Global stability of Cohen-Grossberg neural network with both time-varying and continuous distributed delays

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1 Global stablty of Cohen-Grossberg neural network wth both tme-varyng and contnuous dstrbuted delays José J. Olvera Departamento de Matemátca e Aplcações and CMAT, Escola de Cêncas, Unversdade do Mnho, Campus de Gualtar, Braga, Portugal e-mal: olvera@math.umnho.pt Abstract In ths paper, a generalzed neural network of Cohen-Grossberg type wth both dscrete tme-varyng and dstrbuted unbounded delays s consdered. Based on M-matrx theory, suffcent condtons are establshed to ensure the exstence and global attractvty of an equlbrum pont. The global exponental stablty of the equlbrum s also addressed but for the model wth bounded dscrete tme-varyng delays. A comparson of results shows that these results generalze and mprove some earler publcatons. Keywords: Cohen-Grossberg neural network, unbounded delay, tme-varyng delay, dstrbuted delay, global asymptotc stablty, global exponental stablty Mathematcs Subect Classfcaton: 34K20, 92B20. 1 Introducton The Cohen-Grossberg neural network models, frst proposed and studed by Cohen and Grossberg 4], have been the subect of an actve research due to ther large applcaton n varous engneerng and scentfc areas such as neural-bology, populaton bology, and computng technology. The neural network n 4] can be descrbed by the followng system of ordnary dfferental equatons ẋ t) = a x t)) b x t)) c f x t)) + I, = 1,..., n. 1.1) In order to be more realstc, dfferental equatons descrbng neural networks should ncorporate tme delays to take nto account physcal and bologcal phenomena. It s known that tme delays may lead to oscllaton, dvergence, or nstablty to a system 14]. For over two decades, several generalzatons of model 1.1) wth constant, dscrete tmevaryng, or contnuous dstrbuted delays have been proposed and studed see 2], 3], 18], 19], 20], and references theren). Besdes the cted works, there s an extensve lterature dealng wth stablty of Cohen-Grossberg models wth delays. Recently, neural network models wth both dscrete tme-varyng and contnuous dstrbuted delays have been consdered 17], 21], and the followng Cohen-Grossberg neural network model ẋ t) = a x t)) b x t)) c g x t)) d f x t τ t))) q 1 k s)v x t + s))ds + I ], t 0, 1.2)

2 was studed n 13], 16]. In ths paper we consder a generalzaton of 1.2), see 2.1) below, and shall study the exstence of an equlbrum pont and ts global attractvty and global exponental stablty. We emphasze that, contrary to the usual approach n the lterature, nstead of Lyapunov functonal technques, we apply some qute dfferent technques, presented n 5], 6], 15]) to study the global stablty of an equlbrum pont of the model. Our method allows us to deal wth more general models. Now we set some defntons and notatons. We denote by BC = BC, 0]; R n ) the space of bounded and contnuous functons, φ :, 0] R n, equpped wth the norm φ = sup s 0 φs), where s the maxmum norm n R n,.e. x = max{ x : = 1,..., n} for x = x 1,..., x n ) R n. For a R n, we also use a to denote the constant functon ϕs) = a n BC. A vector c = c 1,..., c n ) R n s sad to be postve f c > 0 for = 1,..., n and n ths case we wrte c > 0. For a real matrx A = a ] n n we denote by A the absolute-value matrx gven by A = a ] n n. For an open set D BC and f : 0, + ) D R n contnuous, consder the functonal dfferental equaton FDE) gven n a general settng by ẋt) = ft, x t ), t 0, 1.3) where, as usual, x t denotes the functon x t :, 0] R n defned by x t s) = xt + s) for s 0. As we are nterested n neural network models, we always consder solutons of 1.3) wth ntal bounded condtons. It s well-known that the Banach space BC s not an admssble phase space for 1.3) n the sense of 10], consequently the standard exstence, unqueness, contnuous dependence type results are not avalable. Instead of BC, we consder the admssble Banach space { } UC g = φ C, 0]; R n φs) φs) ) : sup <, s unformly contnuous on, 0], s 0 gs) gs) equpped wth the norm φ g satsfyng: = sup s 0 φs), where g :, 0] 1, ) s a functon gs) g1) g s a non-ncreasng contnuous functon and g0) = 1; g2) gs + u) lm = 1 unformly on, 0]; u 0 gs) g3) gs) + as s. As BC UC g, then BC s a subspace of UC g, and we denote by BC g the space BC wth the norm g. As UC g s an admssble Banach space, we consder the FDE 1.3), n the phase space UC g, for a convenent functon g and under enough smooth propertes of f, and we have exstence and unqueness of soluton for the ntal value problem see 11]). We denote by xt, 0, ϕ) the soluton of 1.3) wth ntal condton x 0 = ϕ, ϕ UC g. In secton 2, we establsh suffcent condtons for the exstence and global attractvty of an equlbrum pont of a general Cohen-Grossberg neural network model. An equlbrum pont x of a FDE s sad to be globally attractve f any soluton xt) wth bounded ntal condton satsfes xt) x as t +. 2

