Oscillatory retarded functional systems

Transcript

1 J. Math. Anal. Appl Oscillatory retarded functional systems José M. Ferreira a,,1 and Sandra Pinelas b a Instituto Superior Técnico, Departamento de Matemática, Av. Rovisco Pais, Lisboa, Portugal b Universidade dos Açores, Departamento de Matemática, R. Mãe de Deus, Ponta Delgada, Portugal Received 4 December 00 Submitted by S.R. Grace Abstract This note is concerned with the oscillatory behavior of a linear retarded system. Several criteria are obtained for having the system oscillatory. Conditions regarding the existence of nonoscillatory solutions are also given. 003 Elsevier Inc. All rights reserved. 1. Introduction The oscillation theory of delay equations has received a large amount of attention during the last two decades, as one can see through the textbooks [1 ] and references therein. However, excepting discrete difference systems and particular results which can be obtained through some studies regarding differential systems of neutral type, some gaps can be found in the literature with respect to functional retarded systems. The aim of this note is to study the oscillatory behavior of the system xt = x t rθ d [ νθ ], 1 * Corresponding author. address: jose.manuel.da.silva.ferreira@math.ist.utl.pt J.M. Ferreira. 1 Supported in part by FCT Portugal X/$ see front matter 003 Elsevier Inc. All rights reserved. doi: /s00-47x

2 J.M. Ferreira, S. Pinelas / J. Math. Anal. Appl where xt R n,rθis a real continuous function on [, 0], positive on [, 0[, and νθ is a real n-by-n matrix valued function of bounded variation on [, 0], whichincase of having r0 = 0 will be assumed atomic at zero, that is, such that lim γ 0 + γ [ ] d ηθ = 0, where for a given norm,, in the space M n R, of all real n-by-n matrices, by b a d [ ηθ ] we mean the total variation of ν on an interval [a,b] [, 0]. Notice that when rθ is positiveon [, 0], no atomicity assumption on the function ν is necessary. The system 1 for rθ = rθ r > 0 and θ [, 0], is the class of linear retarded functional systems xt = r xt + θd [ ηθ ], where ηθ = νθ/r is assumed to be atomic at zero. This is the most common general linear retarded functional system appearing in the literature see [6]. Our preference on system 1 regards the possibility of understanding more clearly the role of the delays on the oscillatory behavior of functional retarded systems. Considering the value r =max{rθ: θ 0}, a continuous function x : [ r, + [ R, is said a solution of 1 if satisfies this equation for every t 0. A solution of 1, xt =[x 1 t,..., x n t] T, is called oscillatory if every component, x j t, j = 1,...,n, has arbitrary large zeros; otherwise xt is said a nonoscillatory solution. Whenever all solutions of 1 are oscillatory we will say that 1 is an oscillatory system. Both systems 1 and include the important class of the delay difference systems p xt = A j xt r j, 3 j=1 where the A j are n-by-n real matrices and the r j are positive real numbers. This case corresponds to have in 1, ν as a step function of the form p νθ = Hθ θ j A j, 4 j=1 where H denotes the Heaviside function, <θ 1 < <θ p < 0, and rθ is any continuous and positive function on [, 0] such that rθ j = r j,forj = 1,...,p. The oscillatory behavior of this class of systems is studied in [7]. A specific treatment for discrete difference systems is included in [3]. As is well known the systems 1,, and 3 can be looked, respectively, as particular cases of the differential systems of neutral type

