Usually, we are interested in finding the Frechet derivative of a given functional. My problem is the opposite; to find a functional satisfying a condition given in terms of the Frechet derivative of the functional.
Let the Frechet derivative of the unknown, possibly nonlinear, functional $\Xi : L_2(0,1) \rightarrow \mathbb{R} $, be defined as the linear operator $D\Xi_f : L_2(0,1) \rightarrow \mathbb{R} $ such that
\begin{align} \lim_{h\rightarrow 0} \frac{||\Xi[f+h] -\Xi[f] - D \Xi_f[h]||_\mathbb{R}}{||h||_{L_2(0,1)}} = 0, \end{align} and satisfying a given condition \begin{align} D \Xi_f[h] = \left<\eta[f],h\right> \end{align} where $\eta : L_2(0,1) \rightarrow L_2(0,1) $ is a known functional.
Is there any systematic approach to find such a functional $\Xi$, besides trial and error?
More specifically, I am interested in situations where $\eta$ takes the form \begin{align} \eta[f](x) = \int_x^1 (2-x) \phi(f(x)) dx \end{align} and where $\phi : L_2(0,1) \rightarrow \mathbb{R} $ is a low order polynomial functional. For example $\phi(f) = f$ or $\phi(f) = f^2$.
Any suggestions on
- Conditions $\eta$ must satisfy for a functional $\Xi$ to exist
- General solution strategies or tricks
- Specific solutions for the specific $\eta$ provided above
will be appreciated.