This Provisional PDF corresponds to the article as it appeared upon acceptance. Fully formatted PDF and full text (HTML) versions will be made available soon. Biperiodicity in neutral-type delayed difference neural networks Advances in Difference Equations 2012, 2012:5 doi:10.1186/1687-1847-2012-5 Zhenkun Huang (hzk974226@jmu.edu.cn) Youssef N Raffoul (Youssef.Raffoul@notes.udayton.edu) ISSN 1687-1847 Article type Research Submission date 17 October 2011 Acceptance date 31 January 2012 Publication date 31 January 2012 Article URL http://www.advancesindifferenceequations.com/content/2012/1/5 This peer-reviewed article was published immediately upon acceptance. It can be downloaded, printed and distributed freely for any purposes (see copyright notice below). For information about publishing your research in Advances in Difference Equations go to http://www.advancesindifferenceequations.com/authors/instructions/ For information about other SpringerOpen publications go to http://www.springeropen.com Advances in Difference Equations © 2012 Huang and Raffoul ; licensee Springer. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Biperiodicity in neutral-type delayed difference neural networks Zhenkun Huang ∗1 and Youssef N Raffoul 2 1 School of Science, Jimei University, Xiamen 361021, P. R. China 2 Department of Mathematics, University of Dayton, Dayton, OH 45469-2316, USA ∗ Corresponding author: hzk974226@jmu.edu.cn Email address: YNR: Youssef.Raffoul@notes.udayton.edu Abstract In this article we employ Krasnoselskii’s fixed point theorem to obtain new biperiodicity criteria for neutral-type difference neural networks with delays. It is shown that the neutral-type term can leads to biperiodicity results. That is coexistence of a positive p eriodic sequence solution and its anti-sign periodic sequence solution. We illustrate our novel approach the biperiodicity dynamics of biperiodicity for neutral-type delay difference neural networks by two computer numerical examples. Mathematics Subject Classification 2010: 39A23; 39A10. Keywords: difference neural networks; biperiodicity; neutral-type; delayed. 1 Introduction It is well known that neural networks with delays have a rich dynamical behavior that have been recently investigated by Huand and Li [1] and the references therein. It is naturally important that such systems should contain some information regarding the past rate of change since they effectively describe and model the dynamic of the application of neural networks [2–4]. As a consequence, scholars and researchers have paid more attention to the stability of neural networks that are described by nonlinear delay differential equations of the neutral type (see [4–8]) ˙u i (t) = −a i (t)u i (t) + m j=1 b ij (t)g j (u j (t)) + m j=1 c ij ˙u j (t − τ) + m j=1 d ij (t)g j t −∞ h j (t − s)u j (s)ds + I i (t), i ∈ N := {1, 2, · · · , m} (1.1) Cheng et al. first investigated the globally asymptotic stability of a class of neutral-type neural networks with delays [6]. Delay-dependent criterion has been attained in [5] by us- 1 ing Lyapunov stability theory and linear matrix inequality. Recently, a conservative robust stability criteria for neutral-type networks with delays are proposed in [4] by using a new Lyapunov–Krasovskii