Existence of Positive Solutions for a Class of Nonlinear Algebraic Systems

Based on Guo-Krasnoselskii’s fixed point theorem, the existence of positive solutions for a class of nonlinear algebraic systems of the form x = GF(x) is studied firstly, where G is a positive n × n square matrix, x = col(x 1 , x 2 , . . . , x n ), and F(x) = col(f(x 1 ), f(x 2 ), . . . , f(x n )), where, F(x) is not required to be satisfied sublinear or superlinear at zero point and infinite point. In addition, a new cone is constructed in R. Secondly, the obtained results can be extended to some more general nonlinear algebraic systems, where the coefficient matrix G and the nonlinear term are depended on the variable x. Corresponding examples are given to illustrate these results.


Introduction
Many problems arising from economics [1][2][3][4], complex network [5][6][7], mechanical engineering [8], Verschelde's web page [9], and mathematics [10][11][12][13] can be formulated as a system of equations.The existence problem of a solution for a system of equations is called the zero point problem.This problem is associated with a function  :  →   , where  is a subset of the -dimensional Euclidean space   .A point  ∈  is a zero point of  if the image of  is the origin, and  is a fixed point of  if  is a zero point of the function  given by () = () −  and thus the image () of  is  itself.
In applications, positive solutions for a system of equations are important; see [1, 2, 4-7, 9-13, 17-23, 23-29].For example, in the study of discrete dynamical systems [5] or continuous time dynamical systems [6,7], a positive solution for a system of equations represents the existence of stationary states.In another example, the second-order boundary value problem that represents an adiabatic tubular chemical reactor [27] can be changed to the form of a system of equations by using the SINC method for numerical solutions [29].Positive solutions for a system of equations then represent the steady state temperature of the reaction.A positive solution for the system of equations also ensures the solvability of the three-point boundary value problem studied in [13].
In this paper, we will construct a new cone in   and obtain some new existence results for a general nonlinear algebraic system (3).In particular, the obtained results are also sharp even if  = 1.Naturally, we also improve our recent result when system (3) is reduced to problem (1).We will also extend our results to the more general cases (the coefficient matrix is dependent on the state variable  or () = col( 1 (),  2 (), . . .,   ())).In this case, all obtained results are new.
Our main tool is Guo-Krasnoselskii's fixed point theorem.Thus, in the next section, we give the theorem and prove three extended results.Our main results will be discussed in Section 3 and some more general cases will be considered in Section 4. Of course, some explanatory examples and remarks are also given in Sections 3 and 4, respectively.

Preliminaries
Guo-Krasnoselskii's fixed point theorem is mainly adopted in this paper.To this end, we will display this result in the following lemma.Firstly, we give a definition of cone.We let  be a real Banach space.A nonempty closed convex set  ⊂  is called a cone if it satisfies the following two conditions: (i)  ∈  and  ≥ 0 imply that  ∈ , and (ii)  ∈  and − ∈  imply that  = , where  ∈  is called the zero element of .
Now we state Guo-Krasnoselskii's fixed point theorem concerning cone expansion and compression of norm type as follows.
Lemma 1 (see [30, P133]).Let Ω 1 and Ω 2 be two bounded open sets in  such that  ∈ Ω 1 and By using Lemma 1, we immediately obtain the following results.

Existence of Positive Solutions for System (3)
First of all, we give two conditions of Theorem 5.
Remark 11.Note that the function  may have no definition on [0, ).Thus, the function  may have singularities.For example, where  > 0. In [8,13], it is assumed that  is sublinear or superlinear at zero point and infinite point.In this situation, we can also obtain some corresponding results.

More General Results
In this section, we assume that the coefficient matrix  and the nonlinear term of (3) may be dependent on the variable .In this case, we have or where Mathematical Problems in Engineering 5 At this time, we give the following conditions: (S 1 ) There exists a positive , such that   () is continuous and nonnegative 0 ≤   ≤  and ,  ∈ [1, ].
Similarly, we have the following result.
For  ∈  ∩ Ω  , we have or For  ∈  ∩ Ω  , the cases are similar.The proof is complete.
Example 13.Consider the nonlinear algebraic system of the form where  1 ,  2 ∈ [0, 1].We can choose that  = 1 and  = 1/4.At this time, we have  1 =  2 = 2 and  = 1/2 which imply that conditions ( 1 ) and ( 2 ) hold.At the same time, we have max In view of Theorem 12, when max We have max Thus, all conditions of Theorem 12 hold.In [11,23], it is assumed that  is a positive or nonnegative matrix, where   is independent on the variable .In the above section, the coefficient matrix  is dependent on the variable .
Remark 14.Clearly, we can also obtain the existence conditions for the systems or They will be omitted.