# Higher Order Functional Equations

A common point of study is the theory of functional equations first encountered in Calculus and from there built up with the calculus of finite differences (And ultimately functional analysis) which treats problems such as the one below

$$a_0(x) + 2 f(x) + 3 f(x+1) = 0$$

Where the objective is to find the Function $f(x)$ that satisfies the above law.

Consider now the Idea that the above expression can be rewritten as:

$$g(x,f(x)) = 0$$

Where G denotes a 'functional' of both f and x. We could very well have problems such as:

$$g(x,f(x) + 1) + 3*g(x,f(x)) + 3*g(x,f'(x+1)) = 0$$

Where we are attempting to determine the functional g such that over all numbers x, and all functions f, the above is satisfied.

Naturally this leads to increasingly complex notions of 2nd level functionals (who input first level functionals etc...)

And this is where my question arises: What field of math is the theory of all the functionals, functionals of functionals etc... studied under? It seems the Calculus of Variations only explores up to the second level and not nearly as widespread over the range of equations as modern Differential and Partial Differential Equation theory are.

And neither of the above 3 explore much past finite differences to other strange functional equations.

• I'm not very expert in this stuff, but months ago I was reading something about difference algebras (algebras equipped with a function $\sigma(y)=f(x+1)$) and equations inside that structures. I hope can help. PS: I'm searching for the same theory too. – MphLee Mar 7 '15 at 15:14