3 In secton 3, we set suffcent condtons for the equlbrum pont to be globally exponentally stable. An equlbrum pont x of a neural network model s sad to be globally exponentally stable f there are postve constants ε, M such that xt, 0, ϕ) x Me εt ϕ x t 0, ϕ BC. As mentoned n 7], the defnton of global exponental stablty of an equlbrum x s the usual n the lterature on neural networks wth unbounded delays, but s does not even mply the stablty of x n the phase space UC g,.e. relatve to the norm g. To prove the boundedness solutons of the dfferental systems studed n ths paper, we use the followng result establshed n 7]. Lemma ] Consder equaton 1.3) n UC g, and suppose that f transforms closed bounded sets of 0, + ) D nto bounded sets of R n. If the hypothess H) for all t 0 and ϕ BC such that ϕs) < ϕ0) for s, 0), then ϕ 0)f t, ϕ) < 0 for some {1,..., n} such that ϕ0) = ϕ 0) holds, then any soluton xt) of 1.3) wth ntal condton ϕ BC s defned and bounded on 0, + ), and verfes xt) ϕ for all t 0. To prove the exstence and unqueness of an equlbrum pont, we make use of arguments n 15] and the followng lemma: Lemma ] If H : R n R n s a contnuous and nectve functon such that lm Hx) = +, x + then Hx) s a homeomorphsm of R n onto tself. 2 Global attractvty Consder the generalzed Cohen-Grossberg model wth both dscrete tme-varyng and contnuous dstrbuted nfnte delays gven by ẋ t) = a x t)) b x t)) + h p) x t τ p) t)))+ +f p) ))] g p) x t + s))dη p) s), t 0, 2.1) = 1,..., n, where a : R 0, ), b, h p), f p), gp) : R R, τ p) : 0, ) 0, ) are contnuous functons, and η p) :, 0] R are non-decreasng bounded functons, normalzed so that η p) 0) ηp) ) = 1, for all, {1,..., n}, p {1,..., P }. In 7], a functon g :, 0] 1, + ) was defned by ) gs) = 1 on r 1, 0]; ) g r n ) = n, n N; ) g s contnuous and pecewse lnear lnear on ntervals r n+1, r n ]), 3

4 where r n + s a sutable sequence of postve numbers, n such a way that condtons g1), g2), g3) hold and gs)dη p) < +,, = 1,..., n, p = 1,..., P. See more detals n Lemma 4.1 n 7]. Thus, we may consder the dfferental system 2.1) n the phase space UC g. As mentoned above, we only consder solutons of 2.1) wth bounded ntal data,.e., x 0 = ϕ, ϕ BC. 2.2) In the sequel, for 2.1) the followng hypotheses wll be consdered: A1) for each {1,..., n}, there s β > 0 such that b u) b v))/u v) β, u, v R, u v; A2) h p), f p), gp), respectvely, for, = 1,..., n, p = 1,..., P ; σ p) : R R are Lpschtz functons wth Lpschtz constants γ p), µp) A3) t τ p) t) as t, for, = 1,..., n, p = 1,..., P. Now, we defne the square real matrces,, and B = dagβ 1,..., β n ), L = l ] and N = B L, 2.3) where β 1,..., β n are as n A1) and l = P γp) + µ p) σp). We recall here the defnton of a non-sngular M-matrx. For further propertes of M- matrces, we refer the reader to Chapter 5 of 8]. Defnton. If D = d ] s a square matrx wth non-postve off-dagonal entres,.e., d 0 for all, we say that D s a non-sngular M-matrx f all the egenvalues of D have postve real part, or, equvalently, f all the prncpal mnors of D are postve. Theorem 2.1. Assume A1)-A3). For N defned n 2.3), f N s a non-sngular M- matrx, then there s a unque equlbrum pont x = x 1,..., x n) of 2.1) whch s globally attractve. Proof. Snce N s a non-sngular M-matrx, then see 8]) there s d = d 1,..., d n ) > 0 such that Nd > 0,.e., β > d 1 l d, = 1,..., n. 2.4) The change z t) = d 1 x t) transforms 2.1) nto ż t) = ā z t)) b z t)) + h t, z t )], t 0, = 1,..., n, 2.5) 4

