I have been searching the web for a good reference related to solving systems of equations and inequalities of functions rather than subsets of $\mathbb{R}$ and $\mathbb{R}^n$.
I have been thinking in the calculus of variations or functional analysis but i have not found anything on my search.
To be more specific, what I am trying to understand is a general theory for situations like:
Suppose I have a space of functions $\mathcal{F}(x)$ and I am trying to find the set of functions $f(x)$ such that: $f(x)<g(x)$ and $\frac{\partial f(x)}{\partial x}>0$, where $g(x)$ is a known function in the space of functions.
To be even more specific, the particular problem i am trying to solve is:
Suppose there is a known constant $\beta$ between $0$ and $1$, there is also a known functional of $v$, say $f(v)$, I want to find the function $v$ or set of functions $v$ such that: \begin{eqnarray} && \frac{1+ \frac{\partial f(v) }{\partial v}}{2(1-\beta)} \geq 0,\\ && \int^1_0 \frac{1+ \frac{\partial f(v)}{\partial v}}{2(1-\beta)} dv = 1,\\ \end{eqnarray} How can I solve this for any given $f(v)$? Where can i find conditions for existence of such a function $v$?
Does any one has a good book or papers about this? Any information is useful. I know the question is kind of general but the issue is that I am searching for a general theory of this.
Thank you.