Suppose that we wish to solve $x'(t)=f(x(t))+g(x(t)), \; x(0)=x_0\in X,$ where $X$ is an infinite dimensional Banach space and $f , g : X \rightarrow X $ are two nonlinear functions. Furthermore, suppose we know how to solve the unperturbed system $y'(t)=f(y(t)),\; y(0)=y_0$, let's say that $y(t)=\phi(y_0,t)$, for some function $\phi$ of $t$ and $y_0$. In the case where $\dim X<\infty,$ one can express the solution $x(t)$ in term of $\phi$ via the so called nonlinear variation of constants formula (due to V. M. Alekseev, (1961). "An estimate for the perturbations of the solutions of ordinary differential equations" Vestnik Moskov Un. Ser, 1, 28-36).
Is there an analogous formula to variation of parameters in the case $\dim X=+\infty$ ?