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Oscillation results for matrix differential systems with damping
Applied Mathematics and Computation 170 (2005) 545–555 www.elsevier.com/locate/amc
Oscillation results for matrix differential systems with damping Yuan Gong Sun *, Fan Wei Meng Department of Mathematics, Qufu Normal University, Qufu, Shandong 273165, China
Abstract By introducing a class of new parameter functions U(t, s, r), we establish some new oscillation criteria in terms of the coefficients for the matrix differential system with damping (P(t)Y 0 ) 0 + R(t)Y 0 + Q(t)Y = 0 under the hypothesis H: P(t) = P*(t) > 0, Q(t) = Q*(t) and R(t) = R*(t) are n · n matrices of real valued continuous functions on the interval [t0, 1). Our results are not contained in known ones. 2005 Elsevier Inc. All rights reserved. Keywords: Oscillation; Matrix differential system; Damping
1. Introduction Consider the second order linear matrix differential system with damped term ðP ðtÞY 0 Þ0 þ RðtÞY 0 þ QðtÞY ¼ 0;
t P t0 ;
*
Corresponding author. E-mail address: [email protected] (Y.G. Sun).
0096-3003/$ - see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2004.12.012
ð1Þ
546
Y.G. Sun, F.W. Meng / Appl. Math. Comput. 170 (2005) 545–555
where P(t) = P*(t) > 0 (i.e., P(t) is positive definite), Q(t) = Q*(t) and R(t) = R*(t) are n · n matrices of real valued continuous functions on the interval [t0, 1). By M* we mean the transpose of the matrix M. In the absence of damping, i.e., R(t) 0, (1) reduces to the following matrix differential system 0
ðP ðtÞY 0 Þ þ QðtÞY ¼ 0;
t P t0 ;
which is the particular case of the matrix Hamiltonian system 0 U ðtÞ ¼ AðtÞU ðtÞ þ BðtÞV ðtÞ; V 0 ðtÞ ¼ CðtÞU ðtÞ A ðtÞV ðtÞ;
ð2Þ
ð3Þ
with A(t) 0, B1(t) = P(t) and C(t) = Q(t). When R(t) f 0, (1) in general cannot be rewritten to system (3). Therefore, all the existing oscillation results for (2) and (3) generally cannot be applied to (1). By now, there have been many papers (see, for example, [1–22] and the references quoted therein) devoted to the oscillation of systems (2) and (3). It is well known that a successful oscillation theory for matrix differential system can be carried out only for the class of prepared solutions. As usual, a nontrivial solution Y(t) (i.e., detY(t) 5 0 for at lease one t 2 [t0, 1)) of (2) is said to be prepared if for t 2 [t0, 1), 0
Y ðtÞP ðtÞY 0 ðtÞ ðY ðtÞÞ P ðtÞY ðtÞ:
ðaÞ
A nontrivial solution (U(t),V(t)) (i.e., det U(t) 5 0 for at lease one t 2 [t0, 1)) of (3) is said to be prepared if for t 2 [t0, 1), U ðtÞV ðtÞ V ðtÞU ðtÞ 0:
ðbÞ
