Báo cáo hóa học: "OSCILLATORY MIXED DIFFERENCE SYSTEMS" doc

OSCILLATORY MIXED DIFFERENCE SYSTEMS JOS ´ E M. FERREIRA AND SANDRA PINELAS Received 2 November 2005; Accepted 21 February 2006 The aim of this paper is to discuss the oscillatory behavior of difference systems of mixed type. Several criteria for oscillations are obtained. Particular results are included in regard to scalar equations. Copyright © 2006 J. M. Ferreira and S. Pinelas. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, dis- tribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction The aim of this work is to study the oscillatory behavior of the difference system Δx(n) =   i=1 P i x(n −i)+ m  j=1 Q j x(n + j), n =0,1, 2, , (1.1) where x(n) ∈ R d , Δx(n) = x(n +1)−x(n) is the usual difference operator, ,m ∈ N,and for i = 1, , and j = 1, ,mP i and Q j are given d ×d real matrices. For a particular form of the scalar case of (1.1), the same question is studied in [1] (see also [2,Section 1.16]). The system (1.1)isintroducedin[9]. In this paper the authors show that the existence of oscillatory or nonoscillatory solutions of that system determines an identical behavior to the differential system with piecewise constant arguments, ˙ x(t) =   i=1 P i x  [t −i]  + m  j=1 Q j x  [t + j]  , (1.2) where for t ∈ R, x(t) ∈R d and [·] means the greatest integer function (see also [8,Chap- ter 8]). By a solution of (1.1)wemeananysequencex(n), of points in R d ,withn =−, , 0,1, , which satisfy (1.1). In order to guarantee its existence and uniqueness for given Hindawi Publishing Corporation Advances in Difference Equations Volume 2006, Article ID 92923, Pages 1–18 DOI 10.1155/ADE/2006/92923 2 Oscillatory mixed difference systems initial values x − , ,x 0 , ,x m−1 , denoting by I the d ×d identity matrix, we will assume throughout this paper that the matrices P 1 , ,P  ,Q 1 , ,Q m ,aresuchthat det  I −Q 1  = 0, if m =1, detQ m = 0, if m ≥2, P i = 0, for every i = 1, ,, (1.3) with no restrictions in other cases (see [8, Chapter 7] and [9]). We will say that a sequence y(n) satisfies frequently or persistently a given condition, (C), whenever for every ν ∈ N there exists a n>ν such that y(n)verifies(C). When there is a ν ∈ N such that y(n)verifies(C)foreveryn>ν,(C)issaidtobesatisfiedeventually or ultimately. Upon the basis of this terminology, a solution of (1.1), x(n) = [x 1 (n), ,x d (n)] T ,is said to be oscillatory if each real sequence x k (n)(k = 1, ,d) is frequently nonnegative and frequently nonpositive. If for some k ∈{1, ,d} the real sequence x k (n)iseither eventually positive or eventually negative, x(n)issaidtobeanonoscillatory solution of (1.1). Whenever all solutions of (1.1) are oscillatory we will say that (1.1)isanoscillatory system. Otherwise, (1.1)willbesaidnonoscillatory. Systems of mixed-type like (1.1) can be looked as a discretization