Existence of Solutions for a Nonlinear Algebraic System

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,

𝐴𝑥=𝜆𝑓(𝑥), (1.1) will be considered, where 𝜆>0 is a parameter,

𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)^𝑇,𝑓(𝑥)=(𝑓₁(𝑥₁),𝑓₂(𝑥₂),…,𝑓_𝑛(𝑥_𝑛))^𝑇 (1.2) are column vectors with 𝑓_𝑘 is a continuous function defined on 𝐑 and 𝑓_𝑘(−𝑢)=−𝑓_𝑘(𝑢) for 𝑢∈𝐑 and 𝑘∈{1,2,…,𝑛}=[1,𝑛], and 𝑛 is a positive integer. Also 𝐴=(𝑎_𝑖𝑗)_𝑛×𝑛 is an 𝑛×𝑛 square matrix that there exists a positive 𝑛×𝑛 diagonal matrix 𝐷=diag(𝑑₁,𝑑₂,…,𝑑_𝑛) such that 𝐷𝐴 is nonnegative definite. The letter 𝑇 will denote transposition.

For a given 𝜆>0, a column vector 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)^𝑇∈𝐑^𝑛 is said to be a solution of (1.1) corresponding to it if substitution of 𝜆 and 𝑥 into (1.1) renders it an identity. The vector 𝑥 is said to be positive if 𝑥_𝑘>0 for 𝑘∈[1,𝑛], negative if 𝑥_𝑘<0 for 𝑘∈[1,𝑛], and nonzero if 𝑥_𝑘≠0 for 𝑘∈[1,𝑛]. Positive, negative, and (strongly) nonzero vector 𝑥 are denoted by 𝑥>0,𝑥<0 and 𝑥∦0 respectively. If there exists 𝑘₀∈[1,𝑛] such that 𝑥_𝑘0≠0, it will be called nontrivial solution of (1.1). In this case, it is denoted by 𝑥≠0.

First note that 𝑥=0 is always a trivial solution of (1.1). Also note that if 𝑥 is a solution of (1.1), then −𝑥 is a solution of (1.1) as well. Therefore, we always consider solution pairs, ±𝑥.

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 [1–6]), the discrete analogue of the periodic boundary value problems (see [7–11]), the steady-state equations of coupled map lattices (see [12–25]), the discrete periodic boundary value problems (see [26–29]), the steady-state equations of compartmental system (see [30–34]), the steady-state models on complex dynamical network ([35–39]), the steady-state systems of discrete Turing instability models (see [5, 33, 40–68]). 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 [26–29, 36–38] 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:

−(𝑟(𝑡)𝑥^′(𝑡))^′+𝑞(𝑡)𝑥=𝑓(𝑡,𝑥). (2.1) Thus, existence of solutions for second-order differential equation with periodic boundary value condition has been extensively studied by a number of authors (see [8–11]).

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

−(𝑟(𝑡)𝑥^′(𝑡))^′+𝑞(𝑡)𝑥=𝑓(𝑡,𝑥),𝑡∈(0,1),𝑥(0)=𝑥(1),𝑥^′(0)=𝑥^′(1), (2.2) can be written by

−𝑎_𝑘𝑥_𝑘−1+𝑏_𝑘𝑥_𝑘−𝑐_𝑘𝑥_𝑘+1=𝜆𝑓_𝑘(𝑥_𝑘𝑥)for𝑘∈[1,𝑛],₀=𝑥_𝑛,𝑥_𝑛+1=𝑥₁, (2.3) where 𝑟∈𝐶¹[0,1] with 𝑟(𝑡)≥𝑟_min>0, 𝑞(𝑡)∈𝐶[0,1] with 𝑞(𝑡)≥0, 𝑓(𝑡,−𝑥)=−𝑓(𝑡,𝑥) for all (𝑡,𝑥)∈[0,1]×𝐑, 𝑥_𝑘=𝑘ℎ, ℎ=1/𝑛 and

𝑎_𝑘=2𝑟_𝑘−1/2ℎ,𝑐_𝑘=2𝑟_𝑘+1/2ℎ,𝑏_𝑘=2(𝑟_𝑘−1/2+𝑟_𝑘+1/2)ℎ+2ℎ𝑞_𝑘,𝑓_𝑘(𝑥_𝑘)=𝑓(𝑘,𝑥_𝑘),𝜆=2ℎ. (2.4)

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

𝑏𝐴=¹−𝑐₁0⋯−𝑎₁−𝑎₂𝑏₂−𝑐₂⋯0⋯⋯0⋯−𝑎_𝑛−1𝑏_𝑛−1−𝑐_𝑛−1−𝑐_𝑛⋯0−𝑎_𝑛𝑏_𝑛. (2.5) The conditions 𝑟(𝑡)≥𝑟_min>0 and 𝑞(𝑡)≥0 lead to 𝑎_𝑘,𝑏_𝑘,𝑐_𝑘>0, 𝑏_𝑘≥𝑎_𝑘+𝑐_𝑘 for 𝑘∈[1,𝑛]. At the same time, the matrix 𝐴 has a zero eigenvalue and all other eigenvalues are positive, see [7].

Let

𝑑₂=𝑑>0,𝑑_𝑘=^𝑘−1_𝑖=2𝑐_𝑖𝑎_𝑖+1for𝑘∈[3,𝑛−1], (2.6) the conditions

𝑐₁=𝑑𝑎₂,𝑎₁=𝑐_𝑛,𝑎_𝑛=𝑐_𝑛−1^𝑛−2_𝑖=2𝑐_𝑖𝑎_𝑖+1 (2.7) imply that the matrix 𝐷𝐴 is symmetric. As an simiple example, we consider

𝐴=2−1−1−0.51−0.5−1−12, (2.8) then there exists the positive diagonal matrix 𝐷=diag(1,2,1) such that

𝐷𝐴=2−1−1−12−1−1−12 (2.9) which is nonnegative definite.

