Discrete Dynamics in Nature and Society

Discrete Dynamics in Nature and Society / 2009 / Article

Research Article | Open Access

Volume 2009 |Article ID 785068 | https://doi.org/10.1155/2009/785068

Guang Zhang, Liang Bai, "Existence of Solutions for a Nonlinear Algebraic System", Discrete Dynamics in Nature and Society, vol. 2009, Article ID 785068, 28 pages, 2009. https://doi.org/10.1155/2009/785068

Existence of Solutions for a Nonlinear Algebraic System

Academic Editor: Binggen Zhang
Received30 Apr 2009
Accepted15 Aug 2009
Published27 Oct 2009

Abstract

As well known, the existence and nonexistence of solutions for nonlinear algebraic systems are very important since they can provide the necessary information on limiting behaviors of many dynamic systems, such as the discrete reaction-diffusion equations, the coupled map lattices, the compartmental systems, the strongly damped lattice systems, the complex dynamical networks, the discrete-time recurrent neural networks, and the discrete Turing models. In this paper, both the existence of nonzero solution pairs and the nonexistence of nontrivial or nonzero solutions for a nonlinear algebraic system will be considered by using the critical point theory and Lusternik-Schnirelmann category theory. The process of proofs on the obtained results is simple, the conditions of theorems are also easy to be verified, however, some of them improve the known ones even if the system is reduced to the precial cases, in particular, others of them are still new.

1. Introduction

In this paper, the nonlinear algebraic system,

will be considered, where is a parameter,

are column vectors with is a continuous function defined on and for and , and is a positive integer. Also is an square matrix that there exists a positive diagonal matrix such that is nonnegative definite. The letter will denote transposition.

For a given , a column vector is said to be a solution of (1.1) corresponding to it if substitution of and into (1.1) renders it an identity. The vector is said to be positive if for , negative if for , and nonzero if for . Positive, negative, and (strongly) nonzero vector are denoted by and respectively. If there exists such that , it will be called nontrivial solution of (1.1). In this case, it is denoted by .

First note that is always a trivial solution of (1.1). Also note that if is a solution of (1.1), then is a solution of (1.1) as well. Therefore, we always consider solution pairs, .

Nonlinear systems of the form (1.1) arise in many applications such as the discrete models of steady-state equations of reaction–diffusion equations (see [16]), the discrete analogue of the periodic boundary value problems (see [711]), the steady-state equations of coupled map lattices (see [1225]), the discrete periodic boundary value problems (see [2629]), the steady-state equations of compartmental system (see [3034]), the steady-state models on complex dynamical network ([3539]), the steady-state systems of discrete Turing instability models (see [5, 33, 4068]). Thus, the existence and the nonexistence of solutions on (1.1) are very important.

In fact, the special cases of (1.1) have been extensively studied by a number of authors, see [2629, 3638] and the listed references therein. However, our results improve and extend the known ones even if (1.1) is reduced to their cases, in particular, some of them are new.

In this paper, the existence of solution pairs for the nonlinear algebraic system (1.1) will be considered by using the critical point theory and Lusternik-Schnirelmann category theory [69] or [70]. The nonexistence of nontrivial and nonzero solutions of (1.1) will also be established. The present paper is organized as follows. In Section 2, problems in various areas are transformed into system (1.1). In Section 3, we discuss turing instability. Then the nonexistence of solutions for (1.1) will be studied in Section 4, here, all results are new. Furthermore, in Section 5, the existence of solution pairs for (1.1) will be considered, some known results will be extended and improved, in particular, the method of proofs is different from previous ones. Some applications will be presented in Section 6.

2. Problems Expressed by (1.1)

A lot of problems in various areas can be expressed by (1.1). In this section, we will pick some typical examples.

2.1. Periodic Boundary Value Problems

As well known, steady-state equations of many important models in application, such as the nonlinear reaction—diffusion equations [6], the generalized reaction Duffing model [4], and the Fisher equation [1], can be expressed by the following equation:

Thus, existence of solutions for second-order differential equation with periodic boundary value condition has been extensively studied by a number of authors (see [811]).

