Existence of Positive Solutions for a Class of Nonlinear Algebraic Systems

Abstract

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

1. Introduction

Many problems arising from economics [1–4], complex network [5–7], mechanical engineering [8], Verschelde’s web page [9], and mathematics [10–13] 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.

Generally, there are no good methods for solving such systems, even in the simple case of only two equations of the form: and ; see van der Laan et al. [14]. Thus, some existing theorems of zero points or fixed points had been extensively established by a number of authors; see [1, 2, 4, 10–13, 15–23] and so forth.

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].

Recently, in [12] the authors had considered the existence of positive solutions for the discrete Dirichlet boundary value problem of the formby using Guo-Krasnoselskii’s fixed point theorem, where is a positive integer, , and is the forward difference operator; that is, , and .

Let , , and , which is given byThen problem (1) can be rewritten by matrix and vector of the formor

Note that (where is defined by (2)); thus, we can let ; then is a cone, where In this case, the method in [12] is valid for a more general nonlinear algebraic system (3) when the coefficient matrix is positive; that is, all elements of are positive.

The more general nonlinear algebraic system (3) has been shown to have interesting applications in various areas such as difference equations, boundary value problems, dynamical networks, existence of periodic solutions, stochastic processes, and numerical analysis; see [5–7, 10–13, 27, 29] and the references therein.

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 . 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 ). 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.

2. 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 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 and be two bounded open sets in such that and . Suppose that is completely continuous. If either for and for or for and for hold, then has at least one fixed point in .

By using Lemma 1, we immediately obtain the following results.

Corollary 2. Let and be two bounded open sets in such that and . Suppose that is completely continuous. If either for and for or for and for hold, then has at least one fixed point in .

Corollary 3. Let and be two bounded open sets in such that and . Suppose that is completely continuous. If either for and for or for and for hold, then has at least one fixed point in .

Corollary 4. Let and be two bounded open sets in such that and . Suppose that is completely continuous. If either for and for or for and for hold, then has at least one fixed point in .

Indeed, by means of Lemma 1, there exists such that . When all conditions of Corollary 2 hold, we have or for , which implies that . Thus, . If all conditions of Corollaries 3 and 4 hold, the proofs are similar.

3. Existence of Positive Solutions for System (3)

First of all, we give two conditions of Theorem 5.)There exists and a positive sequence , such that for .()There exist , such that the function is continuous and either or hold.

Theorem 5. Assume that conditions and hold; the set , and then problem (3) or (4) has at least a positive solution . Such solution satisfies the condition .

Proof. For , clearly, the set is a cone of , where .
For , we denote that Then, for , we have which implies that On the other hand, for . That is, .
Note that the function is continuous for ; clearly, is completely continuous.
For , we have For , we have for and On the opposite, for we have For , we have According to Lemma 1, we complete the proof.

Remark 6. Theorem 5 improves the corresponding result in [12] even if system (3) is reduced to problem (1). For example, when , we have by using Theorem 5. However, according to Theorem  1 in [12], . Clearly, we have Thus, we improve the recent result when system (3) is reduced to problem (1).

Remark 7. When conditions (6) and (7) of () are, respectively, replaced byorsome corresponding results can be obtained by using Corollaries 2, 3, and 4. The proof will be omitted.

When , (3) is reduced to . In this case, we have the following result.

Corollary 8. Assume that there exist , such that and either and or and , then there is , such that ; that is, the function has a fixed point .

Remark 9. Clearly, Corollary 8 is sharp.

Example 10. Consider the nonlinear algebraic system of the formThen, we have , , and In view of Theorem 5, if there exists such that either and or and hold, then problem (21) has at least a positive solution . Note that implies that ; thus, we have . In fact, problem (21) can be rewritten byLet ; we have . Thus, in view of Corollary 8, problem (21) has at least a positive solution with . Such result is also sharp.

Remark 11. Note that the function may have no definition on . Thus, the function may have singularities. For example, where . 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.

4. 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 haveorwhereAt this time, we give the following conditions:()There exists a positive , such that is continuous and nonnegative and .()There exist and a positive sequence , such that for ,.()There exist , such that the function is continuous and either or hold for .

Similarly, we have the following result.

Theorem 12. Assume that conditions , , and hold; then problem (25) or (26) has at least a positive solution . Such solution satisfies the condition .

Proof. Similarly, we can prove that is completely continuous.
For , we have orFor , the cases are similar. The proof is complete.

Example 13. Consider the nonlinear algebraic system of the formwhere . We can choose that and . At this time, we have and which imply that conditions () and () hold. At the same time, we have In view of Theorem 12, whensystem (32) has at least a positive solution . For example, letWe haveThus, 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 orThey will be omitted.

Remark 15. When conditions (28) and (29) of () are, respectively, replaced byorsome corresponding results can be obtained by using Corollaries 2, 3, and 4.

Remark 16. Problem (25) can be rewritten by In fact, we also obtain the existence of positive zero points for system (41). The existence of zero points for (41) has been extensively discussed; see [9, 20, 21, 24, 25].

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (no. 11371277) and the Natural Sciences and Engineering Research Council of Canada (NSERC).