Journal of Applied Mathematics

Volume 2012 (2012), Article ID 580482, 9 pages

doi:10.1155/2012/580482

Research Article

Type-K Exponential Ordering with Application to Delayed Hopfield-Type Neural Networks

Bin-Guo Wang

School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China

Received 10 December 2011; Accepted 9 April 2012

Academic Editor: Chuanhou Gao

Copyright © 2012 Bin-Guo Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Order-preserving and convergent results of delay functional differential equations without quasimonotone condition are established under type-K exponential ordering. As an application, the model of delayed Hopfield-type neural networks with a type-K monotone interconnection matrix is considered, and the attractor result is obtained.

1. Introduction

Since monotone methods have been initiated by Kamke [1] and Müler [2], and developed further by Krasnoselskii [3, 4], Matano [5], and Smith [6], the theory and application of monotone dynamics have become increasingly important (see [7–18]).

It is well known that the quasimonotone condition is very important in studying the asymptotic behaviors of dynamical systems. If this condition is satisfied, the solution semiflows will admit order-preserving property. There are many interesting results, for example, [6, 8–12, 14–17] for competitive (cooperative) or type-K competitive (cooperative) systems and [6, 7, 13] for delayed systems. In particular, for the scalar delay differential equations of the form

if the quasimonotone condition holds, then (1.1) generates an eventually strongly monotone semiflow on the space , which is one of sufficient conditions for obtaining convergent results. In other words, the right hand side of (1.1) must be strictly increasing in the delayed argument. This is a severe restriction, and so the quasimonotone conditions are not always satisfied in applications. Recently, many researchers have tried to relax the quasimonotone condition by introducing a new cone or partial ordering, for example, the exponential ordering [6, 18, 19]. In particular, Smith [6] and Wu and Zhao [18] considered a new cone parameterized by a nonnegative constant, which is applicable to a single equation. Replacing the previous constant by a quasipositive matrix, the exponential ordering is generalized to some delay differential systems by Smith [6] and Y. Wang and Y. Wang [19]. However, the above results are not suitable to the type-K systems (see [6] for its definition). A typical example is a Hopfield-type neural network model with a type-K monotone interconnection matrix, which implies that the interaction among neurons is not only excitatory but also inhibitory. For this purpose, we introduce a type-K exponential ordering and establish order-preserving and convergent results under the weak quasimonotone condition (see Section 2) and then apply the result to a network model with a type-K monotone interconnection matrix.

This paper is arranged as follows. In next section, the type-K exponential ordering parameterized by a type-K monotone matrix is introduced, and convergent result is established. In Section 3, we apply our results to a delayed Hopfield-type neural network.

2. Type-K Exponential Ordering

In this section, we establish a new cone and introduce some order-preserving and convergent results.

Let , be ordered Banach spaces with . For , , we write if ; if ; if . For , we denote and . Thus, we can define the product space which generates two cones and with nonempty interiors and . The ordering relation on and is defined in the following way:

A semiflow on is a continuous mapping : , , which satisfies (i) and (ii) for . Here, for and . The orbit of is denoted by :

An equilibrium point is a point for which for all . Let be the set of all equilibrium points for . The omega limit set of is defined in the usual way. A point is called a quasiconvergent point if . The set of all such points is denoted by Q. A point is called a convergent point if consists of a single point of . The set of all convergent points is denoted by C.

The semiflow is said to be type-K monotone provided

is called type-K strongly order preserving (for short type-K SOP), if it is type-K monotone, and whenever , there exist open subsets , of with , and , such that

The semiflow is said to be strongly type-K monotone on if is type-K monotone, and whenever and , then . We say that is eventually strongly type-K monotone if it is type-K monotone, and whenever , there exists such that . Clearly, strongly type-K monotonicity implies eventually strongly type-K monotonicity.

An matrix is said to be type-K monotone if it has the following manner:

where satisfies if , similarly for the matrix and , .

In this paper, the following lemma is necessary.

Lemma 2.1. If is a type-K monotone matrix, then remains type-K monotone with diagonal entries being strictly positive for all .

Proof. The product of two type-K monotone matrices remains type-K monotone; the rest is obvious and we omit it here.

Let be fixed and let . The ordering relations on are understood to hold pointwise. Consider the family of sets parameterized by type-K monotone matrix given by

It is easy to see that is a closed cone in and generates a partial ordering on which is written by . Assume that is differentiable on , a similar argument to [18, lemma 2.1] implies that if and only if and for all .

Consider the abstract functional differential equation

where is continuous and satisfies a local Lipschitz condition on each compact subset of and is an open subset of . By the standard equation theory, the solution of (2.7) can be continued to the maximal interval of existence . Moreover, if , then is a classical solution of (2.7) for . In this section, for simplicity, we assume that, for each , (2.7) admits a solution defined on . Therefore, (2.7) generates a semiflow on by , where for and .