Pattern dynamics in coupled map lattices (CMLs) have been extensively studied (see [12–19, 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:

𝑢_𝑛^𝑡+1=𝑢^𝑡_𝑛+𝛼(𝑢^𝑡_𝑛−1−2𝑢^𝑡_𝑛+𝑢^𝑡_𝑛+1)+𝛽𝑓(𝑢^𝑡_𝑛), (2.10) where 𝑡∈𝐍 denotes the time and 𝑛∈𝐙 denotes the spatial coordinate, 𝛽/𝛼 is positive and treated as a parameter. This is a discrete analogue of the well-known Nagumo equation of the form

𝜕𝑢𝜕𝜕𝑡=𝐷²𝑢𝜕𝑥²+𝑓(𝑢),𝑥∈𝐑,𝑡∈𝐑⁺, (2.11) where 𝐷 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 {𝑢^𝑡_𝑛} of (2.10) is said to be stationary wave solution if 𝑢_𝑛^𝑡+1=𝑢^𝑡_𝑛 for all 𝑛∈𝐙 and 𝑡∈𝐍. In view of (2.10), we have

0=𝛼(𝜑(𝑛−1)−2𝜑(𝑛)+𝜑(𝑛+1))+𝛽𝑓(𝜑(𝑛)). (2.12) Now, we consider the existence of 𝜔-periodic solution of (2.12). Clearly, this is equal to the existence of solutions for (1.1), 𝜔×𝜔 nonnegative definite matrix,

𝐴=2−10⋯−1−12−1⋯0⋯⋯⋯0⋯−12−1−1⋯0−12_𝜔×𝜔. (2.13)

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

𝑋_𝑘+1−2𝑋_𝑘+𝑋_𝑘−1+𝑔(𝑘,𝑋_𝑘)=0,𝑘∈𝑍, (2.14) where 𝑔(𝑔₁,𝑔₂,…,𝑔_𝑙)^𝑇∈𝐶(𝑍×𝑅^𝑙,𝑅^𝑙) and 𝑔(𝑘+𝜔,𝑈)=𝑔(𝑘,𝑈) for any (𝑘,𝑈)∈𝑍×𝑅^𝑙. 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

Δ⁴𝑥_𝑘−2−𝜆𝑓(𝑘,𝑥_𝑘)=0,𝑘∈𝐙, (2.15) by using linking theorem, where the function 𝑓(𝑘,𝑢) is defined on 𝐙×𝐑 with 𝑓(𝑘+𝜔,𝑢)=𝑓(𝑘,𝑢) for a given positive integer 𝜔 and 𝑓(𝑘,−𝑢)=−𝑓(𝑘,𝑢) for all (𝑘,𝑢)∈𝐙×𝐑.

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

𝐴=6−410⋯001−4−46−41⋯00011−46−4000001−46⋯00⋯0⋯⋯6−4100000⋯−46−4110001−46−4−4100⋯01−46_𝜔×𝜔 (2.16) which is nonnegative definite.

Clearly, 𝑥_𝑘 and 𝑓 of (2.15) can also be replaced by 𝑋_𝑘 and 𝑔 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 [30–34] and references therein). For example, Jacquez and Simon [32] considered the following system:

𝑞̇_𝑖(𝑡)=^𝑛_𝑗=1𝑓_𝑖𝑗𝑞_𝑗(𝑡)+𝐼_𝑖,𝑖∈[1,𝑛], (2.17) where 𝑞_𝑖 represents the mass of compartment 𝑖. 𝑓_𝑖𝑖=−(𝑓_0𝑖+∑_𝑗≠𝑖𝑓_𝑗𝑖), 𝑓_𝑖𝑗 is the transfer or rate coefficient from compartment 𝑗 to compartment 𝑖, 𝑓_0𝑖 is the transfer coefficient from compartment 𝑖 to environment, 𝐼_𝑖 represents the flows into the compartment 𝑖 from outside the system, or inflow. The entries of the matrix 𝐹=(𝑓_𝑖𝑗)_𝑛×𝑛 have three properties:

𝑓_𝑖𝑖𝑓≤0∀𝑖≥0,_𝑖𝑗≥0∀𝑖≠𝑗,_𝑛𝑖=1𝑓_𝑖𝑗=_𝑖≠𝑗𝑓_𝑖𝑗+𝑓_𝑗𝑗≤0∀𝑗≥0. (2.18) A system of the form of (2.17) for which 𝐹=(𝑓_𝑖𝑗)_𝑛×𝑛 satisfies (2.18) is called a compartmental system.

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

−^𝑛_𝑗=1𝑓_𝑖𝑗𝑞_𝑗=𝑔_𝑖(𝑞_𝑖),𝑖∈[1,𝑛]. (2.19) When 𝑔_𝑖 is an odd function, we can obtain (1.1) with the 𝑛×𝑛 square matrix −𝐹 having the form

−𝑓₁₁−𝑓₁₂⋯−𝑓_1𝑛−𝑓₂₁−𝑓₂₂⋯−𝑓_2𝑛⋯−𝑓_𝑛1−𝑓_𝑛2⋯−𝑓_𝑛𝑛. (2.20) By using (2.18), we can get that the matrix −𝐹 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:

𝑢̈_𝑖+𝑘(𝐵𝑢̇)_𝑖+(𝐴𝑢)_𝑖+ℎ(𝑢̇_𝑖)+𝑓(𝑢_𝑖)=𝑔_𝑖, (2.21) where 𝑖=(𝑖₁,𝑖₂,…,𝑖_𝑛)∈𝐙^𝑛_𝑚=𝐙^𝑛∩{𝑖₁,𝑖₂,…,𝑖_𝑛∈[1,𝑚]}, 𝑘≥0, ℎ,𝑓∈𝐶¹(𝐑,𝐑), 𝑔_𝑖∈𝐑(𝑖∈𝐙^𝑛_𝑚) are given, 𝑢=(𝑢_𝑖)_𝑖∈𝐙^𝑛_𝑚 is a vector with the components 𝑢_𝑖 and can be ordered as the following form of 1-dimensional vector in 𝐑^𝑚𝑛:

𝑢=(𝑢_(1,1,…,1),𝑢_(2,1,…,1),…,𝑢_{(𝑚,1,…,1)},…,𝑢_{(1,𝑚,…,𝑚)},…,𝑢_{(𝑚,𝑚,…,𝑚)})^𝑇=(𝑢₁,𝑢₂,…,𝑢_𝜈,…,𝑢_𝑚𝑛)^𝑇∈𝐑^𝑚𝑛, (2.22) where 𝜈=𝑖₁+𝑚(𝑖₂−1)+⋯+𝑚^𝑛−1(𝑖_𝑛−1), 𝑖₁,𝑖₂,…,𝑖_𝑛∈[1,𝑚], 𝑢̇=(𝑢̇_𝑖)_𝑖∈𝐙^𝑛_𝑚. 𝐴 is a nonnegative definite matrix on 𝐑^𝑚𝑛 with eigenvalues 𝜆_𝑠≥0(0≤𝑠≤𝑚^𝑛−1), and 0 is the simple and minimal eigenvalue with corresponding eigenvector 𝑒=(1,…,1)^𝑇∈𝐑^𝑚𝑛. Also (𝐴𝑢̇)_𝑖, (𝐴𝑢)_𝑖 denote the 𝑖th component of 𝐴𝑢̇, 𝐴𝑢, respectively. An example of 𝐴 is 𝐴=−Δ, 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:

𝑢_𝑡𝑡−𝑘Δ𝑢_𝑡−Δ𝑢+ℎ(𝑢_𝑡)+𝑓(𝑢)=𝑔(𝑥), (2.23) which arises in wave phenomena in various areas in mathematical physics (see [20, 21, 23–25] and references therein).