functional and a novel series compensation technique. For more rela- tive results, we can refer to [4,7] and references cited therein. Difference equations or discrete-time analogs of differential equations can preserve the convergence dynamics of their continuous-time counterparts in some degree [9]. So, due to its usage in computer simulations and applications, these discrete-type or difference networks have been deeply discussed by the authors of [10–15] and extended to periodic or almost periodic difference neural systems [16–21]. However, few papers deal with multiperiodicity of neutral-type difference neural net- works with delays. Stimulated by the articles [22,23], in this article, we should consider corresponding neutral-type difference version of (1.1) as follows: u i (n + 1) = a i (n)u i (n) + m j=1 c ij u j (n − τ) + m j=1 b ij (n)g j (u j (n)) + m j=1 d ij (n)g j ∞ v=1 h j (v)u j (n − v) + I i (n), (1.2) where i ∈ N := {1, 2, . . . , m}. Our main aim is to study biperiodicity of the above neutral- type difference neural networks. Some new criteria for coexistence of a perio dic sequence solution and anti-sign periodic one of (1.2) have been derived by using Krasnoselskii’s fixed point theorem. Our results are completely different from monop eriodicity existing ones in [16–20]. The rest of this article is organized as follows. In Section 2, we shall make some prepa- rations by giving some lemmas and Krasnoselskii’s fixed point theorem. In Section 3, we gives new criteria for biperiodicity of (1.2). Finally, two numerical examples are given to illustrate our results. 2 Preliminaries We begin this section by introducing some notations and some lemmas. Let S T be the set of all real T -periodic sequences defined on Z, where T is an integer with T ≥ 1. Then S T is a Banach space when it is endowed with the norm u = max i∈N sup s∈[0,T ] Z u i (s) . Denote [a, b ] Z := {a, a + 1, . . . , b}, where a, b ∈ Z and a ≤ b. Let C((−∞, 0] Z , R m ) be the set of all continuous and bounded functions ψ(s) = (ψ 1 (s), ψ 2 (s), . . . , ψ m (s)) T mapping (−∞, 0] Z into R m . For any given ψ ∈ C((−∞, 0] Z , R N ), we denote by {u(n; ψ)} the sequence solution of system (1.2). Next, we present the basic assumptions: 2 • Assumption (H 1 ): Each a i (·), b ij (·), d ij (·), and I i (·) are T-periodic functions defined on Z, 0 < a i (n) < 1. The activation g j (·) is strictly increasing and bounded with −g j = lim v→−∞ g j (v) < g j (v) < lim v→+ ∞ g j (v) = g j for all v ∈ R. The kernel h j : N → R + is a bounded sequence with ∞ v=1 h j (v) = 1, where i, j ∈ N . For each i ∈ N and any n ∈ Z, we let G i (n, p) = n+T −1 s=p+1 a i (s) 1 − n+T −1 s=n a i (s) −1 , p ∈ [n, n + T − 1] (2.1) Since 0 < a i (n) < 1 for all n ∈ [0, T − 1], each G i (n, p) is not zero and m i := min{G i (n, p) : n ≥ 0, p ≤ T } = G i (n, n) = G i (0, 0) > 0, M i := max{G i (n, p) : n ≥ 0, p ≤ T } = G i (n, n + T − 1) = G i (0, T − 1) > 0. Lemma 2.1. For each i ∈ N and ∀p ∈ Z + , S p−1 s=0 a −1 i (s) u i (p − τ) + u i (p − τ) p−1 s=0 a −1 i (s) = p−1 s=0 a −1 i (s)u i (p − τ) holds for any sequence solution {u(n)} of (1.2), where S is a shift operator defined as S u i (p) = u i (p + 1) for i ∈ N and p ∈ Z + . Proof. S p−1 s=0 a −1 i (s) u i (p − τ) + u