3 08 J.M. Ferreira, S. Pinelas / J. Math. Anal. Appl d xt dt d xt dt d xt dt r x t rθ d [ νθ ] = xt + θd [ νθ ] = p A j xt r j = j=1 r x t rθ d [ ηθ ], xt + θd [ ηθ ], p B j xt r j. 6 j=1 Several criteria for having 6 oscillatory can be found in [ 4], but in all of them, the matrices B j cannot be null, which excludes necessarily the system 3. In [8], the Theorem 3.3 and the Corollaries 4. and 4.3, are oscillation criteria obtained in the regard of the systems and 6, which can as well include the systems and 3, respectively. However the results which will be presented here are of different kind. According to [9], the analysis of the oscillatory behavior of solutions of the system 1, can be based upon the existence or absence of real zeros of the characteristic equation [ det I exp λrθ d [ νθ ]] = 0, 7 where by I we mean the n-by-n identity matrix. In fact, in this framework, one can conclude that 1 is oscillatory if and only if 7 has no real roots, that is, if and only if 0 1 / σ exp λrθ d [ νθ ] 8 for every λ R; nonoscillatory solutions will exist, whenever 7 has at least a real root. We will denote by BV n the space of all functions of bounded variation, η : [, 0] M n R. The space BV 1, of all real functions of bounded variation on [, 0], will be denoted simply by BV.Forφ BV by dφθ, we will mean the total variation of φ on [, 0]. We notice that if η BV n and with j,k = 1,...,n, ηθ =[η jk θ], then each function η jk BV. The matrix [ 0 η = dηjk θ ]. will also be considered. For any η BV n we can formulate the right and left hand limit matrices at any point θ [, 0], ηθ + and ηθ, as well as the right and left hand oscillation matrices Ω η + θ = ηθ+ ηθ and Ωη θ = ηθ ηθ.

4 J.M. Ferreira, S. Pinelas / J. Math. Anal. Appl Nonnegativematrices enable us to considermonotonicn-by-n matrix valued functions. For this purpose we recall that a n-by-n real matrix C =[c jk ] j, k = 1,...,n is said to be nonnegative positive whenever c jk 0, respectively, c jk > 0 for every j,k = 1,...,n. These properties will be expressed as usual, through the notations C 0and C>0, respectively. More generally given two n-by-n real matrices, C and D, we will say that C DC<Dif D C 0 respectively, D C>0. Therefore we will say that a function η : [, 0] M n R is nondecreasing nonincreasing on a interval J [, 0], ifforeveryθ 1,θ J such that θ 1 <θ one has ηθ 1 ηθ respectively, ηθ ηθ 1 ; η will be said increasing decreasing on J, if η is nondecreasing respectively, nonincreasing on J and there exist θ 1,θ J such that θ 1 <θ and ηθ 1 <ηθ respectively, ηθ <ηθ 1. If for every ε>0, sufficiently small, η is increasing decreasing in [θ ε, θ + ε] [ ε, 0] if θ = 0, [, + ε] if θ = we will say that θ is a point of increase of η respectively, a point of decrease of η. As is well known, any function φ BV can be decomposed as the difference of two nondecreasing functions α and β: φ = α β. This decomposition is not unique and a particular decomposition of φ is given by φ = ϕ ψ, 9 where by ϕ and ψ we denote, respectively, the positive and negative variation of φ, which are defined as follows. For each θ [, 0],letP θ be the set of all partitions P ={ = θ 0,θ 1,...,θ k = θ} of the interval [,θ] and to each P P θ associate the sets AP = { j: φθ j φθ j >0 } and BP = { j: φθ j φθ j <0 }. Then ϕ and ψ are defined as { ϕθ = sup φθj φθ j } : P P θ, j AP { ψθ = sup j BP φθ j φθ j } : P P θ whenever AP or BP are empty, we make ϕθ = 0, ψθ = 0. One easily sees that both ϕ and ψ are nondecreasing functions such that φθ = ϕθ ψθ,foreveryθ [, 0]. These facts can be extended for functions η BV n. In fact, since for each θ [, 0] we have ηθ =[η jk θ] j, k = 1,...,n where η jk BV, foreveryj,k = 1,...,n, then decomposing each function η jk j, k = 1,...,n as the difference of two nondecreasing functions α jk and β jk,then-by-n matrix valued functions Aθ =[α jk θ], Bθ =[β jk θ], are both nondecreasing functions in BV n, and for every θ [, 0], we have ηθ = Aθ Bθ. 10 If for every j,k = 1,...,n, we decompose each function η jk according to 9 then we obtain ηθ = Φθ Ψθ, 11