Steady-state equation of (2.21) is

(𝐴𝑢)_𝑘=𝐹_𝑘(𝑢_𝑘),𝑘∈[1,𝑚^𝑛], (2.24) where

𝐹_𝑘(𝑢_𝑘)=−𝑓(𝑢_𝑘)−ℎ(0)+𝑔_𝑘. (2.25) When 𝐹_𝑘 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 𝑁 identical linearly and diffusively coupled nodes, with each node being an 𝑚-dimensional dynamical system. The state equations of this dynamical network are given by

𝑥^′_𝑖=𝑓(𝑥_𝑖)+^𝑁_{𝑗=1,𝑗≠𝑖}𝑐_𝑖𝑗𝑎_𝑖𝑗Γ(𝑥_𝑗−𝑥_𝑖),𝑖∈[1,𝑁], (2.26) where 𝑥_𝑖=(𝑥_𝑖1,𝑥_𝑖2,…,𝑥_𝑖𝑚)^𝑇∈𝐑^𝑚 are the state variables of node 𝑖, the constant 𝑐_𝑖𝑗>0 represents the coupling strength between node 𝑖 and node 𝑗, Γ=(𝜏_𝑖𝑗)∈𝐑^𝑚×𝑚 is a matrix linking coupled variables, and if some pairs (𝑖,𝑗), 1≤𝑖,𝑗≤𝑚, with 𝜏_𝑖𝑗≠0, then it means two coupled nodes are linked through their 𝑖th and 𝑗th state variables, respectively. The coupling matrix 𝐴=(𝑎_𝑖𝑗)∈𝐑^𝑁×𝑁 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 𝑖 and node 𝑗(𝑖≠𝑗), then 𝑎_𝑖𝑗=𝑎_𝑗𝑖=1; otherwise, 𝑎_𝑖𝑗=𝑎_𝑗𝑖=0(𝑖≠𝑗). If the degree 𝑘_𝑖 of node 𝑖 is defined to be the number of its outreaching connections, then

^𝑁_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗=^𝑁_{𝑗=1,𝑗≠𝑖}𝑎_𝑗𝑖=𝑘_𝑖,𝑖∈[1,𝑁]. (2.27) Let the diagonal elements be 𝑎_𝑖𝑖=−𝑘_𝑖, 𝑖∈[1,𝑁].

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

𝑥₁=𝑥₂=⋯=𝑥_𝑛=𝑥,𝑓(𝑥)=0. (2.28) They suppose that Γ 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

𝑋^′=−𝐷𝑋+𝐹(𝑋), (2.29) where 𝑋=(𝑥₁,𝑥₂,…,𝑥_𝑚𝑁)^𝑇 is the state vector, and the 𝐹(𝑋) denotes the 𝑚𝑁-dimensional functional value vector of 𝑋, and

𝐷=(𝑑_𝑖𝑗)_{𝑚𝑁×𝑚𝑁} (2.30) is nonnegative definite in [35].

When 𝐹(−𝑋)=−𝐹(𝑋), 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 𝑖∈[1,𝑁], 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:

𝑣_𝑖(𝑡+1)=𝑘𝑣_𝑖(𝑡)+^𝑛_𝑗=1𝑤_𝑖𝑗𝑢_𝑗(𝑡)+𝑎_𝑖−𝑤_𝑖𝑖𝑎_0𝑖,𝑖∈[1,𝑛], (2.31) where 𝑣_𝑖 is the internal state of neuron 𝑖, 𝑢_𝑖 is the output of neuron 𝑖, 𝑎_𝑖 is the input bias of neuron 𝑖, 𝑎_0𝑖 is the self-recurrent bias of neuron 𝑖, 𝑘 represents the damping factor of nerve membrane, and 𝑤_𝑖𝑗 is the connection weight from neuron 𝑗 to neuron 𝑖 which is written as

𝑤𝑊=¹¹𝑤₁₂⋯𝑤_1𝑛𝑤₂₁𝑤₂₂⋯𝑤_2𝑛𝑤⋮⋮⋯⋮_𝑛1𝑤_𝑛2⋯𝑤_𝑛𝑛. (2.32) In order to obtain the results on asymptotically stability, authors in [39], assume that the input-output function is 𝑢_𝑖(𝑡)=𝑠(𝑣_𝑖(𝑡)), inverse function 𝑠⁻¹(𝑦) of 𝑠(𝑥) exists, 0<𝑠^′(𝑥)≤𝑀, there exists a matrix 𝐷=diag(𝑑₁,𝑑₂,…,𝑑_𝑛) with 𝑑_𝑖>0 for 𝑖∈[1,𝑛] such that (𝐷𝑊)^𝑇=𝐷𝑊, and ((1+𝑘)/𝑀)𝐷+𝐷𝑊(𝑘≥0) is positive definite, this implies that the matrix 𝐷𝑊 is nonnegative definite.

The steady-state equation of (2.31) is

^𝑛_𝑗=1𝑤_𝑖𝑗𝑢_𝑗=𝑓_𝑖(𝑢_𝑖),𝑖∈[1,𝑛], (2.33) where

𝑓_𝑖(𝑢_𝑖)=(1−𝑘)𝑠⁻¹(𝑢_𝑖)−𝑎_𝑖+𝑤_𝑖𝑖𝑎_0𝑖. (2.34)

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

On the other hand, Wang and Cheng [36–38] considered the existence of steady-state solutions for the discrete neural networks

𝑥_𝑖^𝑡+1−𝑥^𝑡_𝑖=𝑥^𝑡_𝑖−1+𝑥^𝑡_𝑖+1−𝑓(𝑖,𝑥^𝑡_𝑖),𝑥_𝑖^𝑡+1=𝑥^𝑡_𝑖−1+𝑥^𝑡_𝑖+1+𝑔_𝑖(𝑥^𝑡_𝑖), (2.35) with the periodic boundary value conditions:

𝑥^𝑡₀=𝑥^𝑡_𝜔,𝑥^𝑡₁=𝑥^𝑡_𝜔+1. (2.36) Our results are also valid for their problems and improve their theorems. In particular, for the more general system of the form

𝑋_𝑘+1+𝑋_𝑘−1−𝑔(𝑘,𝑋_𝑘)=0, (2.37) our results are also valid, where 𝑋_𝑘 and 𝑔 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, 40–43, 45–47, 49, 53–57, 62–67].

However, the mathematical theory of Turing's patterns is not clear, see the recent papers [44, 50–52, 59–61]. 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

Δ²₁𝑥_{𝑖−1,𝑗}+Δ²₂𝑥_{𝑖,𝑗−1}+𝜆𝑓_𝑖𝑗(𝑥_𝑖𝑗)=0 (3.1) with the periodic boundary value conditions

𝑥_0,𝑗=𝑥_𝑛,𝑗,𝑥_1,𝑗=𝑥_𝑛+1,𝑗𝑥,𝑗∈[1,𝑚],_𝑖,0=𝑥_𝑖,𝑚,𝑥_𝑖,1=𝑥_𝑖,𝑚+1,𝑖∈[1,𝑛]. (3.2) 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, 𝑥_𝑘 and 𝑓 of (3.1) can also be replaced by 𝑋_𝑘 and 𝑔 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 𝐷𝐴 is nonnegative definite, we know that its eigenvalues are nonnegative and denote

0=𝛾₁=𝛾₂=⋯=𝛾_𝑚0<𝛾_𝑚0+1≤⋯≤𝛾_𝑛, (4.1) and the corresponding orthonormal eigenvectors are 𝑣₁,𝑣₂,…,𝑣_𝑛.