5 where, for each t 0, = 1,..., n, φ UC g, and u R, we have h t, φ) = d 1 h p) ā u) = a d u), b u) = d 1 b d u). ) d φ τ p) t)) + f p) g p) d φ s))dη p) )) s), Note that b u) b v))/u v) β for u, v R, u v,.e., condton A1) s satsfed by the functons b, {1,..., n}. Defne the contnuous functon H : R n R n, Hx) = b x ) + d 1 ) ) n h p) d x ) + f p) g p) d x ) =1, x = x 1,, x n ) R n. Then x R n s an equlbrum of the neural network 2.5) f and only f Hx ) = 0. Consequently, to prove the exstence and unqueness of an equlbrum pont, t s suffcent to prove that H s a homeomorphsm. Argung as n the proof of Lemma 2.4 n 15], we conclude that H s nectve and lm x Hx) =, thus, from Lemma 1.2, we obtan that H s a homeomorphsm, hence there s a unque equlbrum pont x R n of 2.5). Translatng the equlbrum to the orgn through the change y t) = z t) x, 2.5) becomes where f = f 1,..., f n ) : 0, ) UC g R n s defned by ẏt) = ft, y t ), t 0, 2.6) f t, φ) = ā φ 0) + x ) b φ 0) + x ) + h t, φ + x )]. 2.7) It s easy to see that f transforms closed bounded sets of 0, + ) UC g nto bounded sets of R n. Let t 0 and ϕ BC satsfyng ϕs) < ϕ0), for s, 0), and let {1,..., n} be such that ϕ0) = ϕ 0). Suppose that ϕ 0) > 0 the stuaton s analogous for ϕ 0) < 0). Then ϕ 0) = sup ϕs) and s 0 b ϕ 0) + x ) + h t, ϕ + x ) = b ϕ 0) + x ) b x ) +d 1 +f p) β ϕ 0) d 1 β ϕ 0) d 1 ) h p) d ϕ τ p) t)) + d x ) h p) d x ) ) g p) d ϕ s) + d x )dη p) s) γ p) d ϕ τ p) t)) p) + µ l d sup ϕs) = β d 1 s 0 ] f p) gp) d x )) σp) d ϕ s) dη p) s) ) ) l d sup ϕs) > 0, s 0 5

6 hence f satsfes hypothess H). Consequently, for yt) = y t)) n =1 a soluton of 2.6) wth ntal condton y 0 BC, from Lemma 1.1 we conclude that yt) s defned and bounded on R and verfes Set and yt) y 0, t ) v = lm nf y t), u = lm sup y t), = 1,..., n, t t v = max{v }, u = max{u }. Note that u, v R and v u. It s suffcent to prove that max{u, v} = 0. Assume e.g. that v u, so that max{u, v} = u. The stuaton u v s analogous). Let {1,..., n} such that u = u. Argung as n the proof of Theorem 3.2 n 7], we can conclude that there s a postve real sequence t k ) k N such that t k, y t k ) u and f t k, y tk ) 0, as k. 2.9) For convenence, we prove t here. Case 1. Assume that y t) s eventually monotone. In ths case, lm t y t) = u and, for T large, ether ẏ t) 0 for t T or ẏ t) 0 for t T. Assume e.g. that ẏ t) 0 for all t > T large the stuaton ẏ t) 0 s analogous). Then ẏt) = f t, y t ) 0 for t T large, hence lm sup f t, y t ) := c 0. t If c < 0, then there s t 0 > 0 such that f t, y t ) < c/2 for t t 0, mplyng that t y t) = y t 0 ) + f s, y s )ds y t 0 ) + c t 0 2 t t 0), then y t) as t, whch s not possble because of 2.8). Thus c = 0, whch proves 2.9). Case 2. Assume that y t) s not eventually monotone. In ths case there s a sequence t k ) k N such that t k, ẏ t k ) = 0 and y t k ) u, as k. Then f t k, y tk ) = 0 for all k N, and 2.9) holds. For the sake of contradcton, assume that u > 0. Fx ε > 0 and let T = T ε) > 0 be such that yt) < u ɛ := u + ε, for t T, and T dη p) s) < ε y 0,, {1,..., n}, p {1,..., P }. Snce t τ p) t) as t and y t k ) u > 0, then there s k 0 N such that, for k k 0, t k τ p) t k) > T, t k > 2T, and y t k ) > 0. Hence, from the hypotheses and 2.8) we 6