of the continuous difference system x(t +1) −x(t) =   i=1 P i x(t −i)+ m  j=1 Q j x(t + j). (1.4) When Q m = I, one easily can see that, through a suitable change of variable, this system is a particular case of the delay difference system x(t) = p  i=1 A j x  t −r j  , (1.5) where the A j are d ×d real matrices and the r j are real positive numbers. As is proposed in [8, Section 7.11], we will investigate, here, conditions on the matr ices P i and Q j (i =1, ,,and j = 1, ,m) which make the system (1.1) oscillatory. For that purpose w e will develop the approach made in [3], motivated by analogues methods used in [6, 7] for obtaining oscillation criteria regarding the continuous delay difference system (1.5). We notice that for mixed-t ype differential difference equations and the differential analog of (1.4), those methods seem not to work in general. In fact, for such equations the situation is essentially different since one cannot ensure, a s for (1.5), that the corre- sponding Cauchy problem will be well posed, or guarantee an exponential boundeness for all its solutions (see [11]). J. M. Ferreira and S. Pinelas 3 According to [9](or[8, Chapter 7]) the analysis of the oscillatory behavior of the system (1.1) can be based upon the existence or absence of real positive zeros of the char- acteristic equation det  (λ −1)I −   i=1 λ −i P i − m  j=1 λ j Q j  = 0. (1.6) That is, letting M(λ) =   i=1 λ −i P i + m  j=1 λ j Q j , (1.7) one can say that (1.1) is oscillatory if and only if, for every λ ∈ R + =]0,+∞[, λ −1 /∈ σ  M(λ)  , (1.8) where for any matrix C ∈ M d (R), the space of all d ×d real matrices, by σ(C)wemean its spectral set. Based upon this characterization we will use, as in [3], the so-called logarithmic norms of mat rices. For that purpose, we recall that to each induced norm, ·,inM d (R), we can associate a logarithmic norm μ : M d (R) →R, which is defined through the following derivative: μ(C) =   I + tC   | t=0 , (1.9) where C ∈ M d (R). As is well known, the logarithmic norm of any matrix C ∈ M d (R) provides real bounds of the set Reσ(C) ={Re z : z ∈ σ(C)}, which enables us to handle condition (1.8) in a more suitable way. Those bounds are given in the first of the following elementary properties of any logarithmic norm (see [4, 5]): (i) Reσ(C) ⊂ [−μ(−C),μ(C)] (C ∈ M d (R)); (ii) μ(C 1 ) −μ(−C 2 ) ≤μ(C 1 + C 2 ) ≤μ(C 1 )+μ(C 2 )(C 1 ,C 2 ∈ M d (R)); (iii) μ(γC) = γμ(C), for every γ ≥ 0(C ∈ M d (R)). In regard to a given finite sequence of matrices, C 1 , ,C ν ,inM d (R), and on the basis of a logarithmic norm, μ, we can define other matrix measures with some relevance in the sequel such as a  C k  = μ  k  i=1 C i  , b  C k  = μ  ν  i=k C i  ,fork =1, ,ν. (1.10) In the same context, these measures give rise to the matrix measures α and β considered in [10]asfollows: α  C 1  = a  C 1  = μ  C 1  , α  