Euler's implisit - convergence of each step

Hi,

**problem:**

the trapezoid method of solving differential equations is,

.

Since the method is implicit, we must solve for on each step by some iterative method. If we use the fixed point iteration , we know that the method will converge if for .

Give a condition on such that the fixed point iteration converges at each step.

**attempt:**

First of all, I need to write down an expression for . Fixed-point iteration needs an initial guess. I use as my initial guess and write as,

.

But now

,

which doesn't make much sense.

What am I doing wrong?

Thanks