By using finite differences [7], discrete analogue of the periodic boundary value problem

can be written by

where with , with , for all , , and

In view of (2.3), we can obtain (1.1) with the square matrix having the form

The conditions and lead to , for . At the same time, the matrix has a zero eigenvalue and all other eigenvalues are positive, see [7].

Let

the conditions

imply that the matrix is symmetric. As an simiple example, we consider

then there exists the positive diagonal matrix such that

which is nonnegative definite.

Pattern dynamics in coupled map lattices (CMLs) have been extensively studied (see [1219, 22]). It has been found that CMLs exhibit a variety of space-time patterns such as kink-antikinks, traveling waves, space-time periodic structures, space-time intermittence and spatiotemporal chaos. It is believed that CMLs possess the potential to explain phenomena associated with turbulence and other spatiotemporal systems.

Consider the following coupled map lattice:

where denotes the time and denotes the spatial coordinate, is positive and treated as a parameter. This is a discrete analogue of the well-known Nagumo equation of the form

where is a positive constant. The continuous Nagumo equation (2.11) is used as a model for the spread of genetic traits [2] and for the propagation of nerve pulses in a nerve axon, neglecting recovery [3, 5].

Solution of (2.10) is said to be stationary wave solution if for all and . In view of (2.10), we have

Now, we consider the existence of -periodic solution of (2.12). Clearly, this is equal to the existence of solutions for (1.1), nonnegative definite matrix,

Recently, Zhou et al. [29] consider the discrete time second order dynamical systems

where and for any . Our results are also valid for the problem (2.14). In this case, the corresponding results also improve and extend the main theorem in [29].

Cai et al. [27] considered existence and multiplicity of periodic solution for the fourth-order difference equation

by using linking theorem, where the function is defined on with for a given positive integer and for all .

However, (2.15) is equal to (1.1), where

which is nonnegative definite.

Clearly, and of (2.15) can also be replaced by and of (2.14), respectively. In this case, our all results are new.

2.2. Compartmental System

Dynamic models of many processes in the biological and physical sciences give systems of ordinary differential equations called compartmental systems (see [3034] and references therein). For example, Jacquez and Simon [32] considered the following system:

where represents the mass of compartment . , is the transfer or rate coefficient from compartment to compartment , is the transfer coefficient from compartment to environment, represents the flows into the compartment from outside the system, or inflow. The entries of the matrix have three properties:

A system of the form of (2.17) for which satisfies (2.18) is called a compartmental system.

Considering the compartmental system that the transfer coefficient from compartment to compartment is equal to that from compartment to compartment for all the compartments in the system, and the inflow is determined by the mass of compartment , say, . Then steady-state equation of this kind compartmental system is

When is an odd function, we can obtain (1.1) with the square matrix having the form

By using (2.18), we can get that the matrix is nonnegative definite. To our knowledge, few results on the system (2.19) are found in literature.

2.3. Strongly Damped Lattice System

Recently, Li and Zhou [21] considered the following second-order lattice dynamic system:

where , , , are given, is a vector with the components and can be ordered as the following form of -dimensional vector in :

where , , . is a nonnegative definite matrix on with eigenvalues , and is the simple and minimal eigenvalue with corresponding eigenvector . Also , denote the th component of , , respectively. An example of is , at this time, (2.21) can be regarded as the discrete analogue of the initial-boundary value problem of the following continuous strongly damped wave equation:

which arises in wave phenomena in various areas in mathematical physics (see [20, 21, 2325] and references therein).

Steady-state equation of (2.21) is

where

When is an odd function, (2.24) can be expressed by (1.1). Thus, the existence of (2.24) is important, however, to our knowledge few results are seen in literature.

