THÔNG TIN TÀI LIỆU
Long Advances in Difference Equations 2012, 2012:113 http://www.advancesindifferenceequations.com/content/2012/1/113 RESEARCH Open Access Attracting and invariant sets of nonlinear neutral differential equations with delays Shujun Long* * Correspondence: longer207@yahoo.com.cn College of Mathematics and Information Science, Leshan Normal University, Leshan, 614004, P.R China Abstract In this paper, we study the attracting and invariant sets for a class of nonlinear neutral differential equations with delays By using the properties of M-matrix, a new delay differential-difference inequality is established Based on the new inequality, we get the global attracting and invariant sets and the sufficient condition ensuring the exponential stability in Lyapunov sense of nonlinear neutral differential equations with delays Our results are independent of time delays and not require the differentiability, boundedness of the derivative of delay functions and the boundedness of activation functions Two examples are presented to illustrate the effectiveness of our conclusion Keywords: attracting set; invariant set; stability; neutral; differential-difference inequality; delays Introduction Delay effects exist widely in many real-world models such as the SEIRS epidemic model [] and neural networks [–] The existence of time delays may destroy a stable system and cause sustained oscillations, bifurcation or chaos and thus could be harmful Therefore, it is of prime importance to consider the effect of delays on the dynamical behaviors of the system Recently, there are many authors who consider the effect of delays on the stability in Lyapunov sense of the system with time delays [–] In addition, another type of time delays, namely neutral-type time delays, has recently drawn much attention in research [–] In fact, many practical delay systems can be modeled as differential systems of neutral type whose differential expression includes not only the derivative term of the current state but also the derivative of the past state, such as partial element equivalent circuits and transmission lines in electrical engineering, controlled constrained manipulators in mechanical engineering, neural networks models, and population dynamics (see [] and references therein) The works [–] mentioned above are focused on studying the stability in Lyapunov sense of the neutral differential equations, which requires the existence and uniqueness of equilibrium points However, in many real physical systems, especially in nonlinear and non-autonomous dynamical systems, the equilibrium point sometimes does not exist Therefore, an interesting subject is to discuss the stability in Lagrange sense Basically, the goal of the study on global stability in Lagrange sense is to determine global attracting sets Once a global attracting set is found, a rough bound of periodic states and chaotic attractors can be estimated For this reason, some significant works have been done on the © 2012 Long; 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 Long Advances in Difference Equations 2012, 2012:113 http://www.advancesindifferenceequations.com/content/2012/1/113 techniques and methods of determining the invariant set and attracting set for various differential systems [–] In these works mentioned before, there is only one paper [] that considers a positive invariant set and a global attracting set for nonlinear neutral differential systems with delays, but the boundedness of activation functions is required It is well known that differential inequalities are very important tools for investigating the dynamical behavior of differential equations (see [, , , , , –]) Xu et al developed a delay differential inequality with the impulsive initial conditions and derived some sufficient conditions to determine the invariant set and the global attracting set for a class