7 conclude that, for k > k 0, b y t k ) + x ) + h t k, y tk + x ) = = b y t k ) + x ) b x ) + d 1 +f p) β y t k ) d 1 β y t k ) d 1 β y t k ) d 1 β y t k ) d 1 ) g p) d y t k + s) + d x )dη p) s) ) h p) d y t k τ p) t k)) + d x ) h p) d x ) γ p) d y t k τ p) t k)) + µ p) Lettng k and ε 0, we get lm nf k ] f p) gp) d x )) σp) d y t k + s) dη p) s) ) T )] γ p) d u ε + µ p) σp) d y t k + s) dη p) s) + y t k + s) dη p) s) T )] γ p) d u ε + µ p) σp) d ε + u ε dη p) s) T ) γ p) d u ε + µ p) σp) d u 2ε β y t k ) d 1 b y t k ) + x ) + h t k, y tk + x )] β d 1 d l u 2ε. l d )u > 0. On the other hand, snce y t) s bounded and ā s postve and contnuous, there s K > 0 such that ā y t) + x ) > K for all t 0. Together wth the above nequalty, ths mples that f t k, y tk ) 0 as k, whch s a contradcton. Thus u = 0, hence v = 0 as well, and ths ends the proof of the theorem. Example 2.1. Consder the Cohen-Grossberg neural network model wth nfnte dscrete tme-varyng delays see 12]) ẋ t) = a x t)) b x t)) c g x t)) d f x t τ t))), = 1,..., n, 2.10) where c, d R and a : R 0, ), b : R R and τ : 0, ) 0, ) are contnuous functons, g, f : R R are Lpschtz functons, and t τ t) as t, for, = 1,..., n. Note that system 2.10) ncludes known models of cellular neural networks as partcular cases. System 2.10) has the form 2.1) f P = 2, h 1) u) = c g u), h 2) u) = d f u), f 1) u) = f 2) u) = 0, τ 1) t) = 0 and τ 2) t) = τ t), u R, t 0,, = 1,..., n. If 7