i (p − τ) p−1 s=0 a −1 i (s) = p s=0 a −1 i (s) u i (p + 1 − τ) − u i (p − τ) + u i (p − τ) p s=0 a −1 i (s) − p−1 s=0 a −1 i (s) = p s=0 a −1 i (s)u i (p + 1 − τ) − p−1 s=0 a −1 i (s)u i (p − τ) = p−1 s=0 a −1 i (s)u i (p − τ) . The proof is complete. Lemma 2.2. Assume that (H 1 ) hold. Any sequence {u(n)} ∈ S m T := S T × S T × · · · × S T m is a solution of (1.2) if and only if u i (n) = m j=1 c ij u j (n − τ) + n+T −1 p=n G i (n, p) m j=1 b ij (p)g j (u j (p)) + m j=1 d ij (p)g j ∞ v=1 h j (v)u j (p − v) + I i (p) − m j=1 c ij u j (p − τ)(1 − a i (p)) , (2.2) 3 where G i (n, p) is defined by (2.1) for i ∈ N and p ∈ Z + . Proof. Rewrite (1.2) as u i (n) n−1 s=0 a −1 i (s) = m j=1 c ij u j (n − τ) + m j=1 b ij (n)g j (u j (n)) + m j=1 d ij (n)g j ∞ v=1 h j (v)u j (n − v) + I i (n) n s=0 a −1 i (s), (2.3) where i ∈ N and n ∈ Z + . Summing (2.3) from n to n + T − 1, we obtain n+T −1 p=n u i (p) p−1 s=0 a −1 i (s) = n+T −1 p=n m j=1 c ij u j (p − τ) + m j=1 b ij (p)g j (u j (p)) + m j=1 d ij (p)g j ∞ v=1 h j (v)u j (p − v) + I i (p) p s=0 a −1 i (s). That is, u i (n + T ) n+T −1 s=0 a −1 i (s) − u i (n) n−1 s=0 a −1 i (s) = n+T −1 p=n m j=1 c ij u j (p − τ) + m j=1 b ij (p)g j (u j (p)) + m j=1 d ij (p)g j ∞ v=1 h j (v)u j (p − v) + I i (p) p s=0 a −1 i (s). Since u i (n + T ) = u i (n), we obtain u i (n) n+T −1 s=0 a −1 i (s) − n−1 s=0 a −1 i (s) = n+T −1 p=n m j=1 c ij u j (p − τ) + m j=1 b ij (p)g j (u j (p)) + m j=1 d ij (p)g j ∞ v=1 h j (v)u j (p − v) + I i (p) p s=0 a −1 i (s). (2.4) 4 It follows from Lemma 2.1 that n+T −1 p=n m j=1 c ij u j (p − τ) p s=0 a −1 i (s) = n+T −1 p=n m j=1 c ij u j (p − τ)S p−1 s=0 a −1 i (s) = n+T −1 p=n m j=1 c ij u j (p − τ) p−1 s=0 a −1 i (s) − u j (p − τ) p−1 s=0 a −1 i (s) = m j=1 c ij n+T −1 p=n u j (p − τ) p−1 s=0 a −1 i (s) − n+T −1 p=n m j=1 c ij u j (p − τ) p−1 s=0 a −1 i (s) = m j=1 c ij u j (n − τ) n+T −1 s=0 a −1 i (s) − n−1 s=0 a −1 i (s) − n+T −1 p=n m j=1 c ij u j (p − τ) u−1 s=0 a −1 i (s) . Therefore, one gets from (2.4) that u i (n) n+T −1 s=0 a −1 i (s) − n−1 s=0 a −1 i (s) = m j=1 c ij u j (n − τ) n+T −1 s=0 a −1 i (s) − n−1 s=0 a −1 i (s) − n+T −1 p=n m j=1 c ij u j (p − τ)(1 − a i (p)) p s=0 a −1 i (s) + n+T −1 p=n m j=1 b ij (p)g j (u j (p)) + m j=1 d ij (p)g j ∞ v=1 h j (v)u j (p − v) + I i (p) p s=0 a −1 i (s). Dividing both sides of the above equation by n+T −1 s=0 a −1 i (s)− n−1 s=0 a −1 i (s) completes the proof. In what follows, we state Krasnoselskii’s theorem. Lemma 2.3. Let M be a closed convex nonempty subset of a Banach space (B, · ). Suppose that C and B map M into B such that (i) x, y ∈ M implies that Cx + By ∈ M, (ii) C is continuous and CM is contained in a compact set and (iii) B is a contraction mapping. Then there exists a z ∈ M with z = Cz + Bz. 3 Biperiodicity of neutral-type difference networks Due to the introduction of the neutral term neutral m j=1 c ij , we must construct two closed convex subsets B L and B R in S m T , which necessitate the use of Krasnoselskii’s fixed point theorem. As a consequence, we are able to derive the new biperiodicity criteria for (1.2). That is there exists a positive T -periodic sequence solution in B R and an anti-sign T -periodic sequence solution in B L . Next, for the case c ij ≥ 0, we present the