Solving simultaneous equations with `min{}` function

Question

I have following system of m number of simultaneous equations with min{} function. These equations are symmetric as well.

x1=min{a+b*x2, c};
x2=min{a+b*x3, c};
.
.
.
xm=min{a+b*x1, c};

Is it necessary that x1=x2=...=xm must hold because of the symmetric nature of equations? Is there any theorem which proves or disproves the equivalence of the variables?

Answer 1

Let's consider simpler similar system of equations

Let b <> 0 (otherwise, if b == 0, x1 = x2 = a)

x1 = a + b*x2

x2 = a + b*x1

from here you can find that

x1 = x2 = a/(1-b), b <> 0

And this solution will be effective for the system of equations:

x1 = a + b*x2

x2 = a + b*x3

...

xn = a + b*x1

and x1 = x2 = x3 = ... = xn = a/(1-b)

Now we can apply the results to your system of equations, and solution is:

x1 = x2 =x3 = ... xn =

= a/(1-b), if b <> 0 and a/(1-b) < c

= c, if b <> 0 and a/(1-b) >= c

= a, if b == 0 and a < c

= c, if b == 0 and a >= c

Hope it helps