2.4. Complex Dynamical Network

Recently, complex dynamical network have been considered by Li et al. in [35]. Suppose that a complex network consists of identical linearly and diffusively coupled nodes, with each node being an -dimensional dynamical system. The state equations of this dynamical network are given by

where are the state variables of node , the constant represents the coupling strength between node and node , is a matrix linking coupled variables, and if some pairs , , with , then it means two coupled nodes are linked through their th and th state variables, respectively. The coupling matrix represents the coupling configuration of the network, which is assumed as a random network described by the E-R model or a scale-free network described by the B-A model. If there is a connection between node and node , then ; otherwise, . If the degree of node is defined to be the number of its outreaching connections, then

Let the diagonal elements be ,

In [35] the authors assumed there exists a generous stationary state for network (2.26) which is defined as

They suppose that is positive semidefinite and apply the pinning control strategy on a small fraction of the nodes to achieves the stabilization control of the goal (2.28). The network (2.26) can be rewritten by the system

where is the state vector, and the denotes the -dimensional functional value vector of , and

is nonnegative definite in [35].

When , steady-state equation of (2.29) can be expressed by (1.1). We think that the assumption (2.28) is not fact for a complex dynamical network. Thus, it is necessary to consider the existence of the other solutions. On the other hand, for any we also obtain a nonlinear algebraic system from (2.26), which is the special case of (1.1).

2.5. Discrete Neural Networks

Recently, Zhou et al. [39] considered the following discrete-time recurrent neural networks, which is thought to describe the dynamical characteristics of transiently chaotic neural network:

where is the internal state of neuron , is the output of neuron , is the input bias of neuron , is the self-recurrent bias of neuron , represents the damping factor of nerve membrane, and is the connection weight from neuron to neuron which is written as

In order to obtain the results on asymptotically stability, authors in [39], assume that the input-output function is , inverse function of exists, , there exists a matrix with for such that , and is positive definite, this implies that the matrix is nonnegative definite.

The steady-state equation of (2.31) is

where

When is an odd function, (2.33) can be expressed by (1.1).

On the other hand, Wang and Cheng [3638] considered the existence of steady-state solutions for the discrete neural networks

with the periodic boundary value conditions:

Our results are also valid for their problems and improve their theorems. In particular, for the more general system of the form

our results are also valid, where and are similar with (2.14).

3. Turing Instability

In 1952, Turing [68] suggested that, under certain conditions, chemicals can react and diffusion in such a way as to produce steady-state heterogeneous spatial patterns of chemical of morphogenic concentration. His idea is a simple but profound one. In view of Turing's theory framework, many Turing's patterns have been obtained by the observations, the numerical simulations, the animal coat patterns, the wavelength of the electrochemical system, the vegetation in many semiarid regions, the skeletal pattern formation of chick limb, and so forth, see [33, 4043, 4547, 49, 5357, 6267].

However, the mathematical theory of Turing's patterns is not clear, see the recent papers [44, 5052, 5961]. In fact, when the diffusion term is added, the steady-state solutions are different with the primary model. Some new solutions will be increased. On the other hand, all numerical simulations will use the discrete analogue of the corresponding reaction diffusion equations or systems. Thus, it is necessary to consider the existence of solutions for the discrete steady-state equations. In general, such equation can be expressed by a partial difference equation of the form

with the periodic boundary value conditions

However, we will give the other explanation in the later.

Because the existence of solutions of (3.1)-(3.2) is equal to (1.1), see Zhang et al. [71] or Zhang and Feng [72]. Clearly, and of (3.1) can also be replaced by and of (2.14), respectively.

4. Nonexistence

Usually, the existence of solutions is important. In fact, the nonexistence is also important because it can give some useful information for the existence of solutions. Thus, in this section, we will firstly give the nonexistence results of nontrivial solutions and nonzero solutions on the system (1.1). The obtained results are new.

When the matrix is nonnegative definite, we know that its eigenvalues are nonnegative and denote

and the corresponding orthonormal eigenvectors are .

First of all, we let be a nontrivial solution of (1.1). Multiplying (1.1) by on the left we get

which implies that

In view of the reference [73], we know that

Thus, we have

which implies that the following nonexistence result is fact.

Theorem 4.1. If there exists such that or then the system (1.1) has no nontrivial solutions.

Now, we assume that and denote

Clearly, the condition implies that the infimum

exists and

The condition implies that the supremum

exists and

Thus, Theorem 4.1 implies that the following results hold.

Corollary 4.2. Assume that . Then the condition implies that the system (1.1) has no nontrivial solutions when there exists such that holds. The condition implies that for any , the system (1.1) has no nontrivial solutions when holds.

