How to find a function in Lambda Calculus? Yesterday I have been trying to complete this exercise. I have to find:
$$ ((map)l)t \simeq \lambda k \lambda x ((k)(t)t_1)....((k)(t)t_n)x $$

where $$l=\lambda k \lambda x ((k)t_1)....((k)t_n)x$$
And $t$ is a generic Lambda term.
I tried but without success.
Maybe it will be something like that :
 $$map=(\lambda a \lambda b (\lambda s \lambda d((b(st))d)))$$
Thank you in advance to all will answer
In your attempt,

$$map=(\lambda a \lambda b (\lambda s \lambda d((b(st))d)))$$

$a$ plays the role of $l$, $b$ the role of $t$, $s$ the role of $k$, and $d$ the role of $x$.

So, $b(st)d$ should actually play the role of $l(kt)x$, but here we see some mismatch: $b$ does not represent $l$, $a$ does that. Also, we can not use $t$, but must use $b$ instead. 

Hence, the correct answer seems to be $a(sb)d$. We get (using the standard notational shortcuts)

$$map=(\lambda a b s d.\, a(sb)d)$$

Try to check whether $map\, l\, t$ does $\beta$ convert to the intended term.

Your original idea, namely using $kt$ instead of $k$, seems to be the right intuition, as far as I can see.