C k  = a  C k  − a  C k−1  ,fork =2, ,ν; β  C ν  = b  Cν  = μ  C ν  , β  C k  = b  C k  − b  C k+1  ,fork =1, ,ν −1. (1.11) 4 Oscillatory mixed difference systems In the sequel whenever the values a( −C k ), b(−C k ), α(−C k ), and β(−C k ) are consid- ered, we are implicitly referring to the values above with respect to the finite sequence −C 1 , ,−C ν . Notice that by the property (ii) above, these measures are related with the correspond- ing logarithmic norm μ in the following way: a  C k  ≤ k  i=1 μ  C i  , b  C k  ≤ ν  i=k μ  C i  , (1.12) α  C k  ≤ μ  C k  , β  C k  ≤ μ  C k  , (1.13) for every k = 1, ,ν. Withrespecttothemeasuresα and β the following lemma holds. Lemma 1.1. Let C 1 , ,C ν , be a finite sequence of d ×d real matrices. (a) If γ 1 ≥···≥γ ν ≥ 0 is a nonincreasing finite sequence of nonnegative real numbe rs, then μ  ν  i=1 γ i C i  ≤ ν  i=1 γ i α  C i  . (1.14) (b) If 0 ≤ γ 1 ≤···≤γ ν is a nondecreasing finite sequence of nonnegative real numbers, then μ  ν  i=1 γ i C i  ≤ ν  i=1 γ i β  C i  . (1.15) Proof. We will prove only inequality (1.14). Analogously one can obtain (1.15). Applying the property (ii) of the logarithmic norms, one has μ  ν  i=1 γ i C i  = μ ⎛ ⎝ γ ν ν  i=1 C i + ν−1  i=1  γ i −γ ν  C i ⎞ ⎠ ≤ γ ν μ  ν  i=1 C i  + μ  ν−1  i=1  γ i −γ ν  C i  . (1.16) On the other hand, since ν−1  i=1  γ i −γ ν  C i =  γ 1 −γ 2  C 1 +  γ 2 −γ 3  C 1 +  γ 3 −γ 4  C 1 + ···+  γ ν−1 −γ ν  C 1 +  γ 2 −γ 3  C 2 +  γ 3 −γ 4  C 2 + ···+  γ ν−1 −γ ν  C 2 + ··· +  γ ν−2 −γ ν−1  C ν−2 +  γ ν−1 −γ ν  C ν−2 +  γ ν−1 −γ ν  C ν−1 , (1.17) J. M. Ferreira and S. Pinelas 5 and γ i+1 ≤ γ i ,foreveryi = 1, ,ν −1, we have by the properties (ii) and (iii) of the loga- rithmic norms, μ  ν  i=1 γ i C i  ≤ γ ν μ  ν  i=1 C i  +  γ ν−1 −γ ν  μ  ν−1  i=1 C i  +  γ ν−2 −γ ν−1  μ  ν−2  i=1 C i  + ···+  γ 2 −γ 3  μ  2  i=1 C i  +  γ 1 −γ 2  μ  C 1  . (1.18) Thus μ  ν  i=1 γ i C i  ≤ γ ν  μ  ν  i=1 C i  − μ  ν−1  i=1 C i  + γ ν−1  μ  ν−1  i=1 C i  − μ  ν−2  i=1 C i  + ···+ γ 2  μ  2  i=1 C i  − μ  C 1   + γ 1 μ  C 1  , (1.19) which is equivalent to (1.14).  In view of the examples which will be given in the sections below we recall the follow- ing well-known logarithmic norms of a matrix C = [c jk ] ∈M d (R): μ 1 (C) = max 1≤k≤d  c kk +  j=k   c jk    , μ ∞ (C) = max 1≤j≤d  c jj +  k=j   c jk    , (1.20) which correspond, respectively, to the induced norms in M d (R)givenby C 1 = max 1≤k≤d  d  j=1   c jk    , C ∞ = max 1≤j≤d  d  k=1   c jk    . (1.21) With respect to the norm C 2 induced by the Hilbert norm in R d , the corresponding logarithmic norm is given by μ 2 (C) = maxσ((B + B T )/2). For this specific logarithmic norm, some oscillation criteria are obtained in [3]. 