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

𝑥^𝑇𝐵𝑥−𝜆^𝑛_𝑘=1𝑑_𝑘𝑥_𝑘𝑓_𝑘(𝑥_𝑘)=0, (4.2) which implies that

𝑥^𝑇𝐵𝑥𝑥^𝑇𝑥∑=𝜆^𝑛𝑘=1𝑑_𝑘𝑥_𝑘𝑓_𝑘(𝑥_𝑘)𝑥^𝑇𝑥. (4.3) In view of the reference [73], we know that

max_𝑥≠0𝑥^𝑇𝐵𝑥𝑥^𝑇𝑥=𝛾_𝑛,min_𝑥≠0𝑥^𝑇𝐵𝑥𝑥^𝑇𝑥=𝛾₁=0. (4.4) Thus, we have

0=𝛾₁=𝛾₂=⋯=𝛾_𝑚0∑≤𝜆^𝑛𝑘=1𝑑_𝑘𝑥_𝑘𝑓_𝑘(𝑥_𝑘)𝑥^𝑇𝑥≤𝛾_𝑛, (4.5) which implies that the following nonexistence result is fact.

Theorem 4.1. If there exists 𝜆>0 such that 𝛾_𝑛<𝜆inf_𝑥≠0∑^𝑛_𝑘=1𝑑_𝑘𝑥_𝑘𝑓_𝑘(𝑥_𝑘)‖𝑥‖² (4.6) or sup_𝑥≠0∑^𝑛_𝑘=1𝑑_𝑘𝑥_𝑘𝑓_𝑘(𝑥_𝑘)‖𝑥‖²<0, (4.7) then the system (1.1) has no nontrivial solutions.

Now, we assume that 𝑓₁=𝑓₂=⋯=𝑓_𝑛=𝑓 and denote

lim_|𝑢|→0𝑓(𝑢)𝑢=𝑙,lim_|𝑢|→∞𝑓(𝑢)𝑢=𝐿. (4.8) Clearly, the condition 𝑙=𝐿=+∞ implies that the infimum

inf_𝑢≠0𝑓(𝑢)𝑢 (4.9) exists and

^𝑛_𝑘=1𝑑_𝑘𝑥_𝑘𝑓_𝑘(𝑥_𝑘)≥min_{𝑘∈[1,𝑛]}{𝑑_𝑘}‖𝑥‖²inf_𝑢≠0𝑓(𝑢)𝑢. (4.10) The condition 𝑙=𝐿=−∞ implies that the supremum

sup_𝑢≠0𝑓(𝑢)𝑢 (4.11) exists and

^𝑛_𝑘=1𝑑_𝑘𝑥_𝑘𝑓_𝑘(𝑥_𝑘)≤max_{𝑘∈[1,𝑛]}{𝑑_𝑘}‖𝑥‖²sup_𝑢≠0𝑓(𝑢)𝑢. (4.12) Thus, Theorem 4.1 implies that the following results hold.

Corollary 4.2. Assume that 𝑓₁=𝑓₂=⋯=𝑓_𝑛=𝑓. Then the condition 𝑙=𝐿=+∞ implies that the system (1.1) has no nontrivial solutions when there exists 𝜆>0 such that 𝛾_𝑛<𝜆inf_𝑢≠0𝑓(𝑢)𝑢min_{𝑖∈[1,𝑛]}{𝑑_𝑖} (4.13) holds. The condition 𝑙=𝐿=−∞ implies that for any 𝜆>0, the system (1.1) has no nontrivial solutions when sup_𝑢≠0𝑓(𝑢)𝑢<0 (4.14) holds.

When 𝑙 and 𝐿 satisfy 0<𝑙,𝐿<∞, we easily obtain the following results.

Corollary 4.3. Assume that 𝑓₁=𝑓₂=⋯=𝑓_𝑛=𝑓 and 0<𝑙,𝐿<∞, then condition (4.13) or (4.14) implies that the system (1.1) has no nontrivial solutions.

Similarly, when 𝑙 and 𝐿 satisfy the conditions: (H1) 𝑙=+∞ and 0<𝐿<∞, (H2) 𝐿=+∞ and 0<𝑙<∞, (H3) 𝑙=−∞ and 0<𝐿<∞, (H4) 𝐿=−∞ and 0<𝑙<∞, we have the following result.

Corollary 4.4. Assume that 𝑓₁=𝑓₂=⋯=𝑓_𝑛=𝑓. 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 𝑥, then we have

𝑑𝐷𝐴𝑥=𝜆diag(¹𝑓₁(𝑥₁)𝑥₁,𝑑₂𝑓₂(𝑥₂)𝑥₂𝑑,…,^𝑛𝑓_𝑛(𝑥_𝑛)𝑥_𝑛𝑣)𝑥,_𝑇𝑖𝐷𝐴𝑥=𝜆𝑣^𝑇_𝑖𝑑diag(¹𝑓₁(𝑥₁)𝑥₁,𝑑₂𝑓₂(𝑥₂)𝑥₂𝑑,…,^𝑛𝑓_𝑛(𝑥_𝑛)𝑥_𝑛)𝑥,(𝐷𝐴𝑣_𝑖)^𝑇𝑑𝑥=𝜆[diag(¹𝑓₁(𝑥₁)𝑥₁,𝑑₂𝑓₂(𝑥₂)𝑥₂𝑑,…,^𝑛𝑓_𝑛(𝑥_𝑛)𝑥_𝑛)𝑣_𝑖]^𝑇𝛾𝑥,_𝑖𝑣_𝑖^𝑇𝑥=[diag(𝜆𝑑₁𝑓₁(𝑥₁)𝑥₁,𝜆𝑑₂𝑓₂(𝑥₂)𝑥₂,…,𝜆𝑑_𝑛𝑓_𝑛(𝑥_𝑛)𝑥_𝑛)𝑣_𝑖]^𝑇𝑥. (4.15)(4.16) Thus, we have the following result.

Theorem 4.5. Assume that there exist 𝑘₀∈[𝑚₀,𝑛−1] and 𝜆>0 such that 𝛾_𝑘0<𝛾_𝑘0+1 and 𝛾_𝑘0<𝜆𝑑_𝑘𝑓_𝑘(𝑢)𝑢<𝛾_𝑘0+1for𝑢∈(0,∞),𝑘∈[1,𝑛]. (4.17) 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 𝑢𝑓_𝑘(𝑢)>0 or 𝑢𝑓_𝑘(𝑢)<0 for 𝑘∈[1,𝑛] and 𝑢≠0 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 𝑓_𝑘 crosses the eigenvectors spaces. This motivates us to use Lusternik-Schnirelmann category theory and leads to new methods compared with previous ones, see [26–29, 36–38].

