Alternative solution to fixed point iteration, Cauchy problem

I am solving some Cauchy problems with implicit Euler and Crank Nicolson method.

Obvioulsy it's necessary, with these methods, solve a non linear equation. So I choose fixed point iteration for some problems, but in this particular case it's not possible.

Problem

Let $y'(t)=-\lambda y(t)$ where $y(0)=1$ and $\lambda > 0$

1) Why doesn't the fixed point iteration work for this Cauchy problem when $\lambda$ is big?

2) What could be a different solution for this problem? And solve it.

My idea

As the field is numerical analysis, The first think I thought is about the propagation of error, so I guessed fixed point iteration doesn't work because the error becomes so big when $\lambda > 0$, but I don't know how this is possible. Also I thought to other method, like Newtown's Method, now I don't know how to solve it and if Newton's Method is a good solution for this problem.