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Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009, Article ID 461757, 13 pages doi:10.1155/2009/461757 Research Article A New Singular Impulsive Delay Differential Inequality and Its Application Zhixia Ma1 and Xiaohu Wang2 College of Computer Science and Technology, Southwest University for Nationalities, Chengdu 610041, China Yangtze Center of Mathematics, Sichuan University, Chengdu 610064, China Correspondence should be addressed to Xiaohu Wang, xiaohuwang111@163.com Received 10 January 2009; Accepted March 2009 Recommended by Wing-Sum Cheung A new singular impulsive delay differential inequality is established Using this inequality, the invariant and attracting sets for impulsive neutral neural networks with delays are obtained Our results can extend and improve earlier publications Copyright q 2009 Z Ma and X 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 Introduction It is well known that inequality technique is an important tool for investigating dynamical behavior of differential equation The significance of differential and integral inequalities in the qualitative investigation of various classes of functional equations has been fully illustrated during the last 40 years 1–3 Various inequalities have been established such as the delay integral inequality in , the differential inequalities in 5, , the impulsive differential inequalities in 7–10 , Halanay inequalities in 11–13 , and generalized Halanay inequalities in 14–17 By using the technique of inequality, the invariant and attracting sets for differential systems have been studied by many authors 9, 18–21 However, the inequalities mentioned above are ineffective for studying the invariant and attracting sets of impulsive nonautonomous neutral neural networks with timevarying delays On the basis of this, this article is devoted to the discussion of this problem Motivated by the above discussions, in this paper, a new singular impulsive delay differential inequality is established Applying this equality and using the methods in 10, 22 , some sufficient conditions ensuring the invariant set and the global attracting set for a class of neutral neural networks system with impulsive effects are obtained 2 Journal of Inequalities and Applications Preliminaries Throughout the paper, En means n-dimensional unit matrix, Rthe set of real numbers, N Δ the set of positive integers, and N {1, 2, , n} A ≥ B A > B means that each pair of corresponding elements of A and B satisfies the inequality “≥ > ” Especially, A is called a nonnegative matrix if A ≥ C X, Y denotes the space of continuous mappings from the topological space X to the Δ topological space Y In particular, let C C −τ, , Rn , where τ > is a constant P C a, b , Rn denotes the space of piecewise continuous functions ψ s : a, b → Rn with at most countable discontinuous points and at this points ψ s are right continuous Δ n Especially, let P C P C −τ, , Rn Furthermore, put P C a, b , Rn c∈ a,b P C a, c , R n n n {ψ s : a, b → R | ψ s , ψ s ∈ P C a, b , R }, where ψ s ˙ ˙ P C a, b , R Δ denotes the derivative of ψ s In particular, let P C1 P C1 a, b , Rn H {h t : R → R | h t is a positive integrable function and satisfies t t supa≤t 2.3 It is a cone without conical