2. Criteria involving the measures α and β By (1.8) and the property (i) of the logarithmic norms, we have that (1.1) is oscillatory whenever, for every real positive λ, λ −1 /∈  −μ  − M(λ)  ,μ  M(λ)  . (2.1) This means that (1.1) is oscillatory if either μ  M(λ)  <λ−1, ∀λ ∈R + , (2.2) or μ  − M(λ)  < 1−λ, ∀λ ∈R + . (2.3) 6 Oscillatory mixed difference systems Depending upon the choice of the matrix measures proposed, one can obtain several different conditions regarding the oscillatory behavior of (1.1). Theorem 2.1. If for every i = 1, ,,and j =1, ,m, α  P i  ≤ 0, β  Q j  ≤ 0, (2.4) β  P i  ≤ 0, α  Q j  ≤ 0, (2.5)   i=1 (i +1) i+1 i i β  P i  < −1, (2.6) then (1.1)isoscillatory. Proof. By the property (ii) of the logarithmic norms, one has μ  M(λ)  ≤ μ    i=1 λ −i P i  + μ  m  j=1 λ j Q j  . (2.7) For every real λ ∈]1,+∞[, inequalities (1.14)and(1.15)andassumption(2.4)imply that μ  M(λ)  ≤   i=1 λ −i α  P i  + m  j=1 λ j β  Q j  ≤ 0. (2.8) Then, for every real λ>1, we conclude that μ  M(λ)  <λ−1, (2.9) since in that case λ −1 > 0. Let now 0 <λ ≤ 1. From (2.7) and inequalities (1.14)and(1.15), we obtain μ  M(λ)  ≤   i=1 λ −i β  P i  + m  j=1 λ j α  Q j  , (2.10) and by assumption (2.5)wehave μ  M(λ)  ≤   i=1 λ −i β  P i  . (2.11) But as max λ>1  λ −i λ −1  =− (i +1) i+1 i i , (2.12) we conclude that, for every real 0 <λ ≤ 1,   i=1 λ −i β  P i  ≤− (λ −1)   i=1 (i +1) i+1 i i β  P i  . (2.13) J. M. Ferreira and S. Pinelas 7 Thus by (2.6), μ  M(λ)  ≤− (λ −1)   i=1 (i +1) i+1 i i β  P i  <λ−1, (2.14) also for every real 0 <λ ≤ 1.  As a corollary of Theorem 2.1, we obtain the following statement. Corollary 2.2. Under (2.4)and(2.5), if   i=1 β  P i  < − 1 4 , (2.15) then (1.1)isoscillatory. Proof. Since (i +1) i+1 /i i ≥ 4 for every positive integer, the condition (2.15) implies (2.6).  The condition (2.15)isaresultof(2.6) through a substitution involving the lower index of the family of matrices P i . A condition involving the largest index, m, of the family of matrices Q j is stated in the following theorem. Theorem 2.3. Under (2.4)and(2.5), if β(P i ) = 0,forsomei =1, ,,and  m  m j =1 α  Q j    i =1 β  P i   1/(m+1)    i=1 β  P i    1 m +1  ≤− 1, (2.16) then (1.1)isoscillatory. Proof. As in the proof of Theorem 2.1,wehave μ  M(λ)  <λ−1, (2.17) for every real λ>1. Recalling inequality (2.10), we obtain by (2.5), for every real 0 <λ ≤ 1, μ  M(λ)  ≤ λ −1   i=1 β  P i  + λ m m  j=1 α  Q j  , (2.18) since λ −i ≥ λ −1 and λ j ≥ λ m . The function f (λ) = λ −1   i=1 β  P i  + λ m m  j=1 α  Q j  (2.19) is strictly concave and f (λ) ≤  m  m j =1 α  Q j    i =1 β  P i   1/(m+1)    i=1 β  P i    1 m +1  . (2.20) 8 Oscillatory mixed difference systems By (2.16)wehavethen,foreveryreal0<λ ≤ 1, μ(M(λ)) ≤−1 <λ−1, and consequently condition (2.2) is fulfilled and system (1.1) is oscillatory.  