In the following, we will seek a sufficient condition for the solution of (2.7) to preserve the ordering .

Whenever , then

Theorem 2.2. Suppose that holds. If , then for all .

Proof. Let . For any , define for , and let be a unique solution of the following equation:

Let and define

Since , is closed and nonempty. We first prove the following two claims.

Claim 1. If , there exists such that .

According to the integral expression of (2.9) we have

Since and hold, we have

By the characteristic of a cone, there is such that

By Lemma 2.1, we have

which, together with the definition of , implies that

Claim 2. Let . Then .

If , then there is a sequence such that as . From the closeness of we have . By Claim 1, for some , which contradicts the definition of . Therefore, , which implies .

Since uniformly on bounded subset of as , then

Letting in , we have , which implies that .

By the definition of the semiflow , it is easy to see from that is monotone with respect to in the sense that whenever for all .

As we all know the strongly order-preserving property is necessary for obtaining some convergent results. However, it is easy to check that the cone has empty interior on ; we cannot, therefore, expect to show that the semiflow generated by (2.7) is eventually strongly type-K monotone in . Let and define

It is easy to check that is a Banach space, is a cone with nonempty interior Int (see [20]), and is continuous. Using the smoothing property of the semiflow on and fundamental theory of abstract functional differential equations, we deduce that for all , , is continuous, and for any with . Thus, from Theorem 2.2, type-K strongly order-preserving property can be obtained.

Theorem 2.3. Assume that holds. If , then in for all .

In order to obtain the main result of this paper, which says that the generic solution converges to equilibrium, the corresponding compactness assumption will be required.

(A1) maps bounded subset of to bounded subset of . Moreover, for each compact subset of , there exists a closed and bounded subset of such that for each and all large .

Theorem 2.4. Assume that and (A1) hold. Then the set of convergent points in contains an open and dense subset. If consists of a single point, it attracts all solutions of (2.7). If the initial value and consists of two points or more, we conclude that all solutions converge to one of these.

Proof. By Theorem 2.3, the semiflow is eventually strongly monotone in . Let , where denotes a constant mapping defined on ; that is, for all . Obviously, . For any , either the sequence of points or is eventually contained in and approaches as , and, hence, each point of can be approximated either from above or from below in with respect to . The assumption (A1) implies the compactness; that is, has compact closure in for each (see [6]). Therefore, from [6, Theorem  1.4.3], we deduce that the set of quasiconvergent points contains an open and dense subset of . From the proof of [6, Theorem  6.3.1], we know that the set is totally ordered by . Reference [6, Remark  1.4.2] implies that the set of convergent points contains an open and dense subset of . The last two assertions can be obtained from [6, Theorems  2.3.1 and 2.3.2].

Remark 2.5. The above theorem implies that there exists an equilibrium attracting all solutions with initial values in the cone . If consists of a single element, the equilibrium attracts all solutions with initial values in .

3. Delayed Hopfield-Type Neural Networks

In this section, we will apply our main result to the following system of delayed differential equations:

where and are constant, . The interconnection matrix is type-K monotone with the elements in the diagonal being nonnegative. In this situation, the interaction among neurons is not only excitatory but also inhibitory. The external input functions are constants or periodic. The activation functions , where is an open subset of with , satisfy (A1) and following property.

(A2) There exist constants such that for .

First, we consider the case that the external input functions are constants.

Theorem 3.1. Equation (3.1) has an equilibrium which attracts all its solutions coming from the initial value with being bounded.

Proof. From [21, Theorem  1], we deduce that (3.1) admits at least an equilibrium; that is, the equilibrium points set is nonempty.

For , we define

Choosing with , and denoting , and . Since is bounded, for , with , there exist and with such that

From (A2) and the definition of , if , then

for all . By a similar argument we have

for all . Let and let , and define . If , we have for . If and , we deduce that reaches its positive maximum value at . Thus, there exists a positive constant such that holds; the conclusion can be obtained by Remark 2.5.

For the case of the external input functions being periodic functions, we have following result.

Theorem 3.2. For any periodic external input function , , , (3.1) admits a unique periodic solution and all other solutions which come from the initial value with being bounded converge to it as .

Proof. Let be the solution of (3.1) for with for . From the properties of the solution semiflow we have

From the proof of Theorem 3.1, we know that there exists a type-K monotone matrix such that holds; Theorem 2.4 tells us that every orbit of (3.1) is convergent to a same equilibrium, denoted by , and then,

We have, therefore,

From (3.6) and (3.8) we deduce that

Therefore, is a unique periodic solution of (3.1). Using the conclusion of Theorem 2.4 again, we have

Since is a periodic solution, the proof is complete.

