The following is paraphrased from Introduction to integral equations with applications by Abdul Jerri:
Let $F$ be a functional with domain $D$. Start with $F(x,t,u(t))$. If $F$ has a continuous partial derivative $\partial F / \partial u$ in $D$, then we can use the mean-value-theorem:
For any $u_1(t)$ and $u_2(t)$ in $D$, there is an $\eta(t)$ between them, $u_1(t) < \eta(t) < u_2(t)$, such that $\displaystyle\frac{F(x,t,u_1(t)) - F(x,t,u_2(t))}{u_1(t) - u_2(t)} = \frac{\partial F}{\partial u}(x, t, \eta(t))$.
I have two questions:
What precisely is meant by the notation $\displaystyle\frac{\partial F}{\partial u}(x, t, \eta(t))$? (specifically, how is $\displaystyle\frac{\partial F}{\partial u}$ defined?)
How can I generalise this result to suit a functional of the form $F(x, t_1, t_2, u(t_1), u(t_2)$) (where $F$ takes values in $\mathbb R$ and $u_1, u_2 : \mathbb R \to \mathbb R$, if this makes any difference)?
Any explanations, insight or references to relevant sources of information would be greatly appreciated.