For a given symmetric matrix 𝐵, the index of the corresponding quadratic form on 𝐑^𝑛, 𝑞(𝑥)=𝑥^𝑇𝐵𝑥, is the largest dimension of a subspace 𝑆⊂𝐑^𝑛 such that 𝑞(𝑥)<0 for all 𝑥∈𝑆, 𝑥≠0. The following result is specialized to our cases, see the references [69] or [70].

Lemma 5.1. If 𝐻(𝑥) with 𝐻(0)=0 is a 𝐶¹ even function on 𝐑^𝑛 of the form 𝐻(𝑥)=𝑞(𝑥)+𝑣(𝑥), where 𝑞(𝑥) is a quadratic form of index 𝑚, and such that 𝐻(𝑥)≥0 for large ‖𝑥‖(where ‖𝑥‖=𝑥²₁+𝑥²₂+⋯+𝑥²_𝑛) and that 𝑣(𝑥)=𝑜(‖𝑥‖²) as ‖𝑥‖→0, then 𝐻(𝑥) has at least 𝑚 pairs, ±𝑥, nonzero critical points.

For using Lemma 5.1, we reformulate our problem as a critical point problem. For any 𝑘∈[1,𝑛] and 𝑢∈𝐑, we let

𝐹_𝑘(𝑢)=^𝑢₀𝑓_𝑘(𝑠)𝑑𝑠. (5.1) At this time, we can define the functions 𝐻∶𝐑^𝑛→𝐑 by

1𝐻(𝑥)=−2𝑥^𝑇𝐵𝑥+𝜆^𝑛_𝑘=1𝑑_𝑘𝐹_𝑘(𝑥_𝑘), (5.2) where 𝐵=𝐷𝐴. Since

𝜕𝐻(𝑥)𝜕𝑥_𝑘=−(𝐵𝑥)_𝑘+𝜆𝑑_𝑘𝑓_𝑘(𝑥_𝑘)for𝑘∈[1,𝑛], (5.3) we see that a column vector 𝑤=(𝑤₁,𝑤₂,…,𝑤_𝑛)^𝑇 is a critical point of the functional 𝐻 corresponding to 𝜆 if and only if 𝑤 is a solution of (1.1) corresponding to 𝜆.

Let

𝑓_𝑖0=lim_|𝑢|→0𝑓_𝑖(𝑢)𝑢,𝜉_𝑖∈𝐶(𝐑,𝐑)for𝑖∈[1,𝑛], (5.4) such that

𝑓_𝑖(𝑢)=𝑓_𝑖0𝑢+𝜉_𝑖(𝑢)for𝑖∈[1,𝑛], (5.5) where

lim_|𝑢|→0𝜉_𝑖(𝑢)𝑢=0for𝑖∈[1,𝑛]. (5.6) In this case, we have

1𝐻(𝑥)=−2𝑥^𝑇𝐵𝑥+𝜆^𝑛_𝑘=1𝑑_𝑘^𝑥𝑘₀𝑓_𝑘=1(𝑠)𝑑𝑠2𝑥^𝑇(𝜆𝐹₀−𝐵)𝑥+𝜆^𝑛_𝑘=1𝑑_𝑘^𝑥𝑘₀𝜉_𝑘1(𝑠)𝑑𝑠,−𝐻(𝑥)=2𝑥^𝑇𝐵𝑥−𝜆^𝑛_𝑘=1𝑑_𝑘^𝑥𝑘₀𝑓_𝑘=1(𝑠)𝑑2𝑥^𝑇(𝐵−𝜆𝐹₀)𝑥−𝜆^𝑛_𝑘=1𝑑_𝑘^𝑥𝑘₀𝜉_𝑘(𝑠)𝑑𝑠, (5.7) where 𝐹₀=diag(𝑑₁𝑓₁₀,𝑑₂𝑓₂₀,…,𝑑_𝑛𝑓_𝑛0).

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

±𝜆^𝑛_𝑘=1𝑑_𝑘^𝑥𝑘₀𝜉_𝑘(𝑠)𝑑𝑠=𝑜(‖𝑥‖²)as‖𝑥‖→0. (5.8) For the functional −𝐻(𝑥), we ask that −𝐻(𝑥)≥0 for large ‖𝑥‖, which implies that 𝐹_𝑘(𝑥_𝑘)≤0 for 𝑘∈[1,𝑛] and large |𝑥_𝑘|. In this case, if there exists 𝑚∈[𝑚₀,𝑛−1] and 𝜆>0 such that

𝛾_𝑚<𝜆𝑑_𝑘𝑓_𝑘0<𝛾_𝑚+1, (5.9) then the matrix 𝐵−𝜆𝐹₀ has exactly 𝑚 negative eigenvalues. Lemma 5.1 implies that the following result holds.

Theorem 5.2. If there exist 𝑚∈[𝑚₀,𝑛−1] and 𝜆>0 such that 𝛾_𝑚<𝛾_𝑚+1 and that 𝛾_𝑚<𝜆𝑑_𝑘𝑓_𝑘0<𝛾_𝑚+1for𝑘∈[1,𝑛], (5.10) further suppose that there is 𝑅_𝜆>0 such that 𝐹_𝑘(𝑥_𝑘)≤0 for |𝑥_𝑘|>𝑅_𝜆 and 𝑘∈[1,𝑛]. Then the system (1.1) has at least 𝑚 nonzero solution pairs.

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

Corollary 5.3. For any 𝑘∈[1,𝑛], if there exists 𝜆>0 such that 𝜆𝑑_𝑘𝑓_𝑘0>𝛾_𝑛 and there is 𝑅_𝜆>0 such that 𝐹_𝑘(𝑥_𝑘)≤0 for |𝑥_𝑘|>𝑅_𝜆, then the system (1.1) has at least 𝑛 nonzero solution pairs.

Corollary 5.4. If there exists 𝜆>0 such that 0<𝜆𝑑_𝑘𝑓_𝑘0<𝛾_𝑚0+1for𝑘∈[1,𝑛], (5.11) further suppose that there is 𝑅_𝜆>0 such that 𝐹_𝑘(𝑥_𝑘)≤0 for |𝑥_𝑘|>𝑅_𝜆 and 𝑘∈[1,𝑛]. Then the system (1.1) has at least 𝑚₀ nonzero solution pairs, particularly, the system (1.1) has at least one positive-negative solution pair.

Corollary 5.5. For any 𝑘∈[1,𝑛] and 𝜆>0, the conditions 𝑓_𝑘0=+∞ and 𝑓_𝑘∞=−∞ imply that the system (1.1) has at least 𝑛 nonzero solution pairs.

Now, we consider the functional 𝐻(𝑥). Similarly, we have the following result by using Lemma 5.1.