When and satisfy , we easily obtain the following results.

Corollary 4.3. Assume that and , then condition (4.13) or (4.14) implies that the system (1.1) has no nontrivial solutions.

Similarly, when and satisfy the conditions: (H1) and , (H2) and , (H3) and , (H4) and , we have the following result.

Corollary 4.4. Assume that . If condition (H1) or (H2) holds, then (4.13) implies that the system (1.1) has no nontrivial solutions. If the condition (H3) or (H4) holds, then (4.14) implies that the system (1.1) has no nontrivial solutions.

Now, we again assume that the system (1.1) has a nonzero solution , then we have

Thus, we have the following result.

Theorem 4.5. Assume that there exist and such that and Then the system (1.1) has no nonzero solutions.

In view of Theorem 4.5, we can also obtain some corollaries, here, we only give a clear result.

Corollary 4.6. The conditions or for and imply that the system (1.1) has no positive-negative solutions.

Corollary 4.6 can be immediately obtained by (4.16).

5. Existence

In view of Section 4, we find that the existence of solutions may become fact when the function crosses the eigenvectors spaces. This motivates us to use Lusternik-Schnirelmann category theory and leads to new methods compared with previous ones, see [2629, 3638].

For a given symmetric matrix , the index of the corresponding quadratic form on , , is the largest dimension of a subspace such that for all , . The following result is specialized to our cases, see the references [69] or [70].

Lemma 5.1. If with is a even function on of the form , where is a quadratic form of index , and such that for large (where ) and that as , then has at least pairs, , nonzero critical points.

For using Lemma 5.1, we reformulate our problem as a critical point problem. For any and , we let

At this time, we can define the functions by

where . Since

we see that a column vector is a critical point of the functional corresponding to if and only if is a solution of (1.1) corresponding to .

Let

such that

where

In this case, we have

where .

By using the condition (5.6), we easily get that

For the functional , we ask that for large , which implies that for and large . In this case, if there exists and such that

then the matrix has exactly negative eigenvalues. Lemma 5.1 implies that the following result holds.

Theorem 5.2. If there exist and such that and that further suppose that there is such that for and . Then the system (1.1) has at least nonzero solution pairs.

By using Theorem 5.2, we clearly obtain the following results.

Corollary 5.3. For any , if there exists such that and there is such that for , then the system (1.1) has at least nonzero solution pairs.

Corollary 5.4. If there exists such that further suppose that there is such that for and . Then the system (1.1) has at least nonzero solution pairs, particularly, the system (1.1) has at least one positive-negative solution pair.

Corollary 5.5. For any and , the conditions and imply that the system (1.1) has at least nonzero solution pairs.

Now, we consider the functional . Similarly, we have the following result by using Lemma 5.1.

Theorem 5.6. If there exist and such that and that further suppose that there is such that Then the system (1.1) has at least nonzero solution pairs.

Corollary 5.7. Assume that the condition holds for any , and there exist and such that the condition (5.13) is valid, then the system (1.1) has at least nonzero solution pairs.

Corollary 5.8. For any and , the conditions and imply that the system (1.1) has at least nonzero solution pairs.

Consider the algebraic equations

which is the special case of (1.1). Obviously, for any all conditions of Theorem 5.6 hold. Thus, (5.14) has at least one nonzero solution pair. In fact, it has the exact nonzero solution pair . Thus, the conditions of Theorem 5.6 are sharp for (5.14).

6. Applications

Clearly, the theorems and corollaries established earlier are useful to solve the problems listed in Section 2. Some simple illustrative examples and remarks will be listed in this section.

6.1. Periodic Solutions

Guo and Yu [28] considered the existence of -periodic solution for

The main result they derived is described as follows.

Assume satisfies the following condition:

(i), and there exists positive integer such that for any , ;(ii)for any , , and ;(iii)there exists , such that for any , then for any positive integer , (6.1) has at least three -periodic solutions.

Now, we consider the existence of -periodic solution of the following nonlinear second-order difference equation:

where is a given positive integer, where is a parameter, and for , and there exists a positive integer such that for any , , . A solution of (6.3) is called to be nonzero if its every term is not zero.

