Find a norm of the inverse of the operator $A: \{ f \in C^1[0,1] \mid i f(0) = -f'(1) \} \to C[0,1]$, $$ A: f \mapsto f' + f. $$ We consider a supremum norm in both $C[0,1]$ and $C^1[0,1]$.
I can show that $A$ provides a bijection and the inverse map is $$ (A^{-1}g)(x) = \left( \int_0^x g(t) e^t \, dt + \frac{\int_0^1 g(t) e^{t-1} \, dt - g(1)}{i - e^{-1}} \right) e^{-x}. $$ It's easy to see that $A^{-1}$ is bounded (unlike $A$). But I have difficulties with finding its norm. I have a hypothesis that $$ \max_{x \in [0,1]} |(A^{-1}g)(x)| = |(A^{-1}g)(0)|. $$ If it's true, then we can consider a sequence of functions $g_n$ such that $g_n(x) = 1$ for $x \in [0, 1 - \frac1n]$, $g_n(1) = -1$ and linear at $[1-\frac1n, 1]$ to show that the norm of $A^{-1}$ is $\frac{2 - e^{-1}}{|i - e^{-1}|}$. But I have difficulties with showing it (and actually, not sure whether it's true).