Theorem 5.6. If there exist 𝑚∈[𝑚₀+1,𝑛] and 𝜆>0 such that 𝛾_𝑚−1<𝛾_𝑚 and that 𝛾_𝑚−1<𝜆𝑑_𝑘𝑓_𝑘0<𝛾_𝑚for𝑘∈[1,𝑛], (5.12) further suppose that there is 𝑅_𝜆>0 such that 𝜆𝑑_𝑘𝐹_𝑘(𝑥_𝑘1)≥2𝛾_𝑘𝑥²_𝑘for|𝑥_𝑘|>𝑅_𝜆,𝑘∈[1,𝑛]. (5.13) Then the system (1.1) has at least 𝑛−𝑚+1 nonzero solution pairs.

Corollary 5.7. Assume that the condition 𝑓_𝑘0<0 holds for any 𝑘∈[1,𝑛], and there exist 𝜆>0 and 𝑅_𝜆>0 such that the condition (5.13) is valid, then the system (1.1) has at least 𝑛 nonzero solution pairs.

Corollary 5.8. For any 𝑘∈[1,𝑛] and 𝜆>0, the conditions 𝑓_𝑘0=−∞ and 𝑓_𝑘∞=+∞ imply that the system (1.1) has at least 𝑛 nonzero solution pairs.

Consider the algebraic equations

𝑥−𝑦=𝜆𝑥³,−𝑥+𝑦=𝜆𝑦³, (5.14) which is the special case of (1.1). Obviously, for any 𝜆>0 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 ±(2/𝜆,−2/𝜆). 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 𝑝𝑚-periodic solution for

Δ²𝑥_𝑘−1+𝑓(𝑘,𝑥_𝑘)=0for𝑘∈𝐙. (6.1) The main result they derived is described as follows.

Assume 𝑓(𝑡,𝑧) satisfies the following condition:

(i)𝑓∈𝐶(𝐙×𝐑,𝐑), and there exists positive integer 𝑚 such that for any (𝑡,𝑧)∈𝐙×𝐑, 𝑓(𝑡+𝑚,𝑧)=𝑓(𝑡,𝑧);(ii)for any 𝑧∈𝐑, ∫^𝑧₀𝑓(𝑡,𝑠)𝑑𝑠≥0, and 𝑓(𝑡,𝑧)=𝑜(𝑧),(𝑧→0);(iii)there exists 𝑅>0,𝛽>2, such that for any |𝑧|≥𝑅, 𝑧𝑓(𝑡,𝑧)≥𝛽^𝑧₀𝑓(𝑡,𝑠)𝑑𝑠>0, (6.2) then for any positive integer 𝑝, (6.1) has at least three 𝑝𝑚-periodic solutions.

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

Δ²𝑥_𝑘−1+𝜆𝑓_𝑘(𝑥_𝑘)=0,𝑘∈𝐙, (6.3) where 𝑝 is a given positive integer, where 𝜆>0 is a parameter, 𝑓_𝑘∈𝐶(𝐑,𝐑) and 𝑓_𝑘(−𝑢)=−𝑓_𝑘(𝑢) for 𝑢∈𝐑, and there exists a positive integer 𝑇 such that for any 𝑘∈𝐙, 𝑢∈𝐑, 𝑓_𝑘+𝑇(𝑢)=𝑓_𝑘(𝑢). 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 𝜔=𝑝𝑇 and has the eigenvalues

𝛿_𝑘=4sin²(𝑘−1)𝜋𝑝𝑇,𝑘∈[1,𝑝𝑇]. (6.4) Note that

sin²𝑘𝜋𝑝𝑇=sin²(𝑝𝑇−𝑘)𝜋,𝑝𝑇 (6.5) then we have

0=𝛿₁<𝛿₂=𝛿_𝑝𝑇<𝛿₃=𝛿_𝑝𝑇−1<⋯<𝛿_{(𝑝𝑇+2)/2} (6.6) when 𝑝𝑇 is even and

0=𝛿₁<𝛿₂=𝛿_𝑝𝑇<𝛿₃=𝛿_𝑝𝑇−1<⋯<𝛿_{(𝑝𝑇+1)/2}=𝛿_{(𝑝𝑇+3)/2} (6.7) when 𝑝𝑇 is odd.

Again let

𝛾₁=𝛿₁<𝛾₂=𝛾₃=𝛿₂,…. (6.8) By using Theorems 5.2 and 5.6, we can obtain the following facts.

Theorem 6.1. For any 𝑘∈[1,𝑝𝑇], if there exist 𝑚∈[1,𝑝𝑇2],when𝑝𝑇iseven,[1,(𝑝𝑇−1)2],when𝑝𝑇isodd, (6.9) and 𝜆>0 such that 4sin²(𝑚−1)𝜋𝑝𝑇<𝜆𝑓_𝑘0<4sin²𝑚𝜋𝑝𝑇for𝑘∈[1,𝑝𝑇], (6.10) further suppose that there is 𝑅_𝜆>0 such that 𝐹_𝑘(𝑥_𝑘)≤0 for |𝑥_𝑘|>𝑅_𝜆 and 𝑘∈[1,𝑝𝑇]. Then the problem (6.3) has at least 4𝑚−2 nonzero 𝑝𝑇-periodic solutions.

Theorem 6.2. When 𝑝𝑇 is a even positive integer and there exist 𝑚∈[2,𝑝𝑇/2+1] and 𝜆>0 such that (6.10) holds, further suppose that there is 𝑅_𝜆>0 such that 𝜆𝐹_𝑘(𝑥_𝑘)≥2sin²(𝑘−1)𝜋𝑥𝑝𝑇^2𝑘for|𝑥_𝑘|>𝑅_𝜆,𝑘∈[1,𝑝𝑇], (6.11) then the system (6.3) has at least 2(𝑝𝑇−2𝑚) nonzero 𝑝𝑇-periodic solutions; If 𝑝𝑇 is an odd positive integer and there exist 𝑚∈[2,(𝑝𝑇+1)/2] and 𝜆>0 such that (6.10) and (6.11) hold, then the system (6.3) has at least 2(𝑝𝑇−2𝑚)+2 nonzero 𝑝𝑇-periodic solutions.

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

Corollary 6.3. For any 𝑘∈[1,𝑝𝑇],𝑓_𝑘0=0,𝑓_𝑘∞=∞, (6.12) or 𝑓_𝑘0=∞,𝑓_𝑘∞=0 (6.13) hold, then the problem (6.3) has at least 2(𝑝𝑇−1) nonzero 𝑝𝑇-periodic solutions.

