I'm looking to study the existence solutions of the following coupled equation:


\begin{equation}
\left\{\begin{matrix}
x(t)&=&\int\_{0}^{t} K\big(t, s\big) f\big(s, x(s),y(s)\big) d s, \quad t \in[0,1) \\ 
y(t)&=&\int\_{0}^{t} K\big(t, s\big) f\big(s, y(s),x(s)\big) d s, \quad t \in[0,1) 
\end{matrix}\right.
\end{equation}
where $K \in L^{1}$ is an scalar kernel, $B$ is a Banach space with the norm $\|.\|$ and $g:[0, T] \times$ $B \rightarrow B$.


The integral here is understood to be the Pettis integral and solutions will be sought in $B:=C([0, T], X)$, where $X$ is a Banach space.


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.



Applications of coupled Volterra-Hammerstein in Banach space
[Solvability of Coupled Systems of Generalized Hammerstein-Type Integral Equations in the Real Line](https://www.mdpi.com/2227-7390/8/1/111/htm) (2020): section 4 gives an application in mechanics, a study of the coupling between bending and torsion of two coupled beams on an elastic foundation, motivated by the dynamics of ice plates on waves.


