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 $u(0)=v(0)=0$.
 A: 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.  
A: I assume that you are already familiar with the concept of dynamical systems and their state space representation. As Robert Israel said, your system is described by
$$\eqalign{\dot u &= f(t,u(t),v(t))\\
            \dot v &= f(t, v(t),u(t))\cr}$$
This very much reminds me of the Competitive Lotka–Volterra model:
$$\begin{cases}\dot u=r_1 u-k_1(u^2+\alpha_1 uv)\\ \dot v=r_2 v-\;k_2(v^2+\alpha_2 uv)\end{cases}$$
But the differences are: the Lotka-Volterra (AKA predator-prey) model is an autonomous system while your model is non-autonomous (i.e. explicitly dependent on $t$). Furthermore, the origin is a fixed point (equilibrium point) of Lotka-Volterra, so the initial state cannot be $u(0)=v(0)=0$. You may also be interested in reading about Lanchester's model but that's quite similar to the LV.
Roughly speaking, I agree with prof. Israel that your equations describe some kind of population dynamics. The predator-prey is a special case of these dynamical systems. In your equation, $u$ can be regarded as the population of a species that is related to another one, $v$. And the relationship between these two can be described by the same non-autonomous function.
Addendum: I try to give a more thorough analysis of the competitive Lotka-Volterra model, since it seems to be the only one that has been suggested so far.
This system has four equilibrium points in the $(u,v)$ plane:
$$(0,0),\quad(r_1/k_1,0),\quad(0,r_2/k_2),\quad\left(\frac{\alpha _1 k_1 r_2-k_2 r_1}{\left(\alpha _1 \alpha _2-1\right) k_1 k_2},\frac{\alpha _2 k_2 r_1-k_1 r_2}{\left(\alpha _1 \alpha _2-1\right) k_1 k_2}\right)$$
For positive values of $\alpha_i,k_i,r_i,\;(i=1,2)$ the first point is an unstable equilibrium point, the second and third ones are saddle points and the last one is stable.
Here is a stream plot of trajectories for $r_1=k_2=2,\alpha_1=\alpha_2=0.5,k_1=1,r_2=3$

If the same function is used to describe $\dot u$ and $\dot v$, then the only change that we will observe in the trajectories is that they will be symmetrical around the line $u=v$. For example, let $r_1=r_2=2$, $k_1=k_2=1$ and $\alpha_1=\alpha_2=0.5$, then:

