Báo cáo hóa học: " Research Article A New Singular Impulsive Delay Differential Inequality and Its Application" doc

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, 1.746 T } is a positive invariant set of 5.1 Acknowledgments This work was supported by the National Natural Science Foundation of China under Grant no 10671133 and the Scientific Research Fund of Sichuan Provincial Education Department 08ZA044 References W Walter, Differential and Integral Inequalities, Springer, New York, NY, USA, 1970 V Lakshmikantham and S Leela, Differential and Integral Inequalities: Theory and Applications Vol I: Ordinary Differential Equations, vol 55 of Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1969 V Lakshmikantham and S Leela, Differential and Integral Inequalities: Theory and Applications Vol II: Functional, Partial, Abstract, and Complex Differential Equations, vol 55 of Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1969 D Y Xu, “Integro-differential equations and delay integral inequalities,” Tohoku Mathematical Journal, vol 44, no 3, pp 365–378, 1992 L Wang and D Y Xu, “Global exponential stability of Hopfield reaction-diffusion neural networks with time-varying delays,” Science in China Series F, vol 46, no 6, pp 466–474, 2003 Y M Huang, D Y Xu, and Z G Yang, “Dissipativity and periodic attractor for non-autonomous neural networks with time-varying delays,” Neurocomputing, vol 70, no 16–18, pp 2953–2958, 2007 D Y Xu and Z Yang, “Impulsive delay differential inequality and stability of neural networks,” Journal of Mathematical Analysis and Applications, vol 305, no 1, pp 107–120, 2005 D Y Xu, W Zhu, and S Long, “Global exponential stability of impulsive integro-differential equation,” Nonlinear Analysis: Theory, Methods & Applications, vol 64, no 12, pp 2805–2816, 2006 D Y Xu and Z Yang, “Attracting and invariant sets for a class of impulsive functional differential equations,” Journal of Mathematical Analysis and Applications, vol 329, no 2, pp 1036–1044, 2007 10 D Y Xu, Z Yang, and Z Yang, “Exponential stability of nonlinear impulsive neutral differential equations with delays,” Nonlinear Analysis: Theory, Methods & Applications, vol 67, no 5, pp 1426– 1439, 2007 11 R D Driver, Ordinary and Delay Differential Equations, vol 20 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1977 12 K Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, vol 74 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1992 13 A Halanay, Differential Equations: Stability, Oscillations, Time Lags, Academic Press, New York, NY, USA, 1966 14 T Amemiya, “Delay-independent stabilization of linear systems,” International Journal of Control, vol 37, no 5, pp 1071–1079, 1983 Journal of Inequalities and Applications 13 15 E Liz and S Trofimchuk, “Existence and stability of almost periodic solutions for quasilinear delay systems and the Halanay inequality,” Journal of Mathematical Analysis and Applications, vol 248, no 2, pp 625–644, 2000 16 A Ivanov, E Liz, and S Trofimchuk, “Halanay inequality, Yorke 3/2 stability criterion, and differential equations with maxima,” Tohoku Mathematical Journal, vol 54, no 2, pp 277–295, 2002 17 H Tian, “The exponential asymptotic stability of singularly perturbed delay differential equations with a bounded lag,” Journal of Mathematical Analysis and Applications, vol 270, no 1, pp 143–149, 2002 18 K N Lu, D Y Xu, and Z C Yang, “Global attraction and stability for Cohen-Grossberg neural networks with delays,” Neural Networks, vol 19, no 10, pp 1538–1549, 2006 19 D Y Xu, S Li, X Zhou, and Z Pu, “Invariant set and stable region of a class of partial differential equations with time delays,” Nonlinear Analysis: Real World Applications, vol 2, no 2, pp 161–169, 2001 20 D Y Xu and H.-Y Zhao, “Invariant and attracting sets of Hopfield neural networks with delay,” International Journal of Systems Science, vol 32, no 7, pp 863–866, 2001 21 H Zhao, “Invariant set and attractor of nonautonomous functional differential systems,” Journal of Mathematical Analysis and Applications, vol 282, no 2, pp 437–443, 2003 22 Q Guo, X Wang, and Z Ma, “Dissipativity of non-autonomous neutral neural networks with timevarying delays,” Far East Journal of Mathematical Sciences, vol 29, no 1, pp 89–100, 2008 23 A Berman and R J Plemmons, Nonnegative Matrices in the Mathematical Sciences, Computer Science and Applied Mathematic, Academic Press, New York, NY, USA, 1979 ... Inequalities and Applications 13 15 E Liz and S Trofimchuk, “Existence and stability of almost periodic solutions for quasilinear delay systems and the Halanay inequality, ” Journal of Mathematical... 2805–2816, 2006 D Y Xu and Z Yang, “Attracting and invariant sets for a class of impulsive functional differential equations,” Journal of Mathematical Analysis and Applications, vol 329, no 2,... Differential and Integral Inequalities, Springer, New York, NY, USA, 1970 V Lakshmikantham and S Leela, Differential and Integral Inequalities: Theory and Applications Vol I: Ordinary Differential Equations,

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