To best of our knowledge, it seems to us that little has been known about the oscillation of (1) except [21]. In [21], the author studied the particular case of (1) with R(t) = r(t)P(t), where r(t) 2 C([t0, 1),R). The purpose of this paper is to deal with the oscillation of the more general system (1). In this paper, we say a nontrivial solution Y(t) of (1) is prepared if for t 2 [t0, 1), ( Y ðtÞP ðtÞY 0 ðtÞ ðY ðtÞÞ0 P ðtÞY ðtÞ 0; ð4Þ 0 Y ðtÞRðtÞY 0 ðtÞ ðY ðtÞÞ RðtÞY ðtÞ 0; i.e., Y*(t)P(t)Y 0 (t) and Y*(t)R(t)Y 0 (t) are symmetric. A prepared solution Y(t) of (1) is said to be oscillatory, if det Y(t) has arbitrarily large zeros on [t0, 1). As we can see, the important tool in the study of oscillatory behavior of solutions for the scalar equation (as a special case of (1) or (2)) is the averaging technique, which involves a function class X. Say a function H = H(t, s) belongs to the function class X, if H 2 C(D, R+), where D = {(t, s) : t0 6 s 6 t}, which satisfies H(t, t) = 0, H(t, s) > 0 for t > s, and has partial derivative oH/os and oH/ot on D such that
Y.G. Sun, F.W. Meng / Appl. Math. Comput. 170 (2005) 545–555
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi oH ¼ h1 ðt; sÞ H ðt; sÞ and ot
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi oH ¼ h2 ðt; sÞ H ðt; sÞ; os
547
ð5Þ
where h21 ; h22 are locally integrable in D. In this paper, we define another function class Y. We say that a function U = U(t, s, r) belongs to the function class Y, denoted by U 2 Y, if U 2 C(E, R), where E = {(t, s, r) : t P s P r P t0}, which satisfies U(t, t, r) = 0, U(t, r, r) = 0 and has the partial derivative oU on E such that (oU/os)2 is locally integrable os in E. Remark 1. We can construct a function U(t, s, r) 2 Y in terms of two functions in X. For example, let U(t, s, r) = H1(t, s)H2(s, r), where H1, H2 2 X. It is easy to see that U(t, s, r) 2 Y. In Sections 2 and 3 of this paper, we will establish some new oscillation results for system (1) by using the auxiliary function U 2 Y. For the sake of convenience we define the operator T[Æ, Æ, Æ; r, t] in view of U 2 Y as the following: Z t T a ðD; E; F ; r; tÞ ¼ aðsÞ½U2 ðt; s; rÞDðsÞ þ Uðt; s; rÞU0s ðt; s; rÞEðsÞ r
U02 s ðt; s; rÞF ðsÞ ds;
ð6Þ
where a(t) is a positive and continuously differentiable function on [t0, 1), D(t), E(t) and F(t) 2 Rn·n (n · n matrices of real valued continuous functions on the interval [t0, 1)).
2. Oscillation criteria of Kamenev type Now, let us give the main results of this paper. Theorem 1. If there exist U 2 Y and f 2 C1[t0, 1) such that for each r P t0 lim sup k1 ½T a ðD; R; P ; r; tÞ > 0;
t!1
where Z s aðsÞ ¼ exp 2 f ðtÞdt ;
DðsÞ ¼ ðM RP 1 R=4ÞðsÞ
and 0
MðsÞ ¼ QðsÞ þ ðf 2 P ÞðsÞ ðfP Þ ðsÞ ðfRÞðsÞ; then system (1) is oscillatory.
ð7Þ
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Proof. Suppose to the contrary that there exists a prepared solution Y(t) of (1) such that det Y(t) 5 0 on [r, 1) for some r > t0. Let V ðtÞ ¼ aðtÞ½ðPY 0 Y 1 ÞðtÞ þ f ðtÞP ðtÞ
ð8Þ
for t P r;
then V(t) is symmetric for t P r. From (1) we have 2
0
