Euler's implisit - convergence of each step

Hi,

**problem:**

the trapezoid method of solving differential equations is,

$\displaystyle y(t_{i+1}) = y(t_i) + h\frac{f(y(t_{i+1})) + f(y(t_i))}{2}$.

Since the method is implicit, we must solve for $\displaystyle y(t_{i+1})$ on each step by some iterative method. If we use the fixed point iteration $\displaystyle x=g(x)$, we know that the method will converge if $\displaystyle |g'(x)|<1$ for $\displaystyle x\in[y(t_i),y(t_{i+1})]$.

Give a condition on $\displaystyle h$ such that the fixed point iteration converges at each step.

**attempt:**

First of all, I need to write down an expression for $\displaystyle g(x)$. Fixed-point iteration needs an initial guess. I use $\displaystyle y(t_i)$ as my initial guess and write $\displaystyle g$ as,

$\displaystyle g(x) = x + h\frac{f(x)+f(x)}{2} = x + hf(x)$.

But now

$\displaystyle g'(x) = 1 + hf'(x) < 1 \Rightarrow h<0$,

which doesn't make much sense.

What am I doing wrong?

Thanks