Remark 3.3. Neural networks have important applications, such as to content-addressable memory [22], shortest path problem [23], and sorting problem [24]. Generally, the monotonicity is always assumed. Here, we relax the monotone condition, and hence neural networks have more extensive applications.

Acknowledgments

This paper is supported by NSF of China under Grant 10926091 and the Fundamental Research Funds for the Central Universities.

References

E. Kamke, “Zur Theorie der Systeme gewöhnlicher Differentialgleichungen. II,” Acta Mathematica, vol. 58, no. 1, pp. 57–85, 1932.

M. Müller, “Uber das fundamenthaltheorem in der theorie der gewohnlichen differentialgleichungen,” Mathematische Zeitschrift, vol. 26, pp. 619–645, 1926.

M. A. Krasnoselskii, Positive Solutions of Operator Equations, Noordhoff Groningen, Groningen, The Netherlands, 1964.

M. A. Krasnoselskii, The Operator of Translation Along Trajectories of Differential Equations, vol. 19 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, USA, 1968.

H. Matano, “Existence of nontrivial unstable sets for equilibriums of strongly order-preserving systems,” Journal of the Faculty of Science IA, vol. 30, no. 3, pp. 645–673, 1984.

H. L. Smith, Monotone Dynamics Systems: An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical Society, Providence, RI, USA, 1995.

H. I. Freedman and X.-Q. Zhao, “Global asymptotics in some quasimonotone reaction-diffusion systems with delays,” Journal of Differential Equations, vol. 137, no. 2, pp. 340–362, 1997.

M. Gyllenberg and Y. Wang, “Dynamics of the periodic type-K competitive Kolmogorov systems,” Journal of Differential Equations, vol. 205, no. 1, pp. 50–76, 2004.

X. Liang and J. Jiang, “On the finite-dimensional dynamical systems with limited competition,” Transactions of the American Mathematical Society, vol. 354, no. 9, pp. 3535–3554, 2002.

X. Liang and J. Jiang, “The classification of the dynamical behavior of 3-dimensional type-K monotone Lotka-Volterra systems,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 51, no. 5, pp. 749–763, 2002.

X. Liang and J. Jiang, “The dynamical behaviour of type-K competitive Kolmogorov systems and its application to three-dimensional type-K competitive Lotka-Volterra systems,” Nonlinearity, vol. 16, no. 3, pp. 785–801, 2003.

X. Liang and J. Jiang, “Discrete infinite-dimensional type-K monotone dynamical systems and time-periodic reaction-diffusion systems,” Journal of Differential Equations, vol. 189, no. 1, pp. 318–354, 2003.

R. H. Martin Jr. and H. L. Smith, “Reaction-diffusion systems with time delays: monotonicity, invariance, comparison and convergence,” Journal für die Reine und Angewandte Mathematik, vol. 413, pp. 1–35, 1991.

H. L. Smith, “Competing subcommunities of mutualists and a generalized Kamke theorem,” SIAM Journal on Applied Mathematics, vol. 46, no. 5, pp. 856–874, 1986.

C. Tu and J. Jiang, “The coexistence of a community of species with limited competition,” Journal of Mathematical Analysis and Applications, vol. 217, no. 1, pp. 233–245, 1998.

C. Tu and J. Jiang, “Global stability and permanence for a class of type-K monotone systems,” SIAM Journal on Mathematical Analysis, vol. 30, no. 2, pp. 360–378, 1999.

C. Tu and J. Jiang, “The necessary and sufficient conditions for the global stability of type-K Lotka-Volterra system,” Proceedings of the American Mathematical Society, vol. 127, no. 11, pp. 3181–3186, 1999.

J. Wu and X.-Q. Zhao, “Diffusive monotonicity and threshold dynamics of delayed reaction diffusion equations,” Journal of Differential Equations, vol. 186, no. 2, pp. 470–484, 2002.

Y. Wang and Y. Wang, “Global dynamics of reaction-diffusion systems with delays,” Applied Mathematics Letters, vol. 18, no. 9, pp. 1027–1033, 2005.

H. Amann, “Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces,” SIAM Review, vol. 18, no. 4, pp. 620–709, 1976.

H. Huang, J. Cao, and J. Wang, “Global exponential stability and periodic solutions of recurrent neural networks with delays,” Physics Letters A, vol. 298, no. 5-6, pp. 393–404, 2002.

S. Grossberg, “Nonlinear neural networks: principles, mechanisms, and architectures,” Neural Networks, vol. 1, no. 1, pp. 17–61, 1988.

J. Wang, “A recurrent neural network for solving the shortest path problem,” IEEE Transactions on Circuits and Systems I, vol. 43, no. 6, pp. 482–486, 1996.

J. Wang, “Analysis and design of an analog sorting network,” IEEE Transactions on Neural Networks, vol. 6, no. 4, pp. 962–971, 1995.