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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 $u(0)=v(0)=0$.

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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 $r_1=k_2=2,\alpha_1=\alpha_2=0.5,k_1=1,r_2=3$

asymmetric stream

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:

symmetric stream

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  • $\begingroup$ Thank you for your answer Sir, To be honest with you , I knew about competitive Lotka–Volterra model, but as I wrote above, I'm looking for an example in a Banach space other than $\mathbb R$ $\endgroup$ – Motaka Oct 9 '19 at 8:21
  • $\begingroup$ @Motaka Sorry I couldn't be of much help. Based on your reply to prof. Israel, I got the impression that you are not so familiar with these systems. IMO, the two terms Banach space other than R and application in physics, biology, etc. sound a bit contradictory. Because in real world applications, we typically deal with Euclidean space and $\mathbb R$. And I doubt that this specific form of dynamical system in its special form could have any application other than population dynamics. $\endgroup$ – polfosol Oct 9 '19 at 8:47
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    $\begingroup$ @Motaka I finally got the time to do as promised. Hope this helps $\endgroup$ – polfosol Oct 12 '19 at 17:45
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    $\begingroup$ @Motaka This paper might be of interest (and the references therein, of course): doi/10.1080/17513758.2019.1568600 $\endgroup$ – polfosol Oct 14 '19 at 13:38
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    $\begingroup$ Nevermind @Motaka $$\;$$ That's fine ;) $\endgroup$ – polfosol Oct 22 '19 at 16:46
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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.

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    $\begingroup$ Thank you for your reply, could you give me some sources where I can read about this? $\endgroup$ – Motaka Oct 3 '19 at 12:11

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