of nonlinear non-autonomous functional differential systems with impulsive effects [] In [], Eduardo Liz et al developed a generalized Halanay inequality and derived some sufficient conditions for the existence and stability of almost periodic solutions for quasilinear delay systems In [], Xu et al developed the singular impulsive delay differential inequality and transformed the n-dimensional impulsive neutral differential equation to a n-dimensional singular impulsive delay differential equation and derived some sufficient conditions ensuring the global exponential stability in Lyapunov sense of a nonlinear impulsive neutral differential equation with time-varying delays, but they assumed that the discontinuous points of the derivative of the solution belonged to the first kind As we all know, the discontinuous points of the derivative of continuous functions may not be the first kind In addition, we know that LMI method is another effective tool for investigating the dynamical behavior of a differential system [, , ] The results given in the LMI form are dependent on time delays, so we must give additional constraint conditions such as differentiability or boundedness of the derivative of delay functions on the time-varying delays However, the conditions given in the form of M-matrix are usually independent of the time delays, thus, the time delays are harmless Motivated by the before discussions, our objective in this paper is to improve the inequality established in [] and [] so that it is effective for neutral differential equation By establishing a new delay differential-difference inequality, without assuming that the discontinuous points of the derivative of the solution belong to the first kind, the global attracting and invariant sets and the sufficient condition ensuring the global exponential stability in Lyapunov sense of a nonlinear neutral differential equations with delays are obtained Our results are independent of the time delays, and not require the differentiability, boundedness of the derivative of delay functions and the boundedness of activation functions Two examples are presented to illustrate the effectiveness of our conclusion Model description and preliminaries Throughout this paper, we use the following notations Let Rn+ be the space of ndimensional nonnegative real column vectors, Rn be the space of n-dimensional real column vectors, N = {, , , n}, and Rm×n denote the set of m × n real matrices Usually E denotes an n × n unit matrix For A, B ∈ Rm×n , the notation A ≥ B (A > B) means that each pair of corresponding elements of A and B satisfies the inequality ‘≥ (>)’ Especially, A ∈ Rm×n is called a nonnegative matrix if A ≥ , and z is called a positive vector if z > Ar denotes the rth row vector of the matrix A C[X, Y ] denotes the space of continuous mappings from the topological space X to the topological space Y Especially, C = C[[–τ , ], Rn ] denotes the family of all continuous Rn valued functions, where τ > PC[J, Rn ] = {ϕ : J → Rn is continuous for all but at most a finite number of points t ∈ J, and at these points t ∈ J, ϕ(t + ) and ϕ(t – ) exist, ϕ(t + ) = ϕ(t)}, where J ⊂ R is a bounded Page of 13 Long Advances in Difference Equations 2012, 2012:113 http://www.advancesindifferenceequations.com/content/2012/1/113 Page of 13 interval, ϕ(t + ) and ϕ(t – ) denote the right-hand and left-hand limits of the function ϕ(t), respectively Especially, let PC = PC[[–τ , ], Rn ] For A ∈ Rn×n , x ∈ Rn , φ ∈ C and ϕ is a continuous function on [t – τ , +∞), we define |A| = |aij | T [x]+ = |x |, , |xn | , , n×n [φ]+τ = [φ]+ τ , ϕ(t) τ ϕi (t) τ [φi ]τ = sup –τ ≤s≤ ϕ (t) τ , , ϕn (t) = = sup –τ ≤s≤ ϕi (t + s) , T τ φi (t + s) , ϕ(t) , + τ = ϕ(t) + τ , t ≥ t , i ∈ N , and D+ ϕ(t) denotes the upper-right-hand derivative of ϕ(t) at time t For ϕ ∈ C, we introduce the following norm: ϕ τ max ϕi (s) = max ≤i≤n –τ ≤s≤ In this paper, we consider the following nonlinear neutral differential equation with time-varying delays: ⎧ ⎪ ⎪(xi (t) – ⎨ ⎪ ⎪ ⎩ n j= cij xj (t – rij (t))) = –di xi (t) + + xi (t + s) = φi (s), n j= aij fj (xj (t)) n j= bij gj (xj (t – τij (t))) + Ji , t ≥ t , () –τ ≤ s ≤ , i ∈ N , where τ , aij , bij , cij , di and Ji are constants, τij (t), rij (t), fj (t), gj (t) ∈ C[R, R], i, j ∈ N , rij (t) is differentiable, and τij (t), rij (t) satisfy ≤ τij (t) ≤ τ , < rij (t) ≤ τ , () the initial function φ(s) = (φ (s), , φn (s))T ∈ C Throughout this paper, the solution x(t) of () with the initial condition φ ∈ C is denoted by x(t, t , φ) or xt (t , φ), where xt (t , φ) = x(t + s, t , φ), s ∈ [–τ , ] Definition The set S ⊂ C is called a positive invariant set of () if, for any initial value φ ∈ S, we have the solution xt (t , φ) ∈ S for t ≥ t Definition The set S ⊂ C is called a global attracting set of () if, for any initial value φ ∈ C, the solution xt (t , φ) converges to S as t → +∞ That is, dist xt (t , φ), S → as t → +∞, where dist(ϕ, S) = infψ∈S dist(ϕ, ψ), dist(ϕ, ψ) = sups∈[–τ ,] |ϕ(s) – ψ(s)|, for ϕ ∈ C Definition The zero solution of () is said to be globally exponentially stable in Lyapunov sense if there exist constants λ > and M ≥ such that for any solution x(t, t , φ) with the initial condition φ ∈ C, xt (t , φ) τ ≤ M φ τ e–λ(t–t ) , t ≥ t () Long Advances in Difference Equations 2012, 2012:113 http://www.advancesindifferenceequations.com/content/2012/1/113 Page of 13 Definition ([]) Let the matrix D = (dij )n×n have non-positive off-diagonal elements (i.e., dij ≤ , i = j), then each of the following conditions is equivalent to the statement ‘D is a nonsingular M-matrix’ (i) All the leading principle minors of D are positive (ii) D = C – M and ρ(C – M) < , where M ≥ , C = diag{c , , cn } (iii) The diagonal elements of D are all positive and there exists a positive vector d such that Dd > or DT d > For a nonsingular M-matrix D, we denote M (D) = {z ∈ Rn |Dz > , z > } For a nonnegative matrix A ∈ Rn×n , let ρ(A) be the spectral radius of A Then ρ(A) is an eigenvalue of A and its eigenspace is denoted by ρ (A) = z ∈ Rn |Az = ρ(A)z , which includes all positive eigenvectors of A provided that the nonnegative matrix A has at least one positive eigenvector (see Ref []) Lemma ([]) If A ≥ and ρ(A) < , then (a) (E – A)– ≥ ; (b) there is a positive vector z ∈ ρ (A) such that (E – A)z > Main results Based on Lemma in [] and Theorem . in [], we develop the following delay differential-difference inequality with the PC-value initial condition such that it is effective for neutral differential equation with delays p Theorem Let σ < b ≤ +∞, and u ∈ C[[σ , b), Rn+ ], ω ∈ C[[σ , b), R+ ] satisfy ⎧ + ⎪ ⎪ ⎨D u(t) ≤ Pu(t) + Q[u(t)]τ + Gω(t) + H[ω(t)]τ + η, ⎪ ⎪ ⎩ ω(t) ≤ Mu(t) + N[u(t)]τ + R[ω(t)]τ + I, u(t) = φ(t), ω(t) = ϕ(t), t ∈ [σ , b), () t ∈ [σ – τ , σ ], p where φ ∈ PC[[σ – τ , σ ], Rn+ ], ϕ ∈ PC[[σ – τ , σ ], R+ ], P = (pij )n×n , pij ≥ , for i = j, Q = (qij )n×n ≥ , G = (gij )n×p ≥ , H = (hij )n×p ≥ , M = (mij )p×n ≥ , N = (nij )p×n ≥ , R = (rij )p×p ≥ , η = (η , , ηn )T ≥ and I = (I , , Ip )T ≥ Suppose that ρ(R) < and = –(P + Q + (G + H)(E – R)– (M + N)) is an M-matrix, then the solution of () has the following property: ⎧ ⎨u(t) ≤ kze–λ(t–σ ) + η, ˆ ⎩ω(t) ≤ k˜ze–λ(t–σ ) + I, ˆ t ∈ [σ , b), () provided that the initial conditions satisfy ⎧ ⎨u(t) ≤ kze–λ(t–σ ) + η, ˆ ⎩ω(t) ≤ k˜ze–λ(t–σ ) + I, ˆ t ∈ [σ – τ , σ ], () Long Advances in Difference Equations 2012, 2012:113 http://www.advancesindifferenceequations.com/content/2012/1/113 Page of 13 where k ≥ , ηˆ = – z∈ η+ M( – z˜ = E – Reλτ ), – M + Neλτ z, (G + H)(E – R)– I, Iˆ = (E – R)– (M + N) – η + (E – R)– (M + N) – (G + H)(E – R)– + (E – R)– I, and the positive constant λ is determined by the following inequalities: ρ eλτ R < and – λE + P + Qeλτ + G + Heλτ E – Reλτ M + Neλτ z < () Proof Since is an M-matrix, there exists a vector z ∈ M ( ) such that z > , that is (P + Q + (G + H)(E – R)– (M + N))z < By using continuity and combining with ρ(R) < , we know there exists a positive constant λ satisfying () We at first shall prove that for any positive ε ⎧ ⎨u(t) < (k + ε)ze–λ(t–σ ) + ηˆ = ξ (t), ⎩ω(t) < (k + ε)˜ze–λ(t–σ ) + Iˆ = ζ (t), () t ∈ [σ , b) p If inequality () is not true, from () and u ∈ C[[σ , b), Rn+ ], ω ∈ C[[σ , b), R+ ], then there must be a constant t * > σ and some integer m, r such that um t * = ξm t * , ui (t) ≤ ξi (t), D+ um t * ≥ ξ t * , () t ∈ σ – τ , t * , i = , , n or ωr t * = ζr t * , ωj (t) ≤ ζj (t), t ∈ σ – τ , t * , j = , , p () By using (), (), () and (), we have D+ um t * ≤ Pm u t * + Qm u t * –λ(t * –σ ) ≤ Pm (k + ε)ze τ + Gm ω t * + Hm ω t * + Gm (k + ε)˜ze + Iˆ + Hm (k + ε)˜ze e – + P + Q + (G + H)(E – R)– (M + N) (G + H)(E – R)– I – – + (G + H)(E – R) (M + N) –λ(t * –σ ) < –λ(k + ε)zm e + – – + – η – m – + (G + H)(E – R) (M + N) = –λ(k + ε)zm e =ξ t * – – + Iˆ + ηm M + Neλτ z η m * m e–λ(t –σ ) + ηm + (G + H)(E – R)– )I m – (G + H)(E – R) I m m + ηm + (G + H)(E – R)– )I – (G + H)(E – R) (M + N) –λ(t * –σ ) + ηˆ λτ –λ(t * –σ ) = (k + ε) P + Qeλτ + G + Heλτ E – Reλτ – + ηm + ηˆ + Qm (k + ε)ze e –λ(t * –σ ) + (P + Q) τ λτ –λ(t * –σ ) – m – (G + H)(E – R) I m – (G + H)(E – R) I m () Long Advances in Difference Equations 2012, 2012:113 http://www.advancesindifferenceequations.com/content/2012/1/113 Page of 13 This contradicts the second inequality in (), so the first inequality in () holds Therefore, we have to assume that () holds and we shall obtain another contradiction Next, we consider three cases Case The elements of the Mr and Nr are not all zero Without loss of generality, we let mrl > , ≤ l ≤ n Then, by using (), () and the first inequality in (), we have ωr t * ≤ Mr u t * + Nr u t * = mrl ul t * + τ + Rr ω t * τ + Ir mrj uj t * + Nr u t * + Rr ω t * τ τ + Ir j=l < mrl ξl t * + mrj ξj t * + Nr u t * τ + Rr ω t * τ + Ir j=l * * ≤ Mr (k + ε)ze–λ(t –σ ) + ηˆ + Nr (k + ε)zeλτ e–λ(t –σ ) + ηˆ * + Rr (k + ε)˜zeλτ e–λ(t –σ ) + Iˆ + Ir * = (k + ε) M + Neλτ + Reλτ E – Reλτ – M + Neλτ z r e–λ(t –σ ) + M + N + R(E – R)– (M + N) – η r + R(E – R)– I r + Ir + (M + N) – (G + H)(E – R)– + R(E – R)– (M + N) λτ – = (k + ε) E – Re + (E – R) (M + N) –λ(t * –σ ) = (k + ε)˜zr e (G + H)(E – R)– I λτ M + Ne + (E – R)– (M + N) – – – η – r –λ(t * –σ ) z re r (G + H)(E – R)– + (E – R)– I r + Iˆr = ζr t * () Which contradicts the first equality in (), so under this case, the second inequality in () holds Case The elements of the Mr and Nr are all zero, but the elements of the Rr are not all p zero Without loss of generality, we let rrh > , ≤ h ≤ p Combining with ω ∈ C[[σ , b), R+ ] and the monotonicity of ζ (t), from () and [ω(t)]τ = sup–τ ≤s≤ ω(t + s), we know there must exist t * – τ ≤ t , , ≤ t * such that ω t* τ = sup t * –τ ≤t≤t * ω(t) = ω (t ), , ωp (tp ) By using () and (), we have ωr t * ≤ Rr ω t * = rrh ωh t * τ + Ir τ rrj ωj t * + τ + Ir j=h < rrh ζh t * – τ + rrj ζj t * – τ + Ir j=h = R r ζ t * – τ + Ir * = Rr (k + ε)˜zeλτ e–λ(t –σ ) + Iˆ + Ir T < ζ t * – τ , , ζp t * – τ T () Long Advances in Difference Equations 2012, 2012:113 http://www.advancesindifferenceequations.com/content/2012/1/113 + R(E – R)– (M + N) – + R(E – R) (M + N) – = (k + ε) E – Reλτ – + (E – R) (M + N) λτ – (k + ε) E – Re * – = (k + ε) Reλτ E – Reλτ M + Neλτ z r e–λ(t –σ ) – – η r (G + H)(E – R)– I r + R(E – R)– I r + Ir * M + Neλτ z r e–λ(t –σ ) + (E – R)– (M + N) – – λτ – E – Re – – (E – R)(E – R) (M + N) = (k + ε)˜zr e – – η λτ M + Ne – η r – (G + H)(E – R) I r + (E – R) I – (E – R)(E – R)– (M + N) –λ(t * –σ ) Page of 13 r –λ(t * –σ ) z re r (G + H)(E – R)– I r + Iˆr = ζr t , * () which contradicts the first equality in (); so under this case, the second inequality in () holds Case The elements of the Mr , Nr and Rr are all zero, then the conclusion of the second inequality in () is trivial From the above analysis, we know () is true for all t ∈ [σ , b) Letting ε → in (), we can get () The proof is complete Remark Suppose that M = N = , R = , I = in Theorem , then we get Lemma in [] Suppose that J = , I = in Theorem , then we get Theorem . in [] For the model (), we introduce the following assumptions: (A ) The functions fj (·), gj (·) are Lipschitz continuous, i.e., there are positive constants kj , lj , j ∈ N such that for all s , s ∈ R fj (s ) – fj (s ) ≤ kj |s – s |, gj (s ) – gj (s ) ≤ lj |s – s | ˆ + B)(E ˆ – |C|)– ) be a nonsingular M(A ) Let C < and ˆ = –(–D + (D + A ˆ = (|aij |kj )n×n , Bˆ = (|bij |lj )n×n Let Jˆ = |A|[f ()]+ + matrix, where D = diag{d , , dn } > , A + + |B|[g()] + [J] ˆ is a Theorem Assume that (A ), (A ) hold Then S = {φ ∈ C|[φ]+τ ≤ (E – |C|)– ˆ – J} global attracting set of () Proof Under the conditions (A ), (A ), from [, ], we know the solution x(t, t , φ) of () exists globally We denote ⎧ ⎨[x(t) – Cx(t – r(t))]+ , t ≥ t , u(t) = ⎩W [(E – |C|)[φ]+ ]+ , t – τ ≤ t ≤ t , τ + ω(t) = x(t) , t ≥ t – τ , where W = diag{w , , wn } ≥ such that [φ(t ) – Cφ(–r(t ))]+ = W [(E – |C|)[φ]+τ ]+ () Long Advances in Difference Equations 2012, 2012:113 http://www.advancesindifferenceequations.com/content/2012/1/113 Page of 13 Then, for t ≥ t , from () and (A ), we calculate the upper-right-hand derivative D+ u(t) along the solutions of (), n n D+ ui (t) = sgn xi (t) – cij xj t – rij (t) cij xj t – rij (t) –di xi (t) – j= j= n n di cij xj t – rij (t) + – j= aij fj xj (t) – fj () j= n n bij gj xj t – τij (t) – gj () + + aij fj () + bij gj () + Ji j= j= n n ≤ –di xi (t) – n j= j= n |bij |lj xj t – τij (t) |aij |kj xj (t) + cij xj t – rij (t) + j= n |aij | fj () + |bij | gj () + |Ji | di |cij | xj t – rij (t) + + j= j= n n |bij |lj + di |cij | ωj (t) |aij |kj ωj (t) + ≤ –di ui (t) + j= τ j= n i ∈ N , t ≥ t |aij | fj () + |bij | gj () + |Ji |, + () j= So, from () and (A ), we get ˆ D+ u(t) ≤ –Du(t) + Aω(t) + Bˆ + D|C| ω(t) + τ ˆ + J, t ≥ t () On the other hand, we have n ωi (t) = xi (t) = xi (t) – n + cij xj t – rij (t) j= n ≤ xi (t) – cij xj t – rij (t) j= n |cij | xj t – rij (t) + cij xj t – rij (t) j= j= n ≤ ui (t) + |cij | ωj (t) τ , t ≥ t () j= That is, + ω(t) ≤ u(t) + |C| ω(t) τ , t ≥ t () From (A ), Definition and Lemma , we have (E – |C|)– ≥ , ˆ – ≥ , and so = ˆ – Jˆ ≥ , Furthermore, for z ∈ υ = E – |C| M( – ˆ – Jˆ ≥ ˆ ), we have ˆ + B) ˆ E – |C| –D + (D + A – z < () Long Advances in Difference Equations 2012, 2012:113 http://www.advancesindifferenceequations.com/content/2012/1/113 Page of 13 By using continuity, we can find a positive constant λ such that ρ eλτ |C| < and ˆ + Be ˆ λτ E – |C|eλτ λE – D + D + A – for z ∈ z From () and the initial conditions in (): x(t + s) = φ(s), s ∈ [–τ , ], where φ ∈ C, we can get ω(t) ≤ k z˜ , u(t) ≤ k z, k = n j= ςij max≤i≤n {wi φ τ} min≤i≤n,≤j≤n {zi , z˜ j } , t – τ ≤ t ≤ t , () where (ςij )n×n = |(E – |C|)| From (), (), we know u(t) ≤ k ze–λ(t–t ) + , ω(t) ≤ k z˜ e–λ(t–t ) + υ, t – τ ≤ t ≤ t () From (), (), (), (A ) and Theorem , we get u(t) ≤ k ze–λ(t–t ) + , ω(t) ≤ k z˜ e–λ(t–t ) + υ, t ≥ t () From (), we know the conclusion is true The proof is complete If J = , f () = g() = in the model (), then we know the model () has an equilibrium point zero From Theorem , we get the following conclusion Corollary Assume that (A ), (A ) with Jˆ = hold Then the zero solution of () is globally exponentially stable in Lyapunov sense and the exponential convergence rate is determined by () ˆ [φ(t ) – Theorem Assume that (A ), (A ) hold Then S = {φ ∈ C|[φ]+τ ≤ (E – |C|)– ˆ – J, + + + Cφ(–r(t ))] = [(E – |C|)[φ]τ ] } is a positive invariant set and also a global attracting set of () Proof Since [φ]+τ ≤ (E – |C|)– ˆ – Jˆ and [φ(t ) – Cφ(–r(t ))]+ = [(E – |C|)[φ]+τ ]+ , then from the definition