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

When 𝑓_𝑘(𝑢)=𝑢³ for any 𝑘∈𝐙, we have

lim_|𝑢|→0𝑢³𝑢=0,lim_|𝑢|→∞𝑢³𝑢=∞. (6.14) Corollary 6.3 implies that for any positive integer 𝑝 and positive number 𝜆, the equation

Δ²𝑥_𝑘−1+𝜆𝑥³_𝑘=0 (6.15) has at least 2(𝑝−1) nonzero 𝑝-periodic solutions. For example, let 𝑝=3 and 𝜆=1, we can consider the existence of solutions for nonlinear system:

2𝑥₁−𝑥₂−𝑥₃=𝑥³₁,−𝑥₁+2𝑥₂−𝑥₃=𝑥³₂,−𝑥₁−𝑥₂+2𝑥₃=𝑥³₃, (6.16) which has at least 4 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:

𝑥₃=−1.5033,𝑥₁=1.894,𝑥₂𝑥=−1.5033,₃=1.5033,𝑥₁=−1.894,𝑥₂𝑥=1.5033,₃=−1.894,𝑥₁=1.5033,𝑥₂𝑥=1.5033,₃=1.894,𝑥₁=−1.5033,𝑥₂𝑥=−1.5033,₃=−1.5033,𝑥₁=−1.5033,𝑥₂,𝑥=1.894₃=1.5033,𝑥₁=1.5033,𝑥₂=−1.894. (6.17) However, we find that the number of nonzero solutions of (6.16) is more than 4. In fact, (6.16) has also nontrivial solutions

𝑥₃=−1.7321,𝑥₁=1.7321,𝑥₂𝑥=0.0,₃=−1.7321,𝑥₁=0.0,𝑥₂𝑥=1.7321,₃=1.7321,𝑥₁=−1.7321,𝑥₂𝑥=0.0,₃=1.7321,𝑥₁=0.0,𝑥₂𝑥=−1.7321,₃=0.0,𝑥₁=−1.7321,𝑥₂𝑥=1.7321,₃=0.0,𝑥₁=1.7321,𝑥₂=−1.7321. (6.18) Thus, we have the following open problem.

Open Problem  1
Obtain better existence results for the system (1.1) when 𝑛≥3.
When 𝑓_𝑘(𝑢)=𝑢(1−𝑢²) for any 𝑘∈𝐙, we have
lim_|𝑢|→0𝑓(𝑢)𝑢=1,lim_|𝑢|→∞𝑓(𝑢)𝑢=−∞. (6.19) For any 𝜆>4,when𝑛iseven,4cos²𝜋2𝑛,when𝑛isodd, (6.20) Theorem 6.1 implies that the periodic boundary value problem of the form Δ²𝑥_𝑘−1+𝜆𝑥_𝑘(1−𝑥²_𝑘𝑥)=0,𝑘∈[1,𝑛]₀=𝑥_𝑛,𝑥₁=𝑥_𝑛+1 (6.21) has at least 2𝑛 nonzero solutions.
On the other hand, Zhou et al. [29] consider the discrete time second-order dynamical systems:
𝑋_𝑘+1−2𝑋_𝑘+𝑋_𝑘−1+𝑔(𝑘,𝑋_𝑘)=0,𝑘∈𝑍, (6.22) where 𝑔(𝑔₁,𝑔₂,…,𝑔_𝑙)^𝑇∈𝐶(𝑍×𝑅^𝑙,𝑅^𝑙) and 𝑔(𝑘+𝜔,𝑈)=𝑔(𝑘,𝑈) for any (𝑘,𝑈)∈𝑍×𝑅^𝑙. 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 𝐼 be an 𝜔×𝜔 unit matrix, then the problem (6.22) is equal to the system (1.1) where 𝐴=2𝐼−𝐼0⋯−𝐼−𝐼2𝐼−𝐼⋯0⋯⋯⋯0⋯−𝐼2𝐼−𝐼−𝐼⋯0−𝐼2𝐼, (6.23) which has the eigenvalues 𝛿_𝑘𝑖=4sin²(𝑘_𝑖−1)𝜋𝜔,𝑘_𝑖=𝑖∈[1,𝜔]. (6.24)

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

𝑥_𝑖^𝑡+1−𝑥^𝑡_𝑖=𝑥^𝑡_𝑖−1+𝑥^𝑡_𝑖+1−𝑓(𝑖,𝑥^𝑡_𝑖) (6.25) with the periodic boundary value conditions

𝑥^𝑡₀=𝑥^𝑡_𝜔,𝑥^𝑡₁=𝑥^𝑡_𝜔+1. (6.26)

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

𝑥_𝑖−1+𝑥_𝑖+1=𝑓(𝑖,𝑥_𝑖𝑥),𝑖∈[1,𝜔]₀=𝑥_𝜔,𝑥₁=𝑥_𝜔+1 (6.27) which can be rewritten by the system (1.1), where 𝜆=1, the coefficient matrix is defined by(2.13)

𝐹_𝑘(𝑢)=𝑢²−^𝑢₀𝑓𝑓(𝑘,𝑠)𝑑𝑠,_𝑘0=lim_𝑢→0𝑓(𝑘,𝑢)𝑢,𝐹(𝑥)=(2𝑥₁−𝑓(1,𝑥₁),…,2𝑥_𝜔−𝑓(𝜔,𝑥_𝜔))^𝑇. (6.28) In this case, in Theorems 6.1 and 6.2  𝑝𝑇 and 𝑓_𝑘0 are, respectively, replaced by 𝜔 and 2−𝑓_𝑘0, 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

𝑋_𝑘+1+𝑋_𝑘−1−𝑔(𝑘,𝑋_𝑘)=0, (6.29) our results are also valid, where 𝑋_𝑘 and 𝑔 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:

𝑥_𝑖^𝑡+1=𝑥^𝑡_𝑖−1+𝑥^𝑡_𝑖+1+𝑓_𝑖(𝑥^𝑡_𝑖) (6.30) with the periodic boundary value condition (6.26). Similarly, the problem (6.30)–(6.26) be rewritten by the system (1.1), where 𝜆=1, 𝐴 is defined by (2.13), and

𝐹_𝑘1(𝑢)=2𝑢²+^𝑢₀𝑓_𝑘(𝑠)𝑑𝑠. (6.31) 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

𝑋_𝑖^𝑡+1=𝑋^𝑡_𝑖−1+𝑋^𝑡_𝑖+1+𝑓_𝑖(𝑋^𝑡_𝑖), (6.32) where 𝑋^𝑡_𝑖 is a 𝑘-vector for each 𝑖∈𝑍.

6.3. Periodic Solutions for Fourth-Order Difference Equation

When 𝜔=2, the problem (2.15) exists 2-periodic solutions if and only if the nonlinear system

81−1𝑥−11𝑥¹₂=𝜆𝑓(1,𝑥₁)𝑓(2,𝑥₂) (6.33) has nontrivial solutions. The matrix

1−1−11 (6.34) has the eigenvalues 𝛾₁=0 and 𝛾₂=2. When 𝜔=3, the problem (2.15) exists 3-periodic solutions if and only if the nonlinear system

32−1−1𝑥−12−1−1−12¹𝑥₂𝑥₃=𝜆𝑓(1,𝑥₁)𝑓(2,𝑥₂)𝑓(3,𝑥₃) (6.35) has nontrivial solutions. The matrix

2−1−1−12−1−1−12 (6.36) has the eigenvalues 𝛾₁=0,𝛾_2,3=3.

