Existence of Solutions for a Nonlinear Algebraic System

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 LusternikSchnirelmann 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.


Introduction
In this paper, the nonlinear algebraic system,

Ax λf x ,
1.1 will be considered, where λ > 0 is a parameter, 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 d 1 , d 2 , . . ., d n such that DA is nonnegative definite.The letter T will denote transposition.
For a given λ > 0, a column vector x x 1 , x 2 , . . ., 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 k 0 ∈ 1, n such that x k 0 / 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][27][28][29][36][37][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.

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.

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 can be written by

2.4
In view of 2.3 , we can obtain 1.1 with the n × n square matrix A having the form The conditions r t ≥ r min > 0 and At the same time, the matrix A has a zero eigenvalue and all other eigenvalues are positive, see 7 .Let imply that the matrix DA is symmetric.As an simiple example, we consider then there exists the positive diagonal matrix D diag 1, 2, 1 such that 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: 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 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 t 1 n u t n for all n ∈ Z and t ∈ N. In view of 2.10 , we have 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,

2.13
Recently, Zhou et al. 29 consider the discrete time second order dynamical systems where g g 1 , g 2 , . . ., 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 fourthorder difference equation 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 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.

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: where q i represents the mass of compartment i. f ii − f 0i j / i f 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:

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 When g i is an odd function, we can obtain 1.1 with the n × n square matrix −F having the form 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.

Strongly Damped Lattice System
Recently, Li and Zhou 21 considered the following second-order lattice dynamic system: m is a vector with the components u i and can be ordered as the following form of 1-dimensional vector in R m n : A is a nonnegative definite matrix on R m n with eigenvalues λ s ≥ 0 0 ≤ s ≤ m n − 1 , and 0 is the simple and minimal eigenvalue with corresponding eigenvector e 1, . . ., 21 can be regarded as the discrete analogue of the initial-boundary value problem of the following continuous strongly damped wave equation: where 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.

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 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 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 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 where X x 1 , x 2 , . . ., x mN T is the state vector, and the F X denotes the mN-dimensional functional value vector of X, and 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 .

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 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 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 d 1 , d 2 , . . ., 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 where

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

2.35
with the periodic boundary value conditions:

2.36
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 X k and g are similar with 2.14 .

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 with the periodic boundary value conditions x 0,j x n,j , x 1,j x n 1,j , j ∈ 1, m , x 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.

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 and the corresponding orthonormal eigenvectors are v 1 , v 2 , . . ., v n .
First of all, we let x be a nontrivial solution of 1.1 .Multiplying 1.1 by x T D on the left we get which implies that In view of the reference 73 , we know that max x / 0 x T Bx x T x γ n , min x / 0 x T Bx x T x γ 1 0. 4.4 Thus, we have which implies that the following nonexistence result is fact.
Theorem 4.1.If there exists λ > 0 such that then the system 1.1 has no nontrivial solutions.

Now, we assume that
Clearly, the condition l L ∞ implies that the infimum inf u / 0 f u u 4.9 exists and n k 1 The condition l L −∞ implies that the supremum sup exists and Thus, Theorem 4.1 implies that the following results hold.
Then the condition l L ∞ implies that the system 1.1 has no nontrivial solutions when there exists λ > 0 such that holds.The condition l L −∞ implies that for any λ > 0, the system 1.1 has no nontrivial solutions when When l and L satisfy 0 < l, L < ∞, we easily obtain the following results.
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.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

Corollary 4.4. Assume that
4.15 Thus, we have the following result.
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 .

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 T Bx, 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 C 1 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 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

5.1
At this time, we can define the functions H : R n → R by where B DA. Since we see that a column vector w w 1 , w 2 , . . ., 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 where lim |u| → 0 ξ i u u 0 for i ∈ 1, n .

5.6
In this case, we have where By using the condition 5.6 , we easily get that 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 ∈ m 0 , n − 1 and λ > 0 such that γ m < λd k f k0 < γ m 1 , 5.9 then the matrix B − λF 0 has exactly m negative eigenvalues.Lemma 5.1 implies that the following result holds.
Theorem 5.2.If there exist m ∈ m 0 , n − 1 and λ > 0 such that γ m < γ m 1 and that 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 0 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 ∈ m 0 1, n and λ > 0 such that γ m−1 < γ m and that further suppose that there is R λ > 0 such that

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 λx 3 , −x y λy 3 , 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 .

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.

Periodic Solutions
Guo and Yu 28 considered the existence of pm-periodic solution for The main result they derived is described as follows.Assume f t, z satisfies the following condition: , 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, z 0 f 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, 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: 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 Note that sin 2 kπ pT sin 2 pT − k π pT , 6.5 then we have when pT is even and when pT is odd.Again let 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 , when pT is odd, 6.9 and λ > 0 such that 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 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.
When f k u u 3 for any k ∈ Z, we have 6.14 Corollary 6.3 implies that for any positive integer p and positive number λ, the equation 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:

Open Problem 1
Obtain better existence results for the system 1.1 when n ≥ 3. When f k u u 1 − u 2 for any k ∈ Z, we have For any when n is even, 4 cos 2 π 2n , when n is odd, 6.20 Theorem 6.1 implies that the periodic boundary value problem of the form has at least 2n nonzero solutions.
On the other hand, Zhou et al. 29 consider the discrete time second-order dynamical systems: where g g 1 , g 2 , . . ., 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 which has the eigenvalues 6.24

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 λ 1, the coefficient matrix is defined by 2.13

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 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: 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 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 X t i is a k-vector for each i ∈ Z.

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 has the eigenvalues γ 1 0, γ 2,3 3.
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 with the periodic boundary value conditions: x x ω 2 x 2 .6.40

Let
x k t k .

6.41
In view of 6.39 , we have

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.

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 Δ 2 1 x i−1,j Δ 2 2 x 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,j , j ∈ 1, m , x 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 with the periodic boundary value conditions x t 0,j x t n,j , x t 1,j x t n 1,j , j ∈ 1, m , x t i,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 : 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.
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.