of u(t) and ω(t), we get u(t) ≤ ˆ – Jˆ and ω(t) ≤ E – |C| ˆ ˆ – J, – t – τ ≤ t ≤ t () We choose k = in Theorem ; the remaining proof is similar to the proof of Theorem , and we omit it here So we get the conclusion If we further assume that cij = , i, j ∈ N , then the system () becomes ⎧ ⎨x (t) = –d x (t) + i i i ⎩xi (t + s) = φi (s), n j= aij fj (xj (t)) + n j= bij gj (xj (t –τ ≤ s ≤ , i ∈ N – τij (t))) + Ji , t ≥ t , () Long Advances in Difference Equations 2012, 2012:113 http://www.advancesindifferenceequations.com/content/2012/1/113 Page 10 of 13 Therefore, we can get the following corollary Corollary Assume that (A ) and (A ) with cij = , i, j ∈ N hold Then S = {φ ∈ C|[φ]+τ ≤ ˆ – B) ˆ – J} ˆ is a positive invariant set and also a global attracting set of () (D – A Remark The authors in [] consider the special case of the model (), but they require that the activation functions are continuous and monotonically nondecreasing, and the dτ (t) delay functions are satisfying dtij ≤ Examples Example Consider the nonlinear neutral differential equation with delays ⎧ ⎪ ⎪x (t) = –x (t) + g (x (t – τ (t))) – g (x (t – τ (t))) ⎪ ⎪ ⎪ ⎨ + ( + cos t)x (t – r(t)) + J , ⎪ ⎪ x (t) = –x (t) – g (x (t – τ (t))) + g (x (t – τ (t))) ⎪ ⎪ ⎪ ⎩ + ( + cos t)x (t – r(t)) + J , t ≥ , where g (s) = for i, j = , |s+|–|s–| , g (s) = s, < r(t) = – sin t ≤ () < = τ , τij (t) = | sin(i + j)t| ≤ = τ By simple computation, we get D= Bˆ = , ˆ= A |C| = , , ˆ + B) ˆ E – |C| ˆ = – –D + (D + A E – |C| – ˆ – = ˆ ) = (z , z )T > Let z = (, )T ∈ M( – = – – , We can easily observe that ρ(|C|) = M( , < , ˆ is a nonsingular M-matrix and z < z < z ˆ ), and λ = ., which satisfies the inequalities ρ eλτ |C| = . < , ˆ + Be ˆ λτ E – |C|eλτ λE – D + D + A – z = (–., –.)T < Case Let J = , J = –, so by Theorem , we know S = {φ ∈ C|[φ]+τ ≤ (E – |C|)– ˆ – Jˆ = ( , )T } is a global attracting set of (), and by Theorem , we know S* = {φ ∈ C|[φ]+τ ≤ Long Advances in Difference Equations 2012, 2012:113 http://www.advancesindifferenceequations.com/content/2012/1/113 Page 11 of 13 Figure The state trajectory of x1 (t), x2 (t) of the model (27) with J1 = 2, J2 = –2 and initial conditions φ1 (s) = + sin 23 π s, φ2 (s) = –6 + 4| cos 23 π s|, s ∈ [–1, 0] Figure The state trajectory of x1 (t), x2 (t) of the model (27) with J1 = 0, J2 = and initial conditions φ1 (s) = + sin 23 π s, φ2 (s) = + 2| cos 23 π s|, s ∈ [–1, 0] ( , )T and φ() = φ(– ) = ±[φ]+τ } is a positive invariant and global attracting set of () (See Figure .) Remark The authors in [] considered the global attracting set of neutral type system, but the boundedness of activation functions is required, so the Theorem in [] is ineffective for the model () Case If J = J = , from Corollary , we know the zero solution of () is globally exponentially stable in Lyapunov sense and the exponential convergence rate is equal to . (See Figure .) Remark It is evident that the delay functions r(t) = – sin t, τij (t) = | sin(i + j)t| not satisfy the condition supt∈R r˙ (t) < , supt∈R τ˙ij (t) < , i, j = , , so the results in [, ] are invalid for the model () Long Advances in Difference Equations 2012, 2012:113 http://www.advancesindifferenceequations.com/content/2012/1/113 Page 12 of 13 Figure The state trajectory of x1 (t), x2 (t) of the model (28) with initial conditions φ1 (s) = + cos π s, φ2 (s) = –3 + sin π s, s ∈ [–1, 0] Example Consider the nonlinear differential equation with delays ⎧ ⎨x (t) = –x (t) + g (x (t – τ (t))) – g (x (t – τ (t))) + , ⎩x (t) = –x (t) – g (x (t – τ (t))) + g (x (t – τ (t))) + , where g (s) = |s+|–|s–| , g (s) = sin s, τij (t) = | sin(i + j)t| ≤ = τ t ≥ , () for i, j = , Similarly to the computation of Example , from Corollary , we can get the set S = {φ ∈ ˆ – B) ˆ – Jˆ = ( , )T } is an invariant and global attracting set of the model C|[φ]+τ ≤ (D – A () (See Figure .) Remark It is evident that the activation