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I am extending Example 2.2 on this sheet.

Suppose $f(x(s),s)$ is such that $|f(x(s),s)-f(y(s),s)|\leq K |x-y|$ for some $K>0  ---(1)$

and $x,y\in C[0,t_f]:\,\,\,t_f<\infty$

Also, let $T$ be such that $[T(x)](s)=\int f(x(s),s)ds + x_0 --- (2)$. 

But this time let  $\|x\|_c=\sup\limits_{s\in [0,t_f]}|x(s)\exp(-cx)|$ for some $c\in \mathbb R---(3)$

I want to find the strictest condition possible on $c$ such that $T$ is a contraction.

I know that it is a function of $RK$... and that we should plug (1) into (2) then in turn into (3). But I don't know how to deal with the $\exp$ to get the (strictest) conditions.

Added: in other words, forwhat values of $c$ wouldI get $T$ to be a contraction map?

Thanks.

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