8 g, f : R R have Lpschtz constants G, F respectvely, then h 1), h2) are also Lpschtz functons, wth Lpschtz constants l 1) = c G, l 2) = d F respectvely, for all, {1,..., n}. Theorem 2.1 appled to system 2.10) gves the followng result: Corollary 2.2. Assume that A1) holds, τ : 0, ) 0, ) satsfy t τ t) as t, and that g, f : R R are Lpschtz functons wth Lpschtz constants G, F, for all, = 1,..., n. If N := B L, where B = dagβ 1,..., β n ), L = l 1) + l 2) ], 2.11) wth β as n A1) and l 1) = c G, l 2) = d F, s a non-sngular M-matrx, then there s a unque equlbrum pont of 2.10), whch s globally attractve. Remark 2.1. For system 2.10), the global attractvty of an equlbrum pont was already obtaned by T. Huang et al. 12], provded that: the functons f, g are Lpschtzan, the matrx N as above s a non-sngular M-matrx, b u) are dfferentable functons such that b u) β > 0 clearly a stronger hypothess than A1)) and the followng addtonal hypothess s satsfed: for each {1,..., n}, there exst a, a > 0 such that 0 < a a u) a, u R. Thus, the above Corollary 2.2 sgnfcantly mproves the crteron n 12]. We also remark that the global exponental stablty of the equlbrum pont of 2.10), wth bounded delays τ, was studed n 6]. In the followng example, a bdrectonal assocatve memory BAM) neural network model s treated as a partcular case of the generalzed Cohen-Grossberg model 2.1). Example 2.2. Consder the BAM neural networks model presented n 15] see also 1]) ẋ t) = a x t)) b x t)) + m ẏ t) = d y t)) c y t)) + =1 f p) g p) x t ω p) t))) y t τ p) f p) y t ρ p) t))) t))), = 1,..., n, g p) x t ζ p) t))), = 1,..., m, 2.12) for t 0 and n, m, P N, where a, d : R 0, ), b, c, g p), f p), g p), f p) : R R are contnuous functons and ω p), ρ p), τ p), ζp) : 0, ) 0, ) are contnuous functons such that t ω p) t), t ρ p) t), t τ p) t), and t ζp) t) as t, = 1,..., n, = 1,..., m and p = 1,..., P. Contrarly to what happens n 15], we do not assume that the tme-dependent delays ω p) t), ρ p) t), τ p) t), ζp) t) are bounded. Clearly, 2.12) s a partcular case of 2.1). Next result s a drect consequence of Theorem 2.1, and extends the stablty crteron presented n 15] to the stuaton wth unbounded dscrete tme-varyng delays. 8

9 Corollary 2.3. Suppose that b and c satsfy A1) wth constants β and γ respectvely, a u) > 0 and d u) > 0 for all u R, f p), g p), f p), gp) are Lpschtz functons wth Lpschtz constants θ p), ξ p), θ p), ξp) respectvely, and ω p), ρ p), τ p), ζp) are contnuous functons such that { } t mn ω p),,p t), ρ p) t), τ p) t), ζp) t) as t. where Defne B G d N := G F C F d n+m) n+m) B = dagβ 1,..., β n ), C = dagγ 1,..., γ m ) G d = dag ξ p) 1,..., ξ n p), F d = dag θ p) 1,..., G = ξ p) m n, F = θ p), n m. θ m p) If N s a non-sngular M-matrx, then there s a unque equlbrum pont of 2.12), whch s globally attractve. 3 Exponental stablty In ths secton, we address the global exponental stablty of the equlbrum pont of 2.1). Here system 2.1) s consdered n the phase space UC g, wth gs) = e αs, s, 0], φs),c for a convenent α > 0 defned below, equpped wth the norm φ g,c = sup, s 0 gs) where,c s the norm n R n defned by y,c = y 1,..., y n ),c = max{c y } wth c = c 1,..., c n ) > 0. The general famly of functonal dfferental systems n the phase space UC g, ẋ t) = ρ t, x t )b x t)) + f t, x t )], = 1,..., n, t 0, 3.1) where ρ : 0, + ) UC g 0, + ), b : R R, and f : 0, + ) UC g R are contnuous functons, was consdered n 7]. For system 3.1), the followng set of hypotheses was mposed: E1) for each {1,..., n}, there s β > 0 such that b u) b v))/u v) β, u, v R, u v; E2) for each {1,..., n}, there s l > 0 such that f t, ϕ) f t, ψ) l ϕ ψ g,c ϕ, ψ UC g ;, 9