By use of (2.3), the following theorem is stated. Theorem 2.4. If for every i = 1, , and j = 1, ,m, α  − P i  ≤ 0, β  − Q j  ≤ 0, (2.21) α  − Q j  ≤ 0, β  − P i  ≤ 0, (2.22) m  j=1 j j ( j −1) j−1 β  − Q j  < −1, (2.23) then (1.1)isoscillatory. Proof. For every λ ≥ 1, as in (2.8), we have μ  − M(λ)  ≤   i=1 λ −i α  − P i  + m  j=1 λ j β  − Q j  , (2.24) and by (2.21) μ  − M(λ)  ≤ m  j=1 λ j β  − Q j  . (2.25) Since for j>1, max λ>1  λ j 1 −λ  =− j j ( j −1) j−1 , (2.26) and for j = 1, sup λ>1  λ 1 −λ  =− 1, (2.27) wecanconclude(undertheconvention0 0 = 1) that m  j=1 λ j β  − Q j  < (λ −1) m  j=1 j j ( j −1) j−1 β  − Q j  , (2.28) for every real λ ≥1. So by (2.23), we obtain μ  − M(λ)  < (λ −1) m  j=1 j j ( j −1) j−1 β  − Q j  ≤ 1 −λ, (2.29) for every real λ ≥ 1. J. M. Ferreira and S. Pinelas 9 On the other hand, for every 0 <λ<1, as in (2.10), by (2.22), we have μ  − M(λ)  ≤   i=1 λ −i β  − P i  + m  j=1 λ j α  − Q j  ≤ 0 < 1 −λ, (2.30) and consequently system (1.1)isoscillatory.  Corollary 2.5. Under (2.21)and(2.22), if m  j=1 β  − Q j  < −1 (2.31) then (1.1)isoscillatory. Proof. Clearly (2.31) implies (2.23).  Remark 2.6. In case of having m>1, (2.31)canbereplacedby  m j =1 β(−Q j ) ≤−1. We illustrate these results with the follow ing example. Example 2.7. Consider system (1.1)withd =  =m =2, and P 1 =  − 11 −1 −4  , P 2 = ⎡ ⎢ ⎣ − 1 10 −1 0 −1 ⎤ ⎥ ⎦ , Q 1 =  − 9 −2 3 −10  , Q 2 =  − 81 −2 −10  . (2.32) Through the logarithmic norm μ 1 ,wehave a  P 1  = μ 1  P 1  = 0 =μ 1  P 2  = b  P 2  , a  P 2  = μ 1  P 1 + P 2  = b  P 1  =− 1 10 , a  Q 1  = μ 1  Q 1  =− 6 =μ 1  Q 2  = b  Q 2  , a  Q 2  = μ 1  Q 1 + Q 2  = b  Q 1  =− 16, (2.33) and consequently α  P 1  = 0, α  P 2  =− 1 10 , β  Q 1  =− 10, β  Q 2  =− 6, β  P 1  =− 1 10 , β  P 2  = 0, α  Q 1  =− 6, α  Q 2  =− 10. (2.34) Since 3 √ 2 ×160  − 1 10  1 2 +1  ≈− 1.0260 < −1, (2.35) we can conclude, by Theorem 2.3, that the correspondent system (1.1) is oscillatory. 10 Oscillatory mixed difference systems Notice that, as 2  i=1 (i +1) i+1 i i β  P i  = 2 2 ×  − 1 10  − 3 3 2 2 ×0 =− 2 5 , 2  i=1 β  P i  =− 1 10 , (2.36) Theorem 2.1 and Corollary 2.2 cannot be applied to this system. The same holds to The- orem 2.4 and Corollary 2.5 since the respective conditions (2.21)and(2.22) are not ful- filled. Through the application of inequalities (1.13), from Theorem 2.1, Corollary 2.2,The- orem 2.4,andCorollary 2.5, the corollaries below extend results contained in [3,Theorem 2]. Corollary 2.8. Let μ(P i ) ≤0, μ(Q j ) ≤0,foreveryi = 1, ,,andj = 1, ,m.Ifoneof the inequalities   i=1 (i +1) i+1 i i μ  P i  < −1,   i=1 μ  P i  < − 1 4 , (2.37) is satisfied, then system (1.1)isoscillatory. Corollary 2.9. Let for every i = 1, ,,andj = 1, ,m, μ(−P i ) ≤0, μ(−Q j ) ≤0.Ifone of the inequalities m  j=1 j j ( j −1) j−1 μ  − Q j  < −1, m  j=1 μ  − Q j  < −1, (2.38) is verified, then system (1.1)isoscillatory. Example 2.10. Consider system (1.1)withd = 2,  = 3, m =2, P 1 =  − 2 −1 1 −7  , P 2 =  − 12 1 −4  , P 3 =  − 50 −2 −1  , Q 1 =  − 11 0 −5  , Q 2 =  − 20 −1 −1  . (2.39) With respect to the logarithmic norm μ 1 ,wehave μ 1  P 1  =− 1, μ 1  P 2  = 0, μ 1  P 3  = μ 1  Q 1  =− 1, μ 1  Q 2  =− 1, μ 1  P 1  + μ 1  P 2  + μ 1  P 3  =− 2. (2.40) Then the corresponding system (1.1)isoscillatorybyCorollary 2.8.RemarkthatCorol- lary 2.9 cannot be used in this case. [...]