^{4}}dr<1$, we can just solve for f'(1) and use Cauchy Schwartz to estimate it by |f(1)| and ∥f∥. Then I can easily get the estimate I wanted.

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