The above problem is equal to the system (1.1) where the matrix is defined by (2.13) with and has the eigenvalues

Note that

then we have

when is even and

when is odd.

Again let

By using Theorems 5.2 and 5.6, we can obtain the following facts.

Theorem 6.1. For any , if there exist and such that further suppose that there is such that for and . Then the problem (6.3) has at least nonzero -periodic solutions.

Theorem 6.2. When is a even positive integer and there exist and such that (6.10) holds, further suppose that there is such that then the system (6.3) has at least nonzero -periodic solutions; If is an odd positive integer and there exist and such that (6.10) and (6.11) hold, then the system (6.3) has at least nonzero -periodic solutions.

In view of Theorem 6.1, we can immediately obtain the following result.

Corollary 6.3. For any or hold, then the problem (6.3) has at least nonzero -periodic solutions.

Remark 6.4. Corollary 6.3 improves the main result in [28].

When for any , we have

Corollary 6.3 implies that for any positive integer and positive number the equation

has at least nonzero -periodic solutions. For example, let and , we can consider the existence of solutions for nonlinear system:

which has at least nonzero solution pairs in view of Corollary 6.3. However, Theorem in [28] can only obtain two nonzero solutions because they conclude a zero solution. In fact, we can solve its the nonzero numerical solutions:

However, we find that the number of nonzero solutions of (6.16) is more than . In fact, (6.16) has also nontrivial solutions

Thus, we have the following open problem.

Open Problem  1
Obtain better existence results for the system (1.1) when .
When for any , we have
For any Theorem 6.1 implies that the periodic boundary value problem of the form has at least nonzero solutions.
On the other hand, Zhou et al. [29] consider the discrete time second-order dynamical systems:
where and for any . Our results are also valid for the problem (6.22). In this case, the corresponding results also improve the main theorem in [29]. In fact, we let be an unit matrix, then the problem (6.22) is equal to the system (1.1) where which has the eigenvalues

6.2. Steady-State Solutions on Discrete Neural Networks

In [36], Wang and Cheng considered the existence of steady-state solutions for the discrete neural network

with the periodic boundary value conditions

In fact, the steady-state equation can be written by

which can be rewritten by the system (1.1), where , the coefficient matrix is defined by(2.13)

In this case, in Theorems 6.1 and 6.2   and are, respectively, replaced by and , then, they are valid for the problem (6.27). At this time, Theorem 6.1 improves the corresponding result in [36], but Theorem 6.2 is new.

On the other hand, for the more general system of the form

our results are also valid, where and are similar with (2.14) and the method is similar with Section 6.2. Thus, the main results in [36, 38] are also extended and improved.

We also find that Wang and Cheng [37] considered the existence of the steady-state solutions for the discrete neural network:

with the periodic boundary value condition (6.26). Similarly, the problem (6.30)–(6.26) be rewritten by the system (1.1), where , is defined by (2.13), and

By using similar method, we can obtain the improved and extended results. Clearly, we can also consider the existence of steady-state solutions for the general system of the form

where is a -vector for each .

6.3. Periodic Solutions for Fourth-Order Difference Equation

When , the problem (2.15) exists -periodic solutions if and only if the nonlinear system

has nontrivial solutions. The matrix

has the eigenvalues and . When , the problem (2.15) exists -periodic solutions if and only if the nonlinear system

has nontrivial solutions. The matrix

has the eigenvalues .

When , the problem (2.15) exists -periodic solutions if and only if the nonlinear system

has nontrivial solutions. The matrix

has the eigenvalues and .

When , (2.15) exists -periodic solutions if and only if the nonlinear system (1.1) has nontrivial solutions, where the matrix is defined in Subsection 2.1.

We can obtain eigenvalues of from the fourth-order linear difference equation

with the periodic boundary value conditions:

Let

In view of (6.39), we have

which implies that

From (6.40), we see that

thus, we have

for and . In view of (6.43), we have obtained

Note that

then

By using Theorems 5.2 and 5.6, we can clearly obtain similar results as Theorems 6.1 and 6.2 and improve and extend the results in [27]. They will be omitted.

6.4. On Partial Difference Equation

In Turing pattern analy