5 10 J.M. Ferreira, S. Pinelas / J. Math. Anal. Appl with Φθ =[ϕ jk θ] and Ψθ=[ψ jk θ],whereϕ jk and ψ jk are, respectively, the positive and negative variation of η jk j, k = 1,...,n.. Nonoscillatory solutions Nonnegative matrices can play some role on the study of the oscillatory behavior of the system 1. According to the Perron Frobenius theorem, a nonnegative matrix C M n R has several important spectral properties see [10,11]. As a matter of fact, denoting by σc the spectrum of C and by ρc the spectral radius of C, one has that ρc σc. Moreover, ρc> 0ifC>0and0 C D ρc ρd. For a matrix C M n R, considering the upper bound sc = max { Re z: z σc }, of the set Re σc ={Re z: z σc}, then sc = ρc σc whenever C 0. Through that same theorem, one can conclude that sc σc,ifc =[c jk ] j, k = 1,...,n,isessentially nonnegative that is, if the off-diagonal entries of Cc jk for j k are nonnegative real numbers. Therefore, if the matrix exp λrθ d [ νθ ] 1 is essentially nonnegative, for every real λ, the spectral set 0 σ exp λrθ d [ νθ ] is dominated by the value that is, sλ = s sλ σ 0 0 exp λrθ d [ νθ ], exp λrθ d [ νθ ] λ R. 13 On this purpose, we notice that for every real λ, the matrix 1 is nonnegative when the function ν is nondecreasing on [, 0], and is essentially nonnegative when for νθ = [ν jk θ] j, k = 1,...,n, the off-diagonal functions ν jk θ j k are nondecreasing on [, 0]. Moreover the assumption 13 is also fulfilled when for each θ [, 0], νθ is a symmetric real matrix.

6 J.M. Ferreira, S. Pinelas / J. Math. Anal. Appl Under 13, the continuous dependence of the spectrum with respect to parameters enable us to handle condition 8 in a more suitable manner. Theorem 1. Under assumption 13, the system 1 is oscillatory if and only if sλ < 1, for every real λ. Proof. Noticing that 0 sλ exp λrθ d [ νθ ], we claim that lim λ + sλ = 0. In fact, if rθ is a positive function on [, 0], then letting mr = min { rθ: θ 0 }, one has for every real λ 0 exp λrθ d [ νθ ] exp λmr [ ] d νθ, and so sλ 0asλ +. If rθ is a positive function on [, 0[ and r0 = 0, as then νθ is atomic at zero, for every ε>0 there exists a real γ>0 such that γ [ ] d νθ ε <. Thus taking m 0 = min{rθ: θ γ } > 0, we have for every real λ 0 exp λrθ d [ νθ ] γ exp λrθ d [ νθ ] + γ γ 0 [ ] [ ] exp λm 0 d νθ + d νθ, γ exp λrθ d [ νθ ] and by consequence, for λ arbitrarily large, we conclude that exp λrθ d [ νθ ] <ε.

7 1 J.M. Ferreira, S. Pinelas / J. Math. Anal. Appl Hence, also in this case, sλ 0asλ +. Therefore supposing that 1 is oscillatory and that for some real λ 0 it is sλ 0 1, by continuity, there exists a real λ 1 λ 0 such that sλ 1 = 1, which, in view of 13, contradicts 8. Remark. The proof of this theorem enable us to conclude that under the assumption 13, the system 1 has at least a nonoscillatory solution whenever sλ +,asλ. Taking a decomposition of ν BV n according to 10, ν = A B, where A,B BV n are nondecreasing functions, the matrix 1 is decomposed into the difference exp λrθ d [ νθ ] = exp λrθ d [ Aθ ] exp λrθ d [ Bθ ]. The following theorem states the existence of, at least, a nonoscillatory solution. Theorem 3. Let θ 0 [, 0] be such that rθ 0 = r. If either Ω A + θ 0> B or ΩA θ 0> B then 1 has at least a nonoscillatory solution. Proof. Let us assume, for example, that Ω + A θ 0> B. For ε>0 small enough, we have exp λrθ d [ Aθ ] θ 0 +ε θ 0 [ ] exp λrθ d Aθ. By application of a mean value property of the functions of bounded variation, we can obtain for every real λ<0, θ 0 +ε θ 0 [ ] exp λrθ d Aθ exp λrθ0 + δε Aθ 0 + ε Aθ 0, for some δ ]0, 1[, depending upon r, θ 0, ε and A. Then, making ε 0 +,wehave exp λrθ d [ Aθ ] exp λ r Ω + A θ 0. On the other hand, for every real λ<0, the following matrix inequality holds 0 exp λrθ d [ Bθ ] exp λ r B,

