Suppose we have a system of equations: x 1 = f 1 ( x 1 , . . . , x n ) x 2 = f 2 ( x 1 , . . . , x n ) . . . x n = f n ( x 1 , . . . , x n )where the f i are components of the gradient ∇ f for a monotone increasing, concave function f, so f i &gt; 0 and ∂ f i ∂ x i ≤ 0.Does the system of equations necessarily have a solution? And is it unique if it exists?

Question

Suppose we have a system of equations:      x    1    =      f    1    (      x    1    ,  .  .  .  ,      x    n    )        x    2    =      f    2    (      x    1    ,  .  .  .  ,      x    n    )    .  .  .        x    n    =      f    n    (      x    1    ,  .  .  .  ,      x    n    )where the       f    i   are components of the gradient       ∇    f for a monotone increasing, concave function   f, so       f    i    &amp;gt;  0 and                     ∂                    f        i                            ∂                    x        i              ≤  0.Does the system of equations necessarily have a solution? And is it unique if it exists?

Milton Hampton · Accepted Answer

Let   g  =  f  +  h where   h  (      x    1    ,  .  .  .  ,      x    n    )  =  −      1    2        Σ    i        x    i    2  Then   g is a strictly concave function, because it is the sum of a concave and a strictly concave function. And stationary points of concave functions are in general global maxima, so uniqueness follows by taking the derivative of   g.

Suppose we have a system of equations: x_1=f_1(x_1,...,x_n) x_2=f_2(x_1,...,x_n) x_n=f_n(x_1,...,x_n) where the f_i are components of the gradient triangledownf for a monotone increasing, concave function f, so f_i>0 and dfi/dxi<=0.

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