I'd like to do the following exercise from my book. The statement is as follows.
Let $\gamma \epsilon { C }^{ 1 }(R\times { R }^{ n })$,$c>0$ and ${ \gamma }_{ 0 }\epsilon { L }^{ 1 }({ R }^{ n })$ such that $\left| { \gamma }(z,x) \right| \le { \gamma }_{ 0 }(x)+c\left| z \right| $.
And let $G(f);=\int _{ { R }^{ n } }^{ }{ { \gamma }(f(x),x)dx } $
Show that $G({ L }^{ 1 }({ R }^{ n }))\rightarrow R$ is Frechet differentiable and calculate the derivative.
It's the first exercise for Frechet differentiability after it was introduced so im not very fermiliar with it.
We need to show that $\lim _{ h\rightarrow 0 }{ \frac { \left| G(u+h)-G(u)-G'(u)h \right| }{ { \left\| h \right\| }_{ { L }^{ 1 } } } } =0$
I started looking at the numerator so we have
$\left| G(u+h)-G(u)-G'(u)h \right| =\int _{ { R }^{ n } }^{ }{ \gamma (u(x)+h(x),x)-\gamma (u(x),x)+\gamma (u(x),x)h(x)dx } $
=$\iint _{ u(x)+h(x) }^{ u(x) }{ \gamma (s,x)ds-\gamma (u(x),x)h(x)dx} $ =$\iint _{ u(x)+h(x) }^{ u(x) }{ \gamma (s,x)-\gamma (u(x),x)dsdx } $
but how can i now make this expression smaller than an arbitrary $\varepsilon >0$? is this calculation even usefull? And how can i use the Assumtion for $\gamma $? Any Help is greatly appreciated Thank you