function g (s) = sin s is not monotonically nondτ (t) decreasing and the delay functions τij (t) = | sin(i + j)t| not satisfy dtij ≤ , i, j = , , so the results in [] are invalid for the model () Competing interests The author declares that they have no competing interests Author’s contributions SJL carried out the main proof of the theorems and examples in this paper alone The author approved the final manuscript Acknowledgements The author sincerely thanks the editor and the reviewers for the detailed comments and valuable suggestions to improve the quality of this paper The author would like to thank the professor Daoyi Xu of Sichuan University for his help in completing this paper This work is supported by National Natural Science Foundation of China under Grant 10971147, Scientific Research Fund of Sichuan Provincial Education Department under Grant 10ZA032, Mathematics Tianyuan Fund under Grant 11126229 and Fundamental Research Funds for the Central Universities under Grant 2011SCU11111 Received: 13 January 2012 Accepted: July 2012 Published: 20 July 2012 References Gao, S, Chen, L, Nieto, JJ, Torres, A: Analysis of a delayed epidemic model with pulse vaccination and saturation incidence Vaccine 24, 6037-6045 (2006) Mohamad, S, Gopalsamy, K: Exponential stability of continuous-time and discrete-time cellular neural networks with delays Appl Math Comput 135, 17-38 (2003) Song, QK, Cao, JD: Stability analysis of Cohen-Grossberg neural network with both time-varying and continuously distributed delays J Comput Appl Math 197, 188-203 (2006) Long Advances in Difference Equations 2012, 2012:113 http://www.advancesindifferenceequations.com/content/2012/1/113 Zhang, Q, Wei, XP, Xu, J: Delay-dependent exponential stability of cellular neural networks with time-varying delays Chaos Solitons Fractals 23, 1363-1369 (2005) Zhou, DM, Zhang, LM, Cao, JD: On global exponential stability of cellular neural networks with Lipschitz-continuous activation function and variable delays Appl Math Comput 2, 379-392 (2004) Gu, HB, Jiang, HJ, Teng, ZD: Existence and globally exponential stability of periodic solution of BAM neural networks with impulses and recent-history distributed delays Neurocomputing 71, 813-822 (2008) Hale, JK, Verduyn Lunel, SM: Introduction to Functional Differential Equations Springer, New York (1993) Kolmanovskii, V, Myshkis, A: Introduction to the Theory and Applications of Functional Differential Equations Kluwer Academic, London (1999) Long, SJ, Xu, DY: Delay-dependent stability analysis for impulsive neural networks with time varying delays Neurocomputing 71, 1705-1713 (2008) 10 Long, SJ, Xu, DY, Zhu, W: Global exponential stability of impulsive dynamical systems with distributed delays Electron J Qual Theory Differ Equ 10, 1-13 (2007) 11 Xu, DY, Yang, ZC: Impulsive delay differential inequality and stability of neural networks J Math Anal Appl 305, 107-120 (2005) 12 Haddock, JR, Krisztin, T, Terjéki, J, Wu, JH: An invariance principle of Lyapunov-Razumikhin type for neutral functional differential equations J Differ Equ 107, 395-417 (1994) 13 Haddock, JR, Krisztin, T, Wu, JH: Asymptotic equivalence of neutral and infinite retarded differential equations Nonlinear Anal 4, 369-377 (1990) 14 Park, JH, Kwon, OM: Global stability for neural networks of neutral-type with interval time-varying delays Chaos Solitons Fractals 41, 1174-1181 (2009) 15 Park, JH, Kwon, OM, Lee, SM: LMI optimization approach on stability for delayed neural networks of neutral-type Appl Math Comput 196, 236-244 (2008) 16 Qiu, J, Cao, J: Delay-dependent robust stability of neutral-type neural networks with time delays J Math Control Sci Appl 1, 179-188 (2007) 17 Rakkiyappan, R, Balasubramaniam, P, Cao, JD: Global exponential stability results for neutral-type impulsive neural networks Nonlinear Anal., Real World Appl 11, 122-130 (2010) 18 