Norm of the Iterated Operator

Hi **MHF** members,

I have a problem where I can reduce it to an integral equation of the form

$\displaystyle x(t)=(Tx)(t)$ for $\displaystyle t\in[a,b]$,

where

$\displaystyle (Tx)(t):=\int_{a}^{t}\int_{u}^{b}\frac{(t-u)^{k-1}}{(k-1)!}\frac{(v-u)^{n-k-1}}{(n-k-1)!}x(v)\mathrm{d}v\mathrm{d}u$ for $\displaystyle t\in[a,b]$.

Here, $\displaystyle 0\leq k\leq n-1$.

I tried to show that

$\displaystyle \lim_{k\to\infty}|(T^{k}x)(t)|=0$ for $\displaystyle t\in[a,b]$.

But I could **not** find it.

Although I could prove what I need, I wonder to see if this is possible.

Thanks.

**bkarpuz**

Re: Norm of the Iterated Operator

Hey bkarpuz.

Can you look firstly at the absolute value of the integral and show it goes to 0? (i.e. show that the factorials go to infinity faster than the terms in the numerator)

You can look at the norms for the numerators and show that |f|^(k-1) / (k-1)! goes to 0 (or the whole term) where f will be the highest value for the (t-u) term.

You also have norm properties like the triangle inequality and the banach inequality (regarding the product of norms) since the operator is continuous (which means you can use the banach identities).