# Nonlinear variation of constants formula

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 $$\dim X=+\infty$$ ?

• It is not so easy to find the paper you refer to. Can you put the main result in your question, or link to a copy? Jul 2, 2021 at 1:07
• This nonlinear variation of constants formula is quoted in this answer, together with more accessible references to where it is further discussed and proved. I don't see anything specifically finite dimensional about the formula, so it might just work, but I haven't checked the proofs. Jul 2, 2021 at 7:59
• Many thanks for the comments. The equation may be an EDP,.. so $f$ is defined only on a domain, for instance $f(x)=Ax+f_1(x),$ where $A$ generates a continuous semigroup and $f_1$ is locally Lipschitz, so it is not obvious to adopt the proof of finite dimension case. Jul 2, 2021 at 10:12