How to solve this set of equations with definite integrals?

Question

I need to find a solution (exact, numeric or even a rough approximate) to this set of $n$ equations:

$$x_i = \int _0 ^1 \frac{ \frac{x_i}{(m_i - t)^2 + p} }{\sum ^{n}_{k=1} \frac{x_k}{(m_k - t)^2 + p}}dt, $$

where $p, m_1, m_2, ... m_n$ are non-zero constants between 0 and 1, $x_1, x_2, ... x_n$ are unknowns.

Any ideas what can be done with it?

Context: As a part of an algorithm, I need to calculate the surface of a section of an area divided into sub-areas of shapes determined by specific functions of $m_i$ and $p$ and normalised. However, the normalisation leads to this set of equations that I don't know how to solve.

Answer 1

Write $f_i(t) = \frac{1}{(m_i - t)^2 + p}$ and consider the function

$$ F(x_1,\cdots,x_i) = \sum_{i=1}^{n} \left( \int_{0}^{1} \frac{x_i f_i(t)}{\sum_{k=1}^{n} x_k f_k(t)} \, dt \right) \mathbf{e}_i. $$

Here, $\mathbf{e}_1,\cdots,\mathbf{e}_n$ is the standard basis of $\mathbb{R}^n$. We seek to find its fixed points. A straightforward observation is that the image of $F$ lies on the hyperplane $x_1 + \cdots + x_n = 1$, and if you restrict yourself to non-zero solutions, then we may consider the $(n-1)$-simplex

$$\Delta^{n-1} = \{ (x_1, \cdots, x_n) : 0 \leq x_i \leq 1 \text{ and } x_1 + \cdots + x_n = 1 \}$$

Numerical observation. When $n = 3$, a numerical simulation seems to suggest that any fixed point of $F$ should lie on the boundary $\partial \Delta^2$. For instance, the following simulation

$\hspace{8em}$

demonstrates the trajectory of iterations $\mathbf{x}, F(\mathbf{x}), F^{\circ2}(\mathbf{x}), \cdots$ starting from various points $\mathbf{x}$ in $\Delta^2$, where $m = (0.766588, 0.883845, 0.0879867)$ and $p=0.401618$.

Using the same parameter, the following is the graph of $\Delta^2$ where each point $\mathbf{x}\in\Delta^2$ is colored according to the value of $\|F(\mathbf{x}) - \mathbf{x}\|$. (The color becomes darker as the value gets closer to $0$.)

$\hspace{8em}$

Possible approach. Numerical simulations for various choices of parameters $m_i$ and $p$ suggest the following conjecture:

Conjecture. For $n\geq 3$, every fixed point of $F$ in $\Delta^{n-1}$ lies on the boundary $\partial \Delta^{n-1}$.

If this is true, we may recursively apply this claim to lower the dimension to $n = 2$. This amounts to investigating $F$ on each line segment $\overline{\mathbf{e}_i\mathbf{e}_j} = \{ (1-x)\mathbf{e}_i + x\mathbf{e}_j : 0 \leq x \leq 1 \}$ in the skeleton of $\Delta^{n-1}$. Then excluding the trivial fixed points $\mathbf{e}_1, \cdots, \mathbf{e}_n$, it boils down to solving the system

$$ 1 = \int_{0}^{1} \frac{(m_i - t)^2 + p}{(1-x)(m_i - t)^2 + x (m_j - t)^2 + p} \, dt \tag{1} $$

for $x \in (0, 1)$ and $i \neq j$. Denoting the right-hand side by $g(x)$, notice that $g$ is strictly convex and $g(0) = 1$. So the existence of $x \in(0, 1)$ with $g(x) = 1$ is equivalent to having both $g'(0) < 0$ and $g(1) > 1$. Then as the symmetry in the role of $m_i$ and $m_j$ suggests, this two conditions reduces to

$$ (m_j - m_i)\int_{0}^{1} \frac{m_i + m_j - 2t}{(m_i - t)^2 + p} \, dt > 0, \qquad (m_i - m_j)\int_{0}^{1} \frac{m_i + m_j - 2t}{(m_j - t)^2 + p} \, dt > 0. \tag{2} $$

Summarizing,

Conclusion. Assuming that the conjecture is true,

1. $\mathbf{e}_1, \cdots, \mathbf{e}_n$ are fixed points of $F$, and
2. Any other fixed points must lie on the interior of the line segment $\overline{\mathbf{e}_i\mathbf{e}_j}$. Parametrizing interior points of $\overline{\mathbf{e}_i\mathbf{e}_j}$ by $\mathbf{x} = (1-x)\mathbf{e}_i + x\mathbf{e}_j$ for $x \in (0, 1)$, $\mathbf{x}$ is a fixed point exactly when $x$ solves $\text{(1)}$. Also, existence of such $x$ is equivalent to $\text{(2)}$.