following assumption: 5 • Assumption (H 2 ): For each i, j ∈ N, c ij ≥ 0, b ii (n) > 0 and 0 < ˆc i := m j=1 c ij < 1, g j (·) satisfies g j (−v) = −g j (v) for all v ∈ R . Moreover, there exist constants α > 0 and β > 0 with α < β such that for all i ∈ N − 1 − ˆc i m i T α + b ii (n)g i (α) − (1 − a i (n))ˆc i β > P i , − − 1 − ˆc i M i T β + b ii (n)g i (β) + (1 − a i (n))ˆc i α > P i , ∀n ∈ Z where P i := sup n∈Z j=i |b ij (n)|g j + m j=1 |d ij (n)|g j + |I i (n)| , i ∈ N Construct two subsets of S T as follows: B l := w ∈ S T − β ≤ w(n) ≤ −α , B r := w ∈ S T α ≤ w(n) ≤ β . Obviously, B L := B l × B l · · · × B l m and B R := B r × B r · · · × B r m are two closed convex subsets of Banach space S m T . Define the map B Σ : B Σ → S m T by (B Σ u) i (n) = m j=1 c ij u j (n − τ), i ∈ N and the map C Σ : B Σ → S m T by (C Σ u) i (n) = n+T −1 p=n G i (n, p) m j=1 b ij (p)g j (u j (p)) − m j=1 c ij u j (p − τ)(1 − a i (p)) + m j=1 d ij (p)g j ∞ v=1 h j (v)u j (p − v) + I i (p) , i ∈ N (3.1) where Σ = R or L. Due to the fact 0 < ˆc i < 1, B Σ defines a contraction mapping. Proposition 3.1. Under the basic assumptions (H 1 ) and (H 2 ), for each Σ, the operator C Σ is completely continuous on B Σ . Proof. For any given Σ and u ∈ B Σ , we have two cases for the estimation of (C Σ u) i (n). • Case 1: As Σ = R and u ∈ B R , u i (n) ∈ [α, β] holds for each i ∈ N and all n ∈ Z. It 6 follows from (3.1) and (H 2 ) that (C R u) i (n) ≤ n+T −1 p=n G i (n, p) b ii (p)g i (β) + j=i |b ij (p)|g j − m j=1 c ij α(1 − a i (p)) + m j=1 |d ij (p)|g j + |I i (p)| ≤ n+T −1 p=n G i (n, p) − ˆc i (1 − a i (p))α + b ii (p)g i (β) + P i ≤ T M i 1 − ˆc i M i T β = (1 − ˆc i )β and (C R u) i (n) ≥ n+T −1 p=n G i (n, p) b ii (p)g i (α) − j=i |b ij (p)|g j − m j=1 c ij β(1 − a i (p)) − m j=1 |d ij (p)|g j − |I i (p)| ≥ n+T −1 p=n G i (n, p) − ˆc i (1 − a i (p))β + b ii (p)g i (α) − P i ≥ T m i 1 − ˆc i m i T α = (1 − ˆc i )α. • Case 2: As Σ = L and u ∈ B L , u i (n) ∈ [−β, −α] holds for each i ∈ N and all n ∈ Z. It follows from (3.1) and (H 2 ) that (C L u) i (n) ≥ n+T −1 p=n G i (n, p) b ii (p)g i (−β) − j=i |b ij (p)|g j − m j=1 c ij (−α)(1 − a i (p)) − m j=1 |d ij (p)|g j − |I i (p)| ≥ n+T −1 p=n G i (n, p) ˆc i (1 − a i (p))α − b ii (p)g i (β) − P i ≥ T M i 1 − ˆc i M i T (−β) = −(1 − ˆc i )β 7 and (C L u) i (n) ≤ n+T −1 p=n G i (n, p) b ii (p)g i (−α) + j=i |b ij (p)|g j − m j=1 c ij (−β)(1 − a i (p)) + m j=1 |d ij (p)|g j + |I i (p)| ≤ n+T −1 p=n G i (n, p) ˆc i (1 − a i (p))β − b ii (p)g i (α) + P i ≤ T m i 1 − ˆc i m i T (−α) = −(1 − ˆc i )α. It follows from above two cases about the estimation of (C Σ u) i (n) that C Σ u ≤ (1 − min{ˆc i })β ≤ β. This shows that C Σ (B Σ ) is uniformly bounded. Together with the continu- ity of C Σ , for any bounded sequence {ψ n } in B Σ , we know that there exists a subsequence {ψ n k } in B Σ such that {C Σ (ψ n k )} is convergent in C Σ (B Σ ). Therefore, C Σ is compact on B Σ . This completes the proof. Theorem 3.1. Under the basic assumptions (H 1 ) and (H 2 ), for each Σ, (1.2) has a T- periodic solution u Σ satisfying u Σ ∈ B Σ . Proof. Let u, ˆu ∈ B Σ . We should show that B Σ u + C Σ ˆu ∈ B Σ . For simplicity, we only consider the case Σ = R. It follows from (2.2) and (H 2 ) that (B R u) i (n) + (C R