When 𝜔=4, the problem (2.15) exists 4-periodic solutions if and only if the nonlinear system

23−21−2𝑥−23−211−23−2−21−23¹𝑥₂𝑥₃𝑥₄=𝜆𝑓(1,𝑥₁)𝑓(2,𝑥₂)𝑓(3,𝑥₃)𝑓(4,𝑥₄) (6.37) has nontrivial solutions. The matrix

3−21−2−23−211−23−2−21−23 (6.38) has the eigenvalues 𝛾₁=0,𝛾_2,3=2, and 𝛾₄=8.

When 𝜔≥5, (2.15) exists 𝜔-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 𝐴 from the fourth-order linear difference equation

𝑥_𝑘−2−4𝑥_𝑘−1+6𝑥_𝑘−4𝑥_𝑘+1+𝑥_𝑘+2=𝜆𝑥_𝑘for𝑘∈[1,𝜔] (6.39) with the periodic boundary value conditions:

𝑥₋₁=𝑥_𝜔−1,𝑥₀=𝑥_𝜔,𝑥_𝜔+1=𝑥₁,𝑥_𝜔+2=𝑥₂. (6.40)

Let

𝑥_𝑘=𝑡^𝑘. (6.41) In view of (6.39), we have

𝑡^𝑘−2−4𝑡^𝑘−1+6𝑡^𝑘−4𝑡^𝑘+1+𝑡^𝑘+2=𝜆𝑡^𝑘, (6.42) which implies that

𝜆=(1−𝑡)⁴𝑡². (6.43) From (6.40), we see that

𝑡^𝜔=1, (6.44) thus, we have

𝑡_𝑝=exp(2(𝑝−1)𝜋𝜔𝑖),𝑝∈[1,𝜔],1−𝑡_𝑝=2sin((𝑝−1)𝜋𝜔𝜋)exp((2+(𝑝−1)𝜋𝜔+𝑗𝜋)𝑖) (6.45) for 𝑝∈[1,𝜔] and 𝑗∈𝑍. In view of (6.43), we have obtained

𝜆_𝑝=(1−𝑡_𝑝)⁴(𝑡_𝑝)²=2⁴sin⁴((𝑝−1)𝜋𝜔)>0,𝑝∈[1,𝜔]. (6.46) Note that

sin⁴𝑝𝜋𝜔=sin⁴(𝜔−𝑝)𝜋𝜔, (6.47) then

0=𝛿₁<𝛿₂=𝛿_𝜔<𝛿₃=𝛿_𝜔−1<⋯<𝛿_𝜔/2+1(𝜔iseven)0=𝛿₁<𝛿₂=𝛿_𝜔<𝛿₃=𝛿_𝜔−1<⋯<𝛿_[𝜔/2]+1=𝛿_[𝜔/2]+2(𝜔isodd). (6.48)

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 analysis, the positive steady-state solutions are usually needed. The authors in [41] think that persistent puzzle in the field of biological electron transfer is the conserved iron-sulfur cluster motif in both high potential iron-sulfur protein (HiPIP) and ferredoxin (Fd) active sites. However, the voltage in cell can be negative and there exists the negative threshold, see [45]. Thus, the negative steady-state should be also considered for Turing pattern analysis. Our results will likely find important implications in other real evolutionary processes.

By the discussion in Section 2, we have known that the models of many applied problems can be expressed by the partial difference equation of the form

Δ²₁𝑥_{𝑖−1,𝑗}+Δ²₂𝑥_{𝑖,𝑗−1}+𝜆𝑓_𝑖𝑗(𝑥_𝑖𝑗)=0 (6.49) with periodic boundary value conditions

𝑥_0,𝑗=𝑥_𝑛,𝑗,𝑥_1,𝑗=𝑥_𝑛+1,𝑗𝑥,𝑗∈[1,𝑚],_𝑖,0=𝑥_𝑖,𝑚,𝑥_𝑖,1=𝑥_𝑖,𝑚+1,𝑖∈[1,𝑛]. (6.50) In fact, we can also give the other explanation.

Indeed, let us consider 𝑛×𝑚 neuron units placed on a torus. Let 𝑥^𝑡_𝑖𝑗 denote the state value of the 𝑖𝑗th neuron unit during the time periodic 𝑡∈{0,1,2,…}. Assume that each neuron unit is random activated by its four neighbors so that the change of state values between two consecutive time periods is given by

𝑥^𝑡+1_𝑖𝑗−𝑥^𝑡_𝑖𝑗=𝛼(Δ²₁𝑥^𝑡_{𝑖−1,𝑗}+Δ²₂𝑥^𝑡_{𝑖,𝑗−1})+𝑓_𝑖𝑗(𝑥^𝑡_𝑖𝑗) (6.51) with the periodic boundary value conditions

𝑥^𝑡_0,𝑗=𝑥^𝑡_𝑛,𝑗,𝑥^𝑡_1,𝑗=𝑥^𝑡_𝑛+1,𝑗𝑥,𝑗∈[1,𝑚],_𝑡𝑖,0=𝑥^𝑡_𝑖,𝑚,𝑥^𝑡_𝑖,1=𝑥^𝑡_𝑖,𝑚+1,𝑖∈[1,𝑛], (6.52) where 𝛼 is the connection weight, 𝑓_𝑖𝑗 stands for the bias mechanism inherent in the 𝑖𝑗th neuron unit.

In order to utilize the neural network modeled by the aforementioned evolutionary system, it is of interest to predict the existence of steady-state solution {𝑥^𝑡_𝑖𝑗,(𝑖,𝑗)∈[1,𝑛]×[1,𝑚]}^∞_𝑡=0 such that 𝑥^𝑡_𝑖𝑗=𝑥_𝑖𝑗 for (𝑖,𝑗)∈[1,𝑛]×[1,𝑚] and 𝑡≥0. This then leads us to finding solutions of the steady system of (6.49)-(6.50) or more generally (1.1).

We can similarly obtain the eigenvalues of corresponding (6.49)–(6.50):

𝜇_𝑖𝑗=8(sin²(𝑖−1)𝜋𝑛+sin²(𝑗−1)𝜋𝑚) (6.53) for (𝑖,𝑗)∈[1,𝑛]×[1,𝑚], thus, the existence and nonexistence of solutions for (6.49)–(6.50) can also be established. They will omitted.

In the present paper, we ask that the coefficient matrix and the nonlinear term of the system (1.1), respectively, satisfy the symmetry and the odd symmetry. Clearly, a number of application problems are not valid. They will be considered in the further paper.

Acknowledgment

This work is supported by the special Fundation of Tianjin Education Committee (2006ZH91) and Recearch Fundation of Tianjin Education Committee (20050404).