I'm considering the nonlinear ODE
$$f''(t)+f(t)-\int_0^t f^3(s)ds=0$$
for $f\in C^{2,\gamma}([0,1])$, with Cauchy data $f(0)=u_0, f'(0)=u_1$. With $X=C^{2,\gamma}([0,1])$ and $Y=C^{0,\gamma}([0,1])$, we may use the nonlinear operator $F:X\rightarrow Y$ given by the formula
$$F[f](t)=f''(t)+f(t)-\int_0^t f^3(s)ds$$ and take its Fréchet derivative at the origin: $DF[0]:f\mapsto f''+f$. Using a variant of the contraction principle I have that $F$ is locally invertible in a neighborhood around the origin if $F[0]=0$ and $DF[0]$ is invertible. However, in this case $DF[0]$ is surjective, but has a two-dimensional kernel: $\ker DF[0]=\{a\cos t+b\sin t: a,b\in\mathbb{R}\}$. Therefore, since $DF[0]$ is Fredholm I used the decomposition $X=\ker DF[0]\oplus X_1$ to conclude that the restriction of $F$ to $X_1$ is indeed locally invertible around zero.
What is left for me to do is to somehow show that if $|u_0|$ and $|u_1|$ are sufficiently small, then there exists a unique solution near the origin to the original equation. However, after trying for a few days I'm completely stuck. I tried the ansatz $f=f_0+f_1$, where $f_0$ is the part of $f$ in $\ker DF[0]$ and $f_1$ is the part of $f$ in $X_1$, but I just can't see it leading anywhere.
Alternatively, I tried to completely abandon the operator theory approach altogether and just use the original equation to state that if $F[f]=0$, then $f$ must be smooth. Then I derived formulas for $f^{(n)}$, $n>2$, in order to get an estimation of $|f^{(n)}(0)|$ so that I could potentially use convergence of a power series in my argument. However, I couldn't find an effective way to bound $|f^{(n)}(0)|$ sufficiently (although I checked numerically that $|f^{(n)}(0)|$ would indeed be sufficiently bounded for my purposes).
What is it that I'm missing? Is there another approach or detail that I'm seemingly blind to finish the argument off?