10 E3) ρ := nf{ρ t, ϕ) : t 0, ϕ BC g, = 1,..., n} > 0; E4) for each {1,..., n}, β > c l. In 7], the followng result was establshed. Theorem ] Assume E1)-E4). If x = x 1,..., x n) R n s an equlbrum pont of 3.1), then there are postve constants ε, M such that xt, 0, ϕ) x,c Me εt ϕ x,c, t 0, ϕ BC g. In order to apply the exponental stablty crteron presented n the prevous theorem to model 2.1), we now assume that the delayed functons τ p) p) s) are bounded,.e., 0 τ t) τ for some τ > 0, and there exsts ξ > 0 such that all the normalzed non-decreasng and bounded functons η p) satsfy e ξs dη p) s) <. 3.2) Theorem 3.2. Consder 2.1), where a : R 0, + ), b : R R, and τ p) 0, + ) are contnuous, h p), f p), and gp) γ p), µp) η p), and σp) respectvely, and η p) ) = 1,, = 1,..., n, p = 1,..., P. 0) ηp) Assume n addton that: ) E1) s satsfed; ) a := nf{a x) : x R} > 0 for = 1,..., n; : 0, + ) are Lpschtz functons wth Lpschz constants are non-decreasng, bounded and normalzed,.e. ) there exsts a constant ξ > 0 such that η p) satsfy 3.2) for all, = 1,..., n, p = 1,..., P ; v) there exsts τ > 0 such that 0 τ p) t) τ for t 0,, = 1,..., n, p = 1,..., P. If the matrx N defned n 2.3) s a non-sngular M-matrx, then there s a unque equlbrum pont of 2.1), whch s globally exponentally stable. Proof. Snce N s a non-sngular M-matrx, there s d = d 1,..., d n ) > 0 such that 2.4) holds see 8]), hence there s δ > 0 such that β > d 1 l 1 + δ)d, = 1,..., n. 3.3) As n the proof of Theorem 4.3 n 7], from 3.2) we can conclude that there s ς 0, ξ) such that e ςs dη p) s) < 1 + δ,, = 1,..., n, p = 1,..., P. 3.4) { } log1 + δ) Let α := mn ς, and consder system 2.1) n the phase space UC g, where τ gs) = e αs, s 0, equpped wth the norm φ g,c wth c = d 1 1,..., d 1 n ). From Theorem 10

11 2.1, we know that there exsts a unque equlbrum pont, x = x 1,..., x n) R n. System 2.1) has the form 3.1) wth f t, φ) = h p) ) )) φ τ p) t)) + f p) g p) φ s))dη p) s). For ϕ, φ UC g and t 0, snce h p), f p) non-decreasng, we have, gp) are Lpschtz functons and η p) are f t, φ) f t, ϕ) = = +f p) h p) φ τ p) t))) hp) ϕ τ p) t))) ) )) g p) φ s))dη p) s) f p) g p) ϕ s))dη p) s) γ p) γ p) d γ p) Ths means that φ τ p) d 1 t)) ϕ τ p) φ ϕ ) τ p) t)) e ατ p) ατ e t) t)) + µ p) p) t) + µ p) σp) ] ) g p) φ s)) g p) ϕ s)) dη p) s) p) 0 eατ t) d + µ p) σp) d e αs dη p) ) φ s) ϕ g,c e αs d 1 d φ ϕ )s) e αs ) γ p) 1 + δ)d + µ p) σp) d 1 + δ) φ ϕ g,c l 1 + δ)d ) φ ϕ g,c. f t, φ) f t, ϕ) l φ ϕ g,c, = 1,..., n, wth l := n l 1 + δ)d, and from 3.3) we have β > c l, {1,..., n}. Now, the concluson follows from Theorem 3.1. dη p) s) ) Remark 3.1 Snce the non-autonomous terms h p) were not ncluded n the famly of neural network models studed n 7], the prevous result mproves Theorem 4.3 n 7]. Example 3.1. If we take P = 2, h 1) u) = c g u), f 1) 1) 2) u) = 0, τ t) = 0, τ t) = τ t), h 2) u) = d f u), f 2) u) = q u + I, g 2) u) = v u), wth c, d, q, I R, and η 2) s) = s k v)dv, s, 0],, = 1,..., n, 11