8 J.M. Ferreira, S. Pinelas / J. Math. Anal. Appl and by consequence exp λrθ d [ Bθ ] exp λ r B. Therefore, for every real λ<0, we have that exp λrθ d [ νθ ] exp λ r Ω + A θ 0 B. 14 As the matrix Ω + A θ 0 B is positive, this means in particular that also exp λrθ d [ νθ ] > 0, for every λ<0. Thus not only assumption 13 is fulfilled but also 0 sλ = ρ exp λrθ d [ νθ ]. But from 14 we can conclude that 0 ρ exp λrθ d [ νθ ] exp λ r ρ Ω A + θ 0 B and by consequence sλ +,asλ. Hence by the Remark, 1 has at least a nonoscillatory solution. The behavior of the function ν at the point θ 0 [, 0] where is attained the absolute maximum of the delay function rθ, has a specific relevance, as was already shown in [1] for the scalar case of 1. Theorem 4. Let θ 0 [, 0] be such that rθ 0 = r and rθ < r for every θ θ 0.If θ 0 is a point of increase of ν, then1 has at least a nonoscillatory solution. Proof. For a matter of simplicity, let us assume that θ 0 =. Considering the decomposition of ν, given by 11, then for some ε>0, the matrix Ψθis constant on [, + ε] and by consequence, on this interval Φθ = νθ C, for some real n-by-n real matrix C. Therefore exp λrθ d [ νθ ]

9 14 J.M. Ferreira, S. Pinelas / J. Math. Anal. Appl = exp λrθ d [ Φθ ] +ε Take 0 <δ<εsuch that +ε exp λrθ d [ Φθ ] +ε exp λrθ d [ Ψθ ] exp λrθ d [ Ψθ ]. m = min { rθ: θ [, + δ] } >M= max { rθ: θ [ + ε, 0] }. One easily can see that the following matrix relations hold, for every real λ<0: 0 +ε +ε exp λrθ d [ Ψθ ] exp λm Ψ, exp λrθ d [ Φθ ] exp λm Φ + ε Φ. Thus we obtain for every real λ<0, +ε +ε which imply that exp λrθ d [ Φθ ] exp λm ν + ε ν, exp λrθ d [ Ψθ ] exp λm Ψ, exp λrθ d [ νθ ] exp λm [ ν + ε ν exp λm M Ψ ]. 1 Since the nonnegative matrix expλm M Ψ tends to the null matrix, as λ, we can concludethat there exists a real number l>0 sufficiently large, such that, for every λ< l, ν + ε ν expλm M Ψ is a positive matrix. Thus, for every λ< l, we have, in particular, exp λrθ d [ νθ ] > 0,

10 J.M. Ferreira, S. Pinelas / J. Math. Anal. Appl and so assumption 13 is satisfied in a way that sλ = ρ 0 exp λrθ d [ νθ ]. Moreover 1 implies that, for every λ< l, ρ 0 exp λrθ d [ νθ ] exp λmρ ν + ε ν exp λm M Ψ. Since the right hand member of this inequality tends to +,asλ, we can state, as in Theorem 1, that 1 has at least a nonoscillatory solution. Theorem 4 can be applied to the system 3 and the following corollary can easily be obtained. Corollary. If r k = max{r j : j = 1,...,p} and A k > 0 then 3 has at least a nonoscillatory solution. Proof. Asamatteroffact,ifA k > 0thenθ k is a point of increase of νθ = p j=1 Hθ θ j A j. Therefore choosing rθ continuous and positive on [, 0] in manner that rθ k = r and rθ< r for every θ θ k, one can conclude by Theorem 4 that 3 has at least a nonoscillatory solution. Similar arguments enable us to state the following theorem. Theorem 6. If ν is increasing on [, 0] and nonconstant on [, 0[ then 1 has at least a nonoscillatory solution. Proof. Since ν is increasing on [, 0[,thereexistθ 1,θ [, 0[ such that θ 1 <θ and = νθ νθ 1 is a positive matrix. Then the following matrix relation holds for every real λ 0: where exp λrθ d [ νθ ] θ m = min { rθ: θ [θ 1,θ ] } > 0. θ 1 [ ] exp λrθ d νθ exp λm, Thus as before, assumption 13 is fulfilled, sλ +,asλ, and 1 has at least a nonoscillatory solution.