surface in Rn We call it an “M-cone” Singular Impulsive Delay Differential Inequality For convenience, we introduce the following conditions C1 Let the r-dimensional diagonal matrix K ki > 0, C2 Let U − P i ∈ S ⊂ N∗ Δ {1, , r}, diag{k1 , , kr } satisfy ki Q be an M-matrix, where Q pij ≥ 0, i / j, pij 0, 0, qij i ∈ S∗ r×r Δ N∗ − S ≥ and P i / j, i ∈ N∗ , j ∈ S∗ 3.1 pij r×r satisfies 3.2 Journal of Inequalities and Applications L1 , , Lr and u t Theorem 3.1 Assume the conditions C1 and C2 hold Let L u1 t , , ur t T be a solution of the following singular delay differential inequality with the initial conditions u t ∈ P C a − τ, a , Rr : KD u t ≤ h t P u t Q u t t ∈ a, b , L , τ 3.3 where τ > 0, a < b ≤ ∞, and ui t ∈ C a, b , R , i ∈ S, ui t ∈ P C a, b , R , i ∈ S∗ , h t ∈ H Then u t ≤ dze−λ t ah s ds − P Q −1 t ∈ a, b , 3.4 t ∈ a − τ, a , 3.5 L, provided that the initial conditions satisfy u t ≤ dze−λ where d ≥ 0, z z1 , , zr T t ah s ds − P Q −1 L, ∈ ΩM U and the positive number λ satisfies the following inequality: λK P Qeλσ z < 0, t ∈ a, b 3.6 Proof By the conditions C2 and the definition of M-matrix, there is a constant vector z z1 , , zr T such that P Q z < 0, − P Q −1 exists and − P Q −1 ≥ By using continuity, we obtain that there must exist a positive constant λ satisfying the inequality 3.6 , that is, r pij i ∈ N∗ qij eλσ zj < −λki zi , 3.7 j Denote by v t v1 t , , vr t T u t P Q −1 L, t ∈ a − τ, b 3.8 It follows from 3.3 and 3.5 that KD v t ≤ h t P u t Q u t τ ≤ h t Pv t Q v t τ v t ≤ dze−λ t ah s ds , L , t ∈ a, b , 3.9 t ∈ a − τ, a In the following, we will prove that for any positive constant ε, vi t ≤ d ε zi e−λ t ah s ds ωi t , t ∈ a, b , i ∈ N∗ 3.10 Journal of Inequalities and Applications Let i ∈ N∗ | vi t > wi t for some t ∈ a, b , ℘ 3.11 inf t ∈ a, b | vi t > wi t , i ∈ ℘ θi If inequality 3.10 is not true, then ℘ is a nonempty set and there must exist some integer m ∈ ℘ such that θm mini∈℘ {θi } ∈ a, b If m ∈ S, by vm t ∈ C a, b , R and the inequality 3.5 , we can get θm > a, vm θm D vm θm ≥ wm θm , ˙ 3.12 vi θm ≤ wi θm , w m θm , 3.13 t ∈ a − τ, θm , i ∈ N∗ , vi t ≤ wi t , By using C2 , 3.3 , 3.7 , 3.12 , 3.13 , and vi t that km D vm θm ≤ h θm sup−τ≤θ a, vm θm ≥ wm θm , vi t ≤ wi t , vi θm ≤ wi θm , t ∈ a − τ, θm , i ∈ N∗ i ∈ S, 3.16 Journal of Inequalities and Applications By using C2 , 3.3 , 3.7 , 3.16 , and vi t 0≤ θ }, i ∈ N∗ , we obtain that sup−τ≤θ 0, denote d , min1≤i≤2n zi 4.17 then dz∗ ≥ e2n 1, , T ∈ R2n From the property of M-cone, we have, dz∗ ∈ ΩM D ϕ s , s ∈ −τ, , where ϕ ∈ P C1 and t0 ∈ R no For the initial conditions x t0 s loss of generality, we assume t0 ≤ t1 , and t ∈ t0 − τ, t0 , we can get −λ t t0 h s ds −λ t t0 h s ds x t ≤ ϕt τ y t ≤ ϕ t e τ e dz∗ , x dz∗ , y 4.18 Then 4.18 yield u t ≤ dz∗ ϕ 1τ e −λ t t0 h s ds t0 − τ ≤ t ≤ t0 , Let N∗ {1, , 2n}, S {1, , n} N and S∗ {n 1, , 2n} conditions of Theorem 3.1 are satisfied By Theorem 3.1, we have u t ≤ dz∗ ϕ 1τ e −λ t t0 h s ds , t0 ≤ t < t1 4.19 N∗ − S Thus, all 4.20 Journal of Inequalities and Applications Suppose that for all m ≤ u t m−1 1, , k, the inequalities ηj dz∗ ϕ e 1τ j −λ t t0 h m−1 s ds tm−1 ≤ t < tm , t ≥ t0 , μj , 4.21 j hold, where η0 μ0 From 4.21 , H5 , and H7 , we can get ≤ I k x t− k x tk k ≤ ηj dz∗ ϕ t x e 1τ j Since − P Q −1 − tk t0 h k