ˆu) i (n) = m j=1 c ij u j (n − τ) + n+T −1 p=n G i (n, p) m j=1 b ij (p)g j (ˆu j (p)) − m j=1 c ij ˆu j (p − τ)(1 − a i (p)) + m j=1 d ij (p)g j ∞ v=1 h j (v)ˆu j (p − v) + I i (p) ≤ m j=1 c ij β + n+T −1 p=n G i (n, p) b ii (p)g i (β) + j=i |b ij (p)|g j − m j=1 c ij α(1 − a i (p)) + m j=1 |d ij (p)|g j + |I i (p)| ≤ ˆc i β + T M i 1 − ˆc i M i T β = β. 8 On the other hand, (B R u) i (n) + (C R ˆu) i (n) = m j=1 c ij u j (n − τ) + n+T −1 p=n G i (n, p) m j=1 b ij (p)g j (ˆu j (p)) − m j=1 c ij ˆu j (p − τ)(1 − a i (p)) + m j=1 d ij (p)g j ∞ v=1 h j (v)ˆu j (p − v) + I i (p) ≥ m j=1 c ij α + n+T −1 p=n G i (n, p) b ii (p)g i (α) − j=i |b ij (p)|g j − m j=1 c ij β(1 − a i (p)) − m j=1 |d ij (p)|g j − |I i (p)| ≥ ˆc i α + T m i 1 − ˆc i m i T α = α. Therefore, all the hypotheses stated in Lemma 2.3 are satisfied. Hence, (1.2) has a T - periodic solution u R satisfying u R ∈ B R . Almost the same argument can be done for the case Σ = L. The proof is complete. For the case c ij ≤ 0, we present the following assumption: • Assumption ( H 2 ): For each i, j ∈ N , c ij ≤ 0 and −1 < ˆc i := m j=1 c ij < 0. There exist constants α > 0 and β > 0 with α < β such that for all n ∈ Z (1 − a i (n))ˆc i β + β − ˆc i α M i T > Q i , −(1 − a i (n))ˆc i α + ˆc i β − α m i T > Q i . where Q i := sup n∈Z m j=1 (|b ij (n)| + |d ij (n)|)g j + |I i (n)| . Similarly as Proposition 3.1, we can obtain Proposition 3.2. Under the basic assumptions (H 1 ) and ( H 2 ), for each Σ, the operator C Σ is completely continuous on B Σ . Proof For any given Σ and u ∈ B Σ , we have two cases for the estimation of (C Σ u) i (n). • Case 1: As Σ = R and u ∈ B R , u i (n) ∈ [α, β] holds for each i ∈ N and all n ∈ Z. It follows from (3.1) and ( H 2 ) that (C R u) i (n) ≤ n+T −1 p=n G i (n, p) − m j=1 c ij β(1 − a i (p)) + Q i ≤ T M i β − ˆc i α M i T = β − ˆc i α 9 [...]... sin 0.2πn, I3 (n) := 0.2 sin 0.2πn, τ = 5, g(z) := g1 (z) = g2 (z) = tanh(z), m = 3, 0.2 0.1 0.05 C = (cij ) = 0.1 0.25 0 , h1 (10) = h2 (10) = h3 (10) = 1, T = 10, 0.05 0.1 0.2 0 0.05 cos(0.2πn) 0 , 0 0 D(n) = (dij (n)) = 0.1 sin(0.2πn) 0 0 0.01 sin(0.2πn) 7 + sin(0.2πn) 0.1 sin(0.2πn) 0.01 sin(0.2πn) 0 B(n) = (bij (n)) = 0.1 cos(0.2πn) 7 + sin(0.2πn) 0.01 sin(0.2πn) 0 7 + sin(0.2πn)... be relaxed and yet still obtain periodic sequence solutions and whether they are always of anti-sign? To discuss the sign of each cij and consider analytic properties of activation functions is a possible way to investigate these problems Competing interests The authors declare that they have no competing interests Authors’ contributions All the authors have contributed in all the part and they have... bidirectional associative memory neural networks with variable delays Phys Lett A 335, 226–234 (2005) [12] Liu, YR, Wang, ZD, Serrano, A, Liu, XH: Discrete-time recurrent neural networks with time-varying delays: Exponential stability analysis Phys Lett A 362, 480–488 (2007) [13] Mohamad S: Global exponential stability in continuous-time and discrete-time delayed bidirectional neural networks Physica D 159,... Multiperiodicity of discrete-time delayed neural networks evoked by periodic external inputs IEEE Trans Neural Netw 17, 1141–1151 (2006) [18] Zhao, HY, Sun, L, Wang, GL: Periodic oscillation of discrete-time