11 16 J.M. Ferreira, S. Pinelas / J. Math. Anal. Appl Explicit conditions for oscillations As is well known matrix measures are a relevant tool on the oscillation theory of delay systems. For a matter of completeness we recall briefly, its definition and the properties which will be used in the sequel. For each induced norm,,inm n R, we associate a matrix measure : M n R R, which is defined for any C M n R as I + γc C = lim, γ 0 + γ where by I we mean the identity matrix. Well known matrix measures of a matrix C = [ ] c jk Mn R, are { 1 C = max c kk + } c jk : k = 1,...,n, j k { C = max c jj + } c jk : j = 1,...,n, k j which correspond, respectively, to the induced norms in M n R given by: { n } C 1 = max c jk : k = 1,...,n, j=1 { n } C = max c jk : j = 1,...,n. k=1 Independently of the considered induced norm in M n R, a matrix measure has always the following properties see [13]: i sc C C. ii C 1 C C 1 + C C 1 + C C 1,C M n R. iii γ C = γc,foreveryγ 0. If η BV n, the continuity of on M n R implies that η BV; in consequence, with [a, b] [, 0], the following inequalities hold see [8]: iv If φ C[a,b]; R is nonincreasing and positive, then b a φθd [ ηθ ] b a φθd ηθ ηa. v If φ C[a,b]; R is nondecreasing and positive, then b a φθd [ ηθ ] b a φθd ηb ηθ.

12 J.M. Ferreira, S. Pinelas / J. Math. Anal. Appl By i, if for some matrix measure, 0 exp λrθ d [ νθ ] < 1 λ R, 16 we can conclude that system 1 is oscillatory. For any η BV n, the functions η 0 and η 1 of BV n, given, respectively, by η 0 θ = η0 ηθ, η 1 θ = ηθ η θ [, 0], will be considered in the sequel. In the following theorem we obtain conditions for having 1 oscillatory with respect to the family of all monotonic delay functions. Theorem 7. If on [, 0], ν 1 is nonincreasing and ν 0 is nondecreasing then 1 is oscillatory for all monotonic delay functions rθ. Proof. Assume that r :[, 0] R is a monotonic function. By iv and v, if rθ is nonincreasing we have that 0 0 exp λrθ d [ νθ ] exp λrθ d [ νθ ] and if rθ is nondecreasing, 0 0 exp λrθ d [ νθ ] exp λrθ d [ νθ ] exp λrθ d ν 1 θ, if λ 0, exp λrθ d ν 0 θ, if λ 0, 17 exp λrθ d ν 1 θ, if λ 0, Then take, respectively, for λ R, the functions exp λrθ d ν 1θ, if λ<0, gλ = ν, if λ = 0, exp λrθ d ν 0θ, if λ>0, exp λrθ d ν 0θ, if λ<0, hλ = ν, if λ = 0, exp λrθ d ν 1θ, if λ>0, exp λrθ d ν 0 θ, if λ 0. 18

13 18 J.M. Ferreira, S. Pinelas / J. Math. Anal. Appl where ν = ν0 ν. Noticing that and g0 + = h0 = g0 = h0 + = d ν 0 θ = ν 0 = ν0 ν d ν 1 θ = ν 1 0 = ν0 ν, one can conclude that g and h are both continuous in R. Since on [, 0], ν 1 is nonincreasing and ν 0 is nondecreasing, one has gλ 0 λ R, if rθ is nonincreasing and hλ 0 λ R, if rθis nondecreasing. Hence, for rθmonotonic, by 17 and 18 one has 16 satisfied, which achieves the proof. Remark 8. The assumptions in the Theorem 7 of ν 1 be nonincreasing and ν 0 be nondecreasing are fulfilled, if one assumes as in [14], that for θ 1,θ [, 0], θ 1 <θ νθ νθ As a matter of fact, if θ 1 <θ, one has by ii, and ν 1 θ ν 1 θ 1 = νθ ν νθ 1 ν νθ ν νθ 1 + ν = νθ νθ 1 0, ν 0 θ 1 ν 0 θ = ν0 νθ 1 ν0 νθ ν0 νθ 1 ν0 + νθ = νθ νθ 1 0. Following [8] let us define j αa 1 = A 1, αa j = A k k=1 k=j j A k k=1 p p βa p = A p, βa j = A k A k for j = 1,...,p 1. k=j+1 for j =,...,p,

