I have the following equation where I look for a function $L:\mathbb R\to \mathbb R$ $$ f\big(L(x_1)-L(x_2)\big)+c f''\big(L(x_1)-L(x_2)\big) = f(x_1-x_2) $$ Here $f$ is an at least odd smooth function from $\mathbb R\to \mathbb R$, c is a constant. $f$ is also monotone increasing. I guess existence of $L$ is equivalent to the similar question (interchanging the left and right side) or perhaps the more easier version (but I guess equivalent) $$ f(x_1-x_2)+c f''(x_1-x_2) = f(K(x_1)-K(x_2)). $$ Since $f$ is monotone increasing and continuous I can write the latter as $$ f^{-1}\big(f(x_1-x_2)+c f''(x_1-x_2)\big) = K(x_1)-K(x_2). $$ We can add additional assumptions to $f$ it and I am asking myself, when does a function $K$ (or $L$) exists to solve this equation? In the end I don't even know how to start since it is not something like $f\circ K$, to be more precise, I have no clue at all.
Do I need a variaction of Inverse/Implicit Function Theorem? What about power series approach, assuming $K$ and $f$ having a series-representation on $\mathbb R$ I can calculate my $a_n$ recursively but what does this say about existence?
Any other trick? In which direction should I look? Thanks for all tips in advance
Additional information, this is a special case of my problem here