Applications of this integral equation in Banach space I’m writing a paper on fixed point theorem, as an application of my main results, I will study this equation: 
\begin{equation}
 \left\{\begin{matrix}
u(t) &=&\int_{0}^{t} f(s,u(s),v(s))ds\,,\: t\in [0,a] 
\\ 
v(t) &=&\int_{0}^{t} f(s,v(s),u(s))ds\,,\: t\in [0,a] \end{matrix}\right.
\end{equation}
where $a$ is a real number such that $a>0$, $E$ a Banach space and $f :[0,a]\times E\times E\rightarrow  E$ a continuous map.
Since I am new to doing research, I want to know if there is an application in physics, biology, population dynamics.. of this system - or this kind of system-.
Are there any existing textbooks/articles/papers about this kind of equation?
Any help would be very much appreciated.

Edit: As it t mentionned by @Robert in his answer, the system is equivalent to the system of differential equations
$$ \eqalign{u'(t) &= f(t,u(t),v(t))\cr
            v'(t) &= f(t, v(t),u(t))\cr}$$
with initial conditions $u(0)=v(0)=0$.

Your system is equivalent to the system of differential equations
$$ \eqalign{u'(t) &= f(t,u(t),v(t))\cr
            v'(t) &= f(t, v(t),u(t))\cr}$$
with initial conditions $u(0)=v(0)=0$.
There are lots of applications of systems of differential equations.  In particular, 
in population dynamics, for $E = \mathbb R$ your system could describe competition between
two species.  