14 J.M. Ferreira, S. Pinelas / J. Math. Anal. Appl Therefore, with respect to the system 3, from the Theorem 7 we can state the following corollary. Corollary 9. If for j = 1,...,p, αa j 0 and βa j 0, thensystem3 is oscillatory for every family of delays r 1,...,r p R p +. In particular, as by ii αa j A j, βa j A j, we obtain as in [7]: Corollary 10. If A k 0, for every k = 1,...,p,then3 is oscillatory for every family of delays r 1,...,r p R p +. In the following we will analyze what happens when rθ is not a monotonic function on [, 0]. For that purpose we notice that by property ii of the matrix measures, 0 exp λrθ d [ νθ ] θ 0 δ + 0 exp λrθ d [ νθ ] + θ 0 +δ θ 0 +δ θ 0 δ exp λrθ dνθ exp λrθ dνθ, 0 for every θ 0 [, 0] and δ 0, small enough. Theorem 11. Let rθ be differentiable and positive on [, 0], increasing on [,θ 0 δ], constant on [θ 0 δ,θ 0 + δ] and decreasing on [θ 0 + δ,0].if νθ 0 + δ νθ 0 δ 0, 1 νθ 0 δ νθ 0, ν 1 θ 0, for every θ [,θ 0 δ], νθ νθ 0 + δ 0, ν 0 θ 0, for every θ [θ 0 + δ,0], 3 and θ 0 δ ν 1 θ d log rθ ν 0 θ d log rθ <e, 4 θ 0 +δ then 1 is oscillatory.

15 0 J.M. Ferreira, S. Pinelas / J. Math. Anal. Appl Proof. We will prove that 16 holds. For λ = 0, by 0, 1 and the first part of and 3 we can conclude that ν0 ν ν0 νθ 0 + δ + νθ 0 + δ νθ 0 δ + νθ 0 δ ν 0. Let now λ<0. By 0 and the properties iv and v of the matrix measures we obtain 0 exp λrθ d [ νθ ] θ 0 δ exp λrθ d νθ 0 δ νθ + exp λrθ 0 νθ 0 + δ νθ 0 δ + θ 0 +δ exp λrθ d νθ νθ 0 + δ. Therefore, integrating by parts, we have 0 exp λrθ d [ νθ ] exp λr νθ 0 δ ν θ 0 δ λ exp λrθ νθ 0 δ νθ drθ + exp λrθ 0 νθ 0 + δ νθ 0 δ + exp λr0 ν0 νθ 0 + δ + θ 0 +δ λ exp λrθ νθ νθ 0 + δ drθ. Then by 1 and the first part of and 3 we state that, for every λ<0, 0 e λrθ d [ νθ ] 0. Consider now λ>0.bythesamearguments,

16 0 J.M. Ferreira, S. Pinelas / J. Math. Anal. Appl exp λrθ d [ νθ ] exp λrθ 0 δ νθ 0 δ ν + θ 0 δ λ exp λrθ ν 1 θ drθ + exp λrθ 0 νθ 0 + δ νθ 0 δ + exp λrθ 0 + δ ν0 νθ 0 + δ θ 0 +δ θ 0 δ λ exp λrθ ν 0 θ drθ λ exp λrθ ν 1 θ drθ θ 0 +δ λ exp λrθ ν 0 θ drθ. Noticing that the function u exp u has an absolute maximum at u = 1, we obtain for every θ [, 0] λ exp λrθ e rθ. As rθis increasing in [,θ 0 δ[ and decreasing in ]θ 0 +δ,0], by the second part of and 3 we obtain 0 1 e exp λrθ d [ νθ ] [ θ0 δ and 4 implies 0 ν 1 θ rθ drθ exp λrθ d [ νθ ] < 1, for every λ>0. This achieves the proof. θ 0 +δ ν 0 θ rθ Example 1. Let us consider the system 1 for θ 6 θ, if 1 θ< 3, 9 rθ=, if 3 θ, θ 4 θ +, if <θ 0, ] drθ,