s ds μj 4.22 j L, we have U D2 V W J δ 4.23 On the other hand, it follows from H7 that U z∗ x D2 V z∗ x Wz∗ eλσ < δz∗ y y 4.24 Then from 4.21 – 4.24 , we have k ≤ y tk ηj dz∗ ϕ y j e 1τ −λ tk t0 h s ds k μj 2, 4.25 j which together with 4.22 yields that ≤ u tk k ηj dz∗ ϕ j e 1τ −λ tk t0 h s ds k μj 4.26 j Then, it follows from 4.21 and 4.26 that u t ≤ k ηj dz∗ ϕ j k ∗ ηj dz ϕ j e 1τ 1τ e −λ t t0 h k s ds μj j −λ tk t0 h s ds −λ e t tk h s ds 4.27 k μj , j ∀ t ∈ tk − τ, tk 10 Journal of Inequalities and Applications Using Theorem 3.1 again, we have ≤ u t k ηj dz∗ ϕ j k 1τ ∗ ηj dz ϕ j 1τ e tk t0 h −λ s ds −λ e t tk h s ds k μj j e t t0 h −λ k s ds μj , 4.28 tk ≤ t < tk j By mathematical induction, we can conclude that k ≤ u t ηj dz∗ ϕ j Noticing that ηk ≤ e ν tk tk−1 h u t s ds 1τ e −λ t t0 h k s ds μj , tk ≤ t < tk , k ∈ N 4.29 j , by H7 , we can use 4.29 to conclude that ≤ dz∗ ϕ dz∗ ϕ 1τ e e 1τ ν t t0 h − λ−ν s ds −λ e t t0 h t t0 h s ds s ds eμ , eμ 4.30 tk−1 ≤ t < tk , k ∈ N This implies that the conclusion of the theorem holds By using Theorem 4.3 with d 0, we can obtain a positive invariant set of 4.1 , and the proof is similar to that of Theorem 4.3 Theorem 4.4 Assume that H1 – H7 with Ik En hold Then S positive invariant set and also a global attracting set of 4.1 {φ ∈ P C1 | φ τ ≤ 1} is a Remark 4.5 Suppose that cij ≡ 0, i, j ∈ N in H5 , and h t ≡ 1, then we can get Theorems and in x ∈ Rn then 4.1 becomes the nonautonomous neutral neural Remark 4.6 If Ik t, x t− networks without impulses, we can get Theorem 4.1 in 22 Illustrative Example The following illustrative example will demonstrate the effectiveness of our results Example 5.1 Consider nonlinear impulsive neutral neural networks: x1 t ˙ − cos2 t x1 t x2 t ˙ − sin2 t x2 t − cos t x2 t − τ22 t sin t tan x1 t − τ11 t − ˙ cos t x2 t − r12 t sin t tan x1 t − r21 t ˙ − 1.5 cos t, t / k, 2.5 sin t, 5.1 Journal of Inequalities and Applications 11 with x1 t I1 t, x t− , x2 t I2 t, x t− , k, k ∈ N, t 5.2 Δ 1/4 | cos i j t | ≤ 1/4 τ, rij t 1/4 − 1/8 | sin i where τij t Ik t, x a1k t x1 b1k t x2 , a2k t x1 b2k t x2 T , k ∈ N j t |, i, j 1, 2, The parameters of conditions H3 – H9 are as follows: δ 1, ⎛ K ⎜0 ⎜ ⎝0 D1 diag{7, 6}, U h t 0 , 0 0 0 0 ⎞ 0⎟ ⎟, 0⎠ D2 diag{8, 7}, , V W −D1 U D2 U −δE2 P ⎛ ⎜1 ⎜ ⎜0 ⎜ ⎜ ⎜ ⎜1 ⎜ ⎝ V W V W Q ⎛ D − P Q ⎜6 ⎜ ⎜0 ⎜ ⎜ ⎜ ⎜−9 ⎜ ⎝ 0 0 σ ⎛ −7 ⎜0 ⎜ ⎝8 ⎞ , J 1.5, 2.5 T , , 0 −6 0 −1 ⎞ 0⎟ ⎟, 0⎠ −1 4⎟ ⎟ 0⎟ ⎟ ⎟ 1⎟ , ⎟ 4⎟ ⎠ 5.3 ⎞ − ⎟ 4⎟ ⎟ − ⎟ ⎟ 1⎟ − ⎟ 4⎟ ⎠ 1 −9 − It is easy to prove that D is an M-matrix and ΩM D z1 , z2 , z3 , z4 T z3 < z4 < 24z1 , 9z1 > | 9z2 Let z∗ 1, 1, 15, 20 T , then z∗ ∈ ΩM D and z∗ x inequality λK P 1, T Let λ Qeλσ z∗ < z4 < z3 < 16z2 5.4 0.1 which satisfies the 5.5 12 Journal of Inequalities and Applications Now, we discuss the asymptotical behavior of the system 5.1 as follows i If a1k t then b2k t 2k 1/2 e1/5 0, b1k t Ik 2k 1 2k e1/5 sin t , a2k t 2k 1/2 e1/5 − cos t , 5.6 2k Thus ηk μk e1/5 ≥ 1, ln ηk e1/5 ≤ 0.04, ν 0.04 < λ, and μ 1/24 Clearly, all conditions of Theorem 4.3 are satisfied, by Theorem 4.3, S {φ ∈ P C1 | φ τ ≤ e1/24 ≈ e1/24 1.196, 1.746 T } is a global attracting set of 5.1 cos t, b2k t ii If a1k t S {φ ∈ P C1 | φ τ ≤ sin t, b1k t a2k t 0, then Ik E2 By Theorem 4.4, ≈ 1.196, 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