Systems of nonlinear algebraic equations with positive solutions

We are concerned with the positive solutions of an algebraic system depending on a parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha> 0$\end{document}α>0 and arising in economics. For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha> 1$\end{document}α>1 we prove that the system has at least a solution. For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$0<\alpha<1$\end{document}0<α<1 we give three proofs of the existence and a proof of the uniqueness of the solution. Brouwer’s theorem and inequalities involving convex functions are essential tools in our proofs.


Introduction
Algebraic systems with positive solutions appear in a large variety of applications. Nonlinear systems of the form F(x) = Ax + p, x ∈ R n + , where A is a positive matrix and p a non-negative vector, are investigated in []. Generalizing some results from [], the existence of a positive solution is proved with Brouwer's theorem, and the uniqueness is a consequence of suitable inequalities. Algorithms for finding the solution are also presented in []. Several problems can be converted into systems of this form (see also []): second order Dirichlet problems, Dirichlet problems for partial difference equations, third and fourth order difference equations, three point boundary value problems, steady states of complex dynamical networks, etc.
Extending the results of [], systems of the more general form γ i (x i ) = n j= g ij (x j ),  ≤ i ≤ n, are studied in [], together with a supplementary list of applications. The existence of a positive solution is proved by using a monotone iterative method; the proof of the uniqueness is based on an extension of the method used in [].
Several classes of other systems and several methods to investigate the existence/ uniqueness of their positive solutions are described, together with applications, in [-] and the references therein.
In this paper we consider an algebraic system which appears in some problems from economics, for example in establishing uniqueness of equilibrium of some models of trade with increasing returns: see [].
Our proofs use Brouwer's fixed point theorem and properties of minimum points of convex functions. Several inequalities, in particular inequalities related to convexity, are instrumental in these proofs.
The system addressed in this paper is described in what follows. Let N ≥  be an integer. Given the real numbers α > , a ni > , b n > , i, n ∈ {, . . . , N}, consider the system of  .
Denote F := (F  , . . . , F N ); then () is equivalent to is a weighted mean of the numbers from U i , and so for all s = (s  , . . . , s N ) such that s  > , . . . , s N > .
]. In any case, due to () we can consider the continuous function F : V → V . Since V is compact and convex, Brouwer's theorem guarantees the existence of a solution s  ∈ V to (). Then s   , . . . , s  N will satisfy (). Now let is a solution to (), and consequently t  : In the next example we present a system () with three solutions in D. Example . Let a  = a  = a  = a  = b  = b  = . Then (t, t) is a solution of () for all  < t < .

The case 0 < α < 1: existence and uniqueness of the solution
We begin with another proof of the existence of solution in D to the system () if  < α < .
Fix an >  sufficiently small, such that N ≤ b  + · · · + b N and The next theorem offers a third proof of the existence and also a proof of the uniqueness of the solution to () in the case  < α < .
Theorem  If  < α < , the system () has a unique solution in D.
The proof of the theorem is divided into the following steps.
() f is strictly convex. Indeed, let s, t ∈ K , s = t. Since the function u → u α (u ≥ ) is strictly concave, we have with at least one strict inequality. Thus The function log is strictly increasing and strictly concave, so that It follows immediately that which means that f is strictly convex.
() f : K → R has a global minimum point x, and x ∈ D.  For  < r < R, let K r,R := {t ∈ K | r ≤ t ≤ R}. Let s = (, , . . . , ) ∈ K . Due to () and (), there exist r and R such that, for all t ∈ K \ K r,R , Since K r,R is compact, the continuous function f restricted to K r,R has a global minimum point in K r,R ; let us denote it by x ∈ K r,R . Due to (), s ∈ K r,R , and so f (x) ≤ f (s) < f (t) for all t ∈ K \ K r,R . It follows that x is a global minimum point of f : K → R.
Suppose that x / This contradicts the global minimality property of x and the proof of Theorem  is finished.