My question: I have to find a function $y: \mathbb R \rightarrow \mathbb R$ fulfilling $$y^\prime(t) = f(t, y(t)),\ \int_{-\infty}^\infty y(t) dt = c$$ with a given $c \in \mathbb R$. What is the name of this kind of problem in the theory of ordinary differential equations? Can you suggest some literature about this topic (especially how one can numerically solve such a problem)?
I know initial value problems and boundary value problems. Unfortunately I have not heard about problems in ODE where one has an integral as a side condition...
Reason for my question: Imagine on can prove for the density $\phi$ of a random variable, that $\phi^\prime(t) = f(t, \phi(t))$. In order to find $\phi(t)$ one has to solve $$\phi^\prime(t) = f(t, \phi(t)),\ \int_{-\infty}^\infty \phi(t) dt = 1$$
Check this article: :) They call it "ODE with integral boundary conditions"
http://www.sciencedirect.com/science/article/pii/S0377042702003710