One can solve a delay differential equation (like for example $f'(x)=f(x-1)$) if we have a function as a bounded condition (in my example we need to know $f$ on $[0,1)$) and then use a simple forward Euler scheme. This is however not possible if $f'$ also depends on the future. One method which seems to work is to integrate the equation on both sides and employ a fixed point iteration.
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This equation can be solved exactly: https://mathoverflow.net/questions/114875/ – Alexandre Eremenko Jun 10 '18 at 14:01
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Your reference is interesting, but I am not really looking for a solution to $f'(x)=f(x+1)+f(x-1)$ but more a general method to solve equations of this type (dependence on future and past states. Also for non-linear equations like for example: $f'(x)=f(x+1)^2 + f(x-3)^7$ – Darkwizie Jun 10 '18 at 22:20