12 where k : 0, + ) 0, + ) are contnuous functons, the model 2.1) becomes the followng Cohen-Grossberg neural networks model ẋ t) = a x t)) b x t)) c g x t)) d f x t τ t))) q for = 1,..., n, wth the connecton matrces k s)v x t + s))ds + I ], t 0, 3.5) C = c ] n n, D = d ] n n, and Q = q ] n n. Applyng Theorem 3.2 to model 3.5), we have the followng result. Corollary 3.3. Consder 3.5), where a : R 0, + ), b : R R, and τ : 0, + ) 0, + ) are contnuous, g, f, and v are Lpschtz functons wth Lpschz constants G, F, and V respectvely, and k are nonnegatve contnuous functons such that + 0 k t)dt = 1,, = 1,..., n. Assume n addton that: ) E1) s satsfed; ) a := nf{a x) : x R} > 0 for = 1,..., n; ) there exsts a constant ξ > 0 such that + 0 k t)e ξt dt < +,, = 1,..., n; v) there exsts τ > 0 such that 0 τ p) t) τ for t 0,, = 1,..., n, p = 1,..., P. If the matrx N = B C G D F Q V where B = dagβ 1,..., β n ), G = dagg 1,..., G n ), F = dagf 1,..., F n ), and V = dagv 1,..., V n ), s a non-sngular M-matrx, then there s a unque equlbrum pont of 3.5), whch s globally exponentally stable. Remark 3.2 Snce model 3.5) s only a partcular case of model 2.1), our Theorem 3.2 s more general than the man result n 16]. Example 3.2. Consder the followng model: ẋt) = 2 + cos xt)) 6xt) + g 1 xt)) + g 1 yt)) ] +g 1 xt τt))) + k s)yt + s)ds ẏt) = 2 + sn yt)) 4.5yt) + g 2 yt)) + g 2 xt τt))) ] +g 2 yt τt))) + k s)xt + s)ds, 3.6) 12

13 where g 1 u) = 1 2 u + 1 u 1 ), g 2u) = 1 2 u u 1 ), τt) = 3 cos t + 1, and kt) = e t. The model 3.6) satsfes all hypotheses of Theorem 3.2 wth a 1 = a 2 = 1, β 1 = 6, β 2 = 4.5, τ = 4, and g 1, g 2 are Lpschtz functons wth Lpschtz constant 1. Thus, ) ) B =, L =, N = and t easy to see that N s a non-sngular M-matrx. From Theorem 3.2, we know that model 3.6) has a unque equlbrum pont whch s globally exponental stable. Snce there s not a R such that g 1 u) = ag 2 u), for all u R, the man result n 16] cannot be appled to prove the stablty of model 3.6). We used the Mathematca software to plot a numercal smulaton of the behavor of the soluton xt), yt)) of model 3.6) wth ntal condton ϕs) = 3 cos s, 2 sn s), s ) 0, for 4 1 τt) 4 and kt) 0. Note that, n ths smple case, the matrx N = s also a non-sngular M-matrx. Once agan, we cannot apply the result n 16]. x t 0.3 ) t 0.1 Fgure 1: Behavor of the frst neuron xt) n system 3.6) wth τt) 4, kt) 0, and ntal condton ϕs) = 3 cos s, 2 sn s) 13

14 y t t Fgure 2: Behavor of the second neuron yt) n system 3.6) wth τt) 4, kt) 0, and ntal condton ϕs) = 3 cos s, 2 sn s) Acknowledgments. Ths research was supported by FCT Portugal), through the research center CMAT. The author thanks Professor Teresa Fara, for helpful dscussons. The author also thanks the referee for valuable comments. References 1] J. Cao and Q. Song, Stablty n Cohen-Grossberg-type bdrectonal assocatve memory neural networks wth tme-varyng delays, Nonlnearty ) ] Y. Chen, Global asymptotc stablty of delayed Cohen-Grossberg neural networks, IEEE Trans. Crcuts Syst ) ] T. Chen and L. Rong, Robust global exponental stablty of Cohen-Grossberg neural networks wth tme delays, IEEE Trans. Neural Netw ) ] M. Cohen and S. Grossberg, Absolute stablty of global pattern formaton and parallel memory storage by compettve neural networks, IEEE Trans. Systems Man Cybernet ) ] T. Fara and J.J. Olvera, Local and global stablty for Lotka-Volterra systems wth dstrbuted delays and nstantaneous negatve feedbacks. J. Dfferental Equatons ) ] T. Fara and J.J. Olvera, Boundedness and global exponental stablty for delayed dfferental equatons wth applcatons, Appl. Math. Comput ) ] T. Fara and J.J. Olvera, General crtera for asymptotc and exponental stablty of neural network models wth unbounded delays, preprnt 8] M. Fedler, Specal Matrz and Ther Applcatons n Numercal Mathematcs, Martnus Nhoff Publ. Kluwer), Dordrecht, ] M. Fort and A. Tes, New condton for global stablty of neural networks wth applcaton to lnear, quadratc programmng problems, IEEE Trans. Crcuts Syst )

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