I have an existence/uniqueness theorem for the PDE $u_t = a(x,t)u_{xx} + b(x,t)u_x + c(x,t)u - g(x,t).$
Now if I have a Gateaux derivative of a map $F$ at a point $p$ satisfying $DF(p)v = v_t - f_1v_{xx} - f_2v_x - f_3v$
(the $f_i$ are functions of $(x,t)$) then how can I use my PDE result to say that $DF(p)$ is invertible? I thought I could rearrange to get $v_t - f_1v_{xx} - f_2v_x - f_3v - DF(p)v = 0$ and I want to put the $f_3$ and $DF(p)$ together so that it is in the form of the PDE and I can just quote the existence result which in this cases would tell me that there is a unique $v$ satisfying this equation and hence $DF(p)$ is invertible. But all I know is the the derivative is linear and I can't put the $f_3$ and the $DF(p)$ together.
Thank you