What is the type of initial value problem?

Suppose we have system of differential equations:

$$f_i'(x)=A_i(f_1,\cdots,f_n,x),$$

where $A_i$ is nonlinear functional of $f_i$ and $x$.

As initial conditions I have $f_i(x_0)=g_i(x_0,B)$, where $B$ depends on solution: $B=\int^{\infty}_0f_1^2(y)ydy$ (one could take different explicit form).

Is there name for such type of initial value problem? Could you give me any sources where I can read about it?