bidirectional associative memory neural networks Neurocomputing 70, 2924–2930 (2007) [19] Zou, L, Zhou, Z: Periodic solutions for nonautonomous discrete-time neural networks Appl Math Lett 19,... cij (−β)(1 − ai (p)) − Qi j=1 ci α − β ˆ = −β Mi T Therefore, all the hypotheses stated in Lemma 2.3 are satisfied Hence, (1.2) has a T -periodic solution uL satisfying uL ∈ B L By a similar argument, one can prove the case Σ = R This completes the proof 4 Numerical examples Example 1 Consider the following neutral-type difference neural networks with delays 3 ui (n + 1) = ai (n)ui (n) + 3 cij uj (n −... nonlinear delay systems of the neutral type IEEE Trans Autom Control 45, 2326–2331 (2000) [4] Zhang, HG, Liu, ZW, Huang, GB: Novel delay-dependent robust stability analysis for switched neutral-type neural networks with time-varying dalays via SC technique IEEE Trans Syst Man, Cyber B: Cybernetics 40, 1480–1491 (2010) [5] Park, JH, Kwon, OM, Lee, SM: LMI optimization approach on stability for delayed neural. .. exponential stability results for neutral type neural networks with distributed time delays Neurocomputing 71, 1039–1045 (2008) [9] Kelley, W, Perterson, A: Difference Equations: An Introduction with Applications Harcourt Acadamic Press, San Diego (2001) [10] Chen, LN, Aihara, K: Chaos and asymptotical stability in discrete-time neural networks Physica D: Nonlinear Phenomena 104, 286–325 (1997) [11] Liang,... to Figures 2 and 3 Phase view for biperiodicity dynamics of (4.1), we can refer to Figure 4 Example 2 Consider the following neutral-type difference neural networks with delays 2 ui (n + 1) = ai (n)ui (n) + 2 cij uj (n − τ ) + j=1 bij (n)gj (uj (n)) + Ii (n), (4.2) j=1 where a1 (n) := exp − 0.1 − 0.01 cos 0.2πn , a2 (n) := exp − 0.2 − 0.1 sin 0.2πn , I1 (n) := 0.02 sin 0.2πn, I2 (n) := 0.02 cos 0.2πn,... for discrete-time neural networks with variable delays Phys Lett A 358,186-198(2006) [15] Brucoli, M, Carnimeo, L, Grassi, G: Discrete-time cellular neural networks for associative memories with learning and forgetting capabilities IEEE Trans Circ Sys I 42, 396–399 (1995) [16] Wang, L, Zou, X: Capacity of stable periodic solutions in discrete-time bidirectional associative memory neural networks IEEE... ) and (H2 ) indicate that neutral term plays an important role on the dynamics of biperiodicity Such study has not been mentioned in the literature However, there is still more to do For example: (i) If we relax the conditions cij ≤ 0 or cij ≥ 0 for all i, j ∈ N on the neutral term, then is the existence of multiperiodic dynamics still exist? (ii) Evidently, in our work Biperiodicity of neural networks . below). For information about publishing your research in Advances in Difference Equations go to http://www.advancesindifferenceequations.com/authors/instructions/ For information about other SpringerOpen. and full text (HTML) versions will be made available soon. Biperiodicity in neutral-type delayed difference neural networks Advances in Difference Equations 2012, 2012:5 doi:10.1186/1687-1847-2012-5 Zhenkun. cos(0.2πn) 0 0.1 sin(0.2πn) 0 0 0 0 0.01 sin(0.2πn) , B(n) = (b ij (n)) = 7 + sin(0.2πn) 0.1 sin(0.2πn) 0.01 sin(0.2πn) 0.1 cos(0.2πn) 7 + sin(0.2πn) 0 0.01 sin(0.2πn) 0 7 + sin(0.2π n ) . 11 Obviously,