Rojsiraphisal, T, Niamsup, P: Exponential stability of certain neutral differential equations Appl Math Comput 217, 3875-3880 (2010) 19 Wen, LP, Wang, WS, Yu, YX: Dissipativity and asymptotic stability of nonlinear neutral delay integro-differential equations Nonlinear Anal 72, 1746-1754 (2010) 20 Xu, DY, Yang, ZG, Yang, ZC: Exponential stability of nonlinear impulsive neutral differential equations with delays Nonlinear Anal 67, 1426-1439 (2007) 21 Xu, LG, Xu, DY: Exponential stability of nonlinear impulsive neutral integro-differential equations Nonlinear Anal 69, 2910-2923 (2008) 22 Mai, H, Liao, X, Li, C: A semi-free weighting matrices approach for neutral-type delayed neural networks J Comput Appl Math 225, 44-55 (2009) 23 Liao, XX, Luo, Q, Zeng, ZG: Positive invariant and global exponential attractive sets of neural networks with time-varying delays Neurocomputing 71, 513-518 (2008) 24 Tu, ZW, Jian, JG, Wang, BX: Positive invariant sets and global exponential attractive sets of a class of neural networks with unbounded time-delays Commun Nonlinear Sci Numer Simul 16, 3738-3745 (2011) 25 Luo, Q, Zeng, ZG, Liao, XX: Global exponential stability in Lagrange sense for neutral type recurrent neural networks Neurocomputing 74, 638-645 (2011) 26 Huang, YM, Zhu, W, Xu, DY: Invariant and attracting set of fuzzy cellular neural networks with variable delays Appl Math Lett 22, 478-483 (2009) 27 Xu, DY, Li, SY, Zhou, X, Pu, ZL: Invariant set and stable region of a class of partial differential equation with time delays Nonlinear Anal., Real World Appl 2, 161-169 (2001) 28 Xu, DY, Yang, ZC: Attracting and invariant sets for a class of impulsive functional differential equations J Math Anal Appl 329, 1036-1044 (2007) 29 Xu, DY, Zhao, HY: Invariant and attracting sets of Hopfield neural networks with delay Int J Syst Sci 32, 863-866 (2001) 30 Zhao, HY: Invariant set and attractor of nonautonomous functional differential systems J Math Anal Appl 282, 437-443 (2003) 31 Burton, TA: Differential inequalities and existence theory for differential, integral and delay equations In: Comparison Methods and Stability Theory, pp 35-56 (1994) 32 Liz, E, Trofimchuk, S: Existence and stability of almost periodic solutions for quasilinear delay systems and the Halanay inequality J Math Anal Appl 248, 625-644 (2000) 33 Wang, XH, Guo, QY, Xu, DY: Exponential p-stability of impulsive stochastic Cohen-Grossberg neural networks with mixed delays Math Comput Simul 79, 1698-1710 (2009) 34 Wang, XH, Xu, DY: Global exponential stability of impulsive fuzzy cellular neural networks with mixed delays and reaction-diffusion terms Chaos Solitons Fractals 42, 2713-2721 (2009) 35 Rakkiyappan, R, Balasubramaniam, P: LMI conditions for global asymptotic stability results for neutral-type neural networks with distributed time delays Appl Math Comput 204, 317-324 (2008) 36 Berman, A, Plemmons, RJ: Nonnegative Matrices in Mathematical Sciences Academic Press, New York (1979) 37 Horn, R, Johnson, C: Matrix Analysis Cambridge University Press, Cambridge (1985) doi:10.1186/1687-1847-2012-113 Cite this article as: Long: Attracting and invariant sets of nonlinear neutral differential equations with delays Advances in Difference Equations 2012 2012:113 Page 13 of 13 ... Cite this article as: Long: Attracting and invariant sets of nonlinear neutral differential equations with delays Advances in Difference Equations 2012 2012:113 Page 13 of 13 ... impulsive neutral differential equations with delays Nonlinear Anal 67, 1426-1439 (2007) 21 Xu, LG, Xu, DY: Exponential stability of nonlinear impulsive neutral integro-differential equations Nonlinear. .. equations with delays are obtained Our results are independent of the time delays, and not require the differentiability, boundedness of the derivative of delay functions and the boundedness of
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