17 J.M. Ferreira, S. Pinelas / J. Math. Anal. Appl and [ ] θ θ + νθ = 3 4 θ + 3. θ + 1 With respect to the matrix measure, 1 is satisfied since ν ν 3 [ ] 3 = < 0. The same holds to assumptions and 3, since for every θ [, 3/], ν 3 [ 3 νθ = θ θ + ] 0 0 θ + 3 θ + 1 = θ + 3 θ + 1 0, [ θ θ + νθ ν = 3 ] 0 0 θ + 3 θ + 1 = θ θ + 3 0, and, for every θ [ /, 0], νθ ν [ ] θ θ + = 0 0 θ + 3 θ = θ θ + 0, [ ] θ θ + ν0 νθ = θ + 3 θ + 1 = 3 + θ + 3 θ Moreover, as 3/ = νθ ν rθ 3/ drθ / θ + 3 θ + 1 θ + 6 θ θ + 6 dθ + ν0 νθ rθ / drθ θ θ + θ + 4 θ + 1 θ 1 dθ = 8 36 ln ln <e, condition 4 is also verified and so the corresponding system 1 is oscillatory.

18 J.M. Ferreira, S. Pinelas / J. Math. Anal. Appl Making in the Theorem 11, δ = 0, we obtain the following corollary. Corollary 13. Let r : [, 0] R + be differentiable on [, 0], increasing on [,θ 0 ] and decreasing on [θ 0, 0]. If for every θ [,θ 0 ], νθ 0 νθ 0, ν 1 θ 0, for θ [θ 0, 0] νθ νθ 0 0, ν 0 θ 0, 6 and θ 0 ν 1 θ d log rθ ν 0 θ d log rθ <e, 7 θ 0 then 1 is oscillatory. This corollary is illustrated in the following example. Example 14. Consider 1 with rθ= θθ + 1 and [ ] θθ + 1 θ + 1 νθ =. 0 6θ 7 For the matrix measure 1, we have for every θ [, /] 1 ν 1 [ ] θθ + 1 θ + 1 νθ = θ = θθ + 1 θ + 1 0, [ ] θθ + 1 θ νθ ν = θ 6 = θθ + 1 θ + 1 0, and for every θ [/, 0], 1 νθ ν 1 [ ] θθ + 1 θ + 1 = θ = θθ + 1 θ + 1 0, [ ] θθ + 1 θ ν0 νθ = θ = θθ + 1 θ

19 4 J.M. Ferreira, S. Pinelas / J. Math. Anal. Appl Therefore conditions and 6 of the Corollary 13 are fulfilled. The same holds to assumption 7, since / = θθ + 1 θ + 1 d θθ + 1 θθ + 1 / θ + 1 θ + 1dθ + / / Hence the corresponding system 1 is oscillatory. θθ + 1 θ + 1 d θθ + 1 θθ + 1 θ + 1 θ + 1dθ = 1 6 <e. Now, assuming that δ = 0, as before, and further that θ 0 =, the following corollary can be stated. Corollary 1. Let on [, 0],be ν 0 θ 0, ν 1 θ 0 and rθ positive, differentiable and decreasing. Then system 1 is oscillatory if ν 0 θ d log rθ > e. 8 Proof. Just notice that, in this case, is fulfilled, since 0 = 0. On the other hand, assumption 6 becomes equivalent to ν 0 θ 0and ν 1 θ 0, for every θ [, 0], while 7 gives rise to 8. Analogously, for δ = 0andθ 0 = 0, we obtain the following corollary. Corollary 16. Let on [, 0],be ν 0 θ 0, ν 1 θ 0 and rθ positive, differentiable and increasing. Then system 1 is oscillatory if ν 1 θ d log rθ <e. The following example illustrates the Corollary 1. Example 17. Let us consider 1 for [ θ νθ = ] θ and rθ= 1 θ. θ θ We have, for the matrix measure, [ θ ν 0 θ = ] θ θ θ = max{ θ θ,θ}= θθ + 1 0,

