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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.

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Solvability of Coupled Systems of Generalized Hammerstein-Type Integral Equations in the Real Line (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.

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    $\begingroup$ I searched for it on the internet --- I haven't been in a library for a decade. $\endgroup$
    – Carlo Beenakker
    Commented Jan 22, 2021 at 15:05

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