... (3.4) i=1 λ −i 1 − λ −1 a P i + λ − a P , i =1 m−1 m λjβ Qj = j =1 λ j b Q j − b Q j+1 + λm b Qm j =1 m m λjb Qj − = j =1 λ( j −1) b Q j j =2 m = λb Q1 + j =2 λ j 1 − λ −1 b Q j (3.5) 12 Oscillatory mixed difference systems Therefore, for every λ > 1, we have by (3.1) m λ−i α Pi ≤ 0, λ j β Q j ≤ 0, i=1 (3.6) j =1 taking into account that λ−i (1 − λ−1 ) > 0, for i = 1, , − 1, and λ j (1 − λ−1 ) > 0,... Qm (3.18) is strictly concave and g(λ) ≤ m a Qm b P1 1/(m+1) b P1 1 +1 m (3.19) for every real λ Then by (3.15), μ M(λ) ≤ −1 < λ − 1, for every 0 < λ ≤ 1, and (1.1) is oscillatory (3.20) 14 Oscillatory mixed difference systems Theorem 3.3 If for every i = 1, , and j = 1, ,m, a − Pi ≤ 0, b − Q j ≤ 0, (3.21) a − Q j ≤ 0, b − Pi ≤ 0, (3.22) b − Q j ≤ −1, (3.23) m b − Q1 < 0, j =1 then (1.1) is oscillatory... correspondent results involving the measures a and b are more general than those obtained with the measures α and β In fact, notice that for any given finite sequence of real numbers, c1 , ,cν , 16 Oscillatory mixed difference systems we have ν k a ck = b ck = ci , i =1 ν ci , i=k a ck = νc1 + (ν − 1)c2 + · · · + 2cν−1 + cν = k =1 ν (ν − k + 1)ck , (3.34) k =1 ν b ck = νcν + (ν − 1)cν−1 + · · · + 2c2 + c1 = k... Kluwer Academic, Dordrecht, 2000 [3] Q Chuanxi, S A Kuruklis, and G Ladas, Oscillations of linear autonomous systems of difference equations, Applicable Analysis 36 (1990), no 1-2, 51–63 18 Oscillatory mixed difference systems [4] W A Coppel, Stability and Asymptotic Behavior of Differential Equations, D C Heath, Massachusetts, 1965 [5] C A Desoer and M Vidyasagar, Feedback Systems: Input-Output Properties,... systems of neutral differential equations with distributed delay, Differential Equations and Dynamical Systems 3 (1995), no 1, 101–120 [11] T Krisztin, Nonoscillation for functional differential equations of mixed type, Journal of Mathematical Analysis and Applications 245 (2000), no 2, 326–345 Jos´ M Ferreira: Departamento de Matem´ tica, Instituto Superior T´ cnico, Avenida Rovisco Pais, e a e 1049-001 Lisboa, . OSCILLATORY MIXED DIFFERENCE SYSTEMS JOS ´ E M. FERREIRA AND SANDRA PINELAS Received 2 November 2005; Accepted. February 2006 The aim of this paper is to discuss the oscillatory behavior of difference systems of mixed type. Several criteria for oscillations are obtained. Particular results are included in regard to. Difference Equations Volume 2006, Article ID 92923, Pages 1–18 DOI 10.1155/ADE/2006/92923 2 Oscillatory mixed difference systems initial values x − , ,x 0 , ,x m−1 , denoting by I the d ×d identity matrix,