20 J.M. Ferreira, S. Pinelas / J. Math. Anal. Appl for every θ [, 0],and ν 1 θ [ θ = ] 1 θ 1 θ + 1 θ = max{θ + θ, θ 1}=θθ + 1 0, for every θ [, 0].As θθ θ dθ = = 1 + ln 4 0, > e, the corresponding system 1 is then oscillatory. θ + dθ = [ θ ] 0 θ ln1 θ 1 θ Considering in 1, νθ given by 4, for <θ 1 < <θ p < 0, and rθ differentiable, decreasing and positive on [, 0], one obtains the system 3 with r j = rθ j for j = 1,...,p, such that r 1 > >r p. In this situation we can apply the Corollary 1 and consequently obtain the following, corollary. Corollary 18. Let p j A k 0 and A k 0, 9 k=j k=1 for every j = 1,...,p.Thensystem3 is oscillatory if p p A k log r j > e. 30 r j j= k=j Proof. The conditions corresponding to ν 0 θ 0and ν 1 θ 0intheCorollary 1, are in this case, respectively, p k=j A k 0and j k=1 A k 0, for every j = 1,...,p. On the other hand, as then p k=1 A k = 0, 8 becomes, ν 0 θ d log rθ θ p = A k d log rθ θp + + A p d log rθ θ k= 1 θ p + θ p 0d log rθ

21 6 J.M. Ferreira, S. Pinelas / J. Math. Anal. Appl p = A k log r p + + A k log r p + A p log r p r 1 r p r p k= k=p p p = A k log r j > e, r j j= k=j which proves the corollary. An analogous result can be obtained for 3 through the Corollary 16, by considering the system 1 with νθ given by 4, for <θ 1 < <θ p < 0, and rθ differentiable and increasing on [, 0]. In fact, now one obtains the system 3 with r j = rθ j for j = 1,...,p, such that r 1 < <r p. Hence the following corollary can be stated. Corollary 19. Let p j A k 0 and A k 0, k=j k=1 for every j = 1,...,p.Thensystem3 is oscillatory if p j A k log r j+1 <e. r j j=1 k=1 The following example illustrates the Corollary 18. Example 0. Let us consider the system xt = A 1 x t 7 + A x 4 where We have A 1 = [ ] 3, A 1 9 = t 3 1 A 1 = max{, 4}= 0, [ 3 1 A 1 + A = 1 4 [ 3 1 A 1 + A + A 3 = A + A 3 = 1 [ A 3 x t, 31 4 [ ] [ ] 1 1, A 3 =. 4 3 ] = max{, }= 0, ] = max{0, }=0, ] = max{, }= 0, 1 A 3 = max{3, }=3 0. So, 9 is satisfied. The same holds to 30, since

22 J.M. Ferreira, S. Pinelas / J. Math. Anal. Appl A k ln r 3 + A k ln r 3 r 1 r k= k=3 = ln 3/ 7/4 + 3ln/4 3/ = ln ln, 3 > e. 6 Hence the system 31 is oscillatory. References [1] G.S. Ladde, V. Lakshmikantham, B.G. Zhang, Oscillation Theory of Differential Equations with Deviating Arguments, Dekker, [] D.D. Bainov, D.P. Mishev, Oscillation Theory for Neutral Differential Equations with Delay, Hilger, [3] I. Györi, G. Ladas, Oscillation Theory of Delay Differential Equations, Oxford Univ. Press, [4] L. Erbe, Q. Kong, B.G. Zhang, Oscillation Theory for Functional Differential Equations, Dekker, 199. [] R.P. Agarwal, S.R. Grace, D. O Reagan, Oscillation Theory for Difference and Functional Differential Equations, Kluwer, 000. [6] J.K. Hale, S.M.V. Lunel, Introduction to Functional Differential Equations, Springer, [7] J.M. Ferreira, A.M. Pedro, Oscillations of delay difference systems, J. Math. Anal. Appl [8] J. Kirchner, U. Stroinsky, Explicit oscillation criteria for systems of neutral equations with distributed delay, Differential Equations Dynam. Systems [9] T. Krisztyn, Oscillations in linear functional differential systems, Differential Equations Dynam. Systems [10] R.A. Horn, C.R. Johnson, Matrix Analysis, Cambridge Univ. Press, [11] R.S. Varga, Matrix Iterative Analysis, Prentice Hall, 196. [1] J.M. Ferreira, Oscillations and nonoscillations in retarded equations, Portugal. Math [13] C.A. Desoer, M. Vidyasagar, Feedback Systems: Input Output Properties, Academic Press, 197. [14] Q. Kong, Oscillation for systems of functional differential equations, J. Math. Anal. Appl