V 0 ðtÞ ¼ 2ðfV ÞðtÞ þ aðtÞ½RY 0 Y 1 Q P ðY 0 Y 1 Þ þ ðfP Þ ðtÞ ¼ 2ðfV ÞðtÞ ðaRP 1 ÞðtÞ½ða1 V ÞðtÞ ðfP ÞðtÞ aðtÞ½ða1 V ÞðtÞ ðfP ÞðtÞ P 1 ðtÞ½ða1 V ÞðtÞ ðfP ÞðtÞ ðaQÞðtÞ ¼ ðRP 1 V ÞðtÞ ða1 VP 1 V ÞðtÞ aðtÞMðtÞ;
ð9Þ
where MðtÞ ¼ QðtÞ þ ðf 2 P ÞðtÞ ðfP Þ0 ðtÞ ðfRÞðtÞ: From (4) and noting that R(t) = R*(t) we get ðRP 1 V ÞðtÞ ¼ ½aRY 0 Y 1 ðtÞ þ ðafRÞðtÞ
¼ ½aðY 1 Þ ðY 0 Þ R ðtÞ þ ðafRÞðtÞ ¼ ðV P 1 RÞðtÞ ¼ ðVP 1 RÞðtÞ:
ð10Þ
1
Set A(t) = (P R)(t)/2. From (9) and (10) we have V 0 ðtÞ ¼ ðA V þ VAÞðtÞ ða1 VP 1 V ÞðtÞ aðtÞMðtÞ ¼ a1 ðtÞ½ðV þ aPAÞ P 1 ðV þ aPAÞ ðtÞ þ ðaA PAÞðtÞ aðtÞMðtÞ ¼ a1 ðtÞ½ðV þ aPAÞ P 1 ðV þ aPAÞ ðtÞ þ ðaRP 1 R=4ÞðtÞ aðtÞMðtÞ ¼ a1 ðtÞ½ðV þ aPAÞ P 1 ðV þ aPAÞ ðtÞ aðtÞDðtÞ;
ð11Þ
where DðtÞ ¼ ðM A PAÞðtÞ ¼ ðM RP 1 R=4ÞðtÞ: Multiplying (11), with t replaced by s, by U2(t, s, r) and integrating it from r to t, we obtain Z t aðsÞU2 ðt; s; rÞDðsÞds r Z t U2 ðt; s; rÞV 0 ðsÞds ¼ r Z t a1 ðsÞU2 ðt; s; rÞ½ðV þ aPAÞ P 1 ðV þ aPAÞ ðsÞds Z rt ¼2 Uðt; s; rÞU0s ðt; s; rÞV ðsÞds Zr t a1 ðsÞU2 ðt; s; rÞ½ðV þ aPAÞ P 1 ðV þ aPAÞ ðsÞds: ð12Þ r
Y.G. Sun, F.W. Meng / Appl. Math. Comput. 170 (2005) 545–555
549
According to the direct computation, we see that
2Uðt; s; rÞU0s ðt; s; rÞV ðsÞ a1 ðsÞU2 ðt; s; rÞ½ðV þ aPAÞ P 1 ðV þ aPAÞ ðsÞ 1 0 ¼ P 1 1 ðsÞW ðt; s; rÞW ðt; s; rÞP 1 ðsÞ Uðt; s; rÞUs ðt; s; rÞEðsÞ
þ aðsÞU02 s ðt; s; rÞP ðsÞ; where 1=2 1=2 P ðsÞ; P 1 1 ðsÞ ¼ a
EðsÞ ¼ aðsÞðA P þ PAÞðsÞ ¼ aðsÞRðsÞ;
and
W ðt; s; rÞ ¼ Uðt; s; rÞ½P 1 ðV þ aPAÞP 1 ðsÞ U0s ðt; s; rÞI n
ðan n n matrixÞ:
Thus, from (12) and the above computation, we have Z
t 2
aðsÞU ðt; s; rÞDðsÞds ¼ r
Z
t 1 P 1 1 ðsÞW ðt; s; rÞW ðt; s; rÞP 1 ðsÞds
r
Z
t
aðsÞ½Uðt; s; rÞU0s ðt; s; rÞRðsÞ
r U02 s ðt; s; rÞP ðsÞ ds:
Therefore, Z t aðsÞ½U2 ðt; s; rÞDðsÞ þ Uðt; s; rÞU0s ðt; s; rÞRðsÞ U02 s ðt; s; rÞP ðsÞ ds 6 0; r
ð13Þ i.e., T a ðD; R; P ; r; tÞ 6 0: This implies that lim sup k1 ½T a ðD; R; P ; r; tÞ 6 0;
t!1
which contradicts the assumption (7). This completes the proof of Theorem 1. h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi If we choose Uðt; s; rÞ ¼ H ðt; sÞH ðs; rÞ, where H 2 X, then we have pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi h1 ðs; rÞ H ðt; sÞ h2 ðt; sÞ H ðs; rÞ 0 ; Us ðt; s; rÞ ¼ 2 2 where h1, h2 are defined by (5). From Theorem 1, we can easily obtain the following corollary:
550
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Corollary 2. If there exist H 2 X and f 2 C1[t0, 1) such that for each r P t0, Z limt!1 sup k1
t
aðsÞH ðt; sÞH ðs; rÞ½DðsÞ þ /ðt; s; rÞRðsÞ /2 ðt; s; rÞP ðsÞ ds
r
ð14Þ
> 0;
h1 ðs;rÞ h2 ðt;sÞ ffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffi, a(s) and D(s) are defined as in Theorem 1, where /ðt; s; rÞ ¼ p H ðs;rÞ
2
H ðt;sÞ
2
then (1) is oscillatory. Remark 2. Under the appropriate choice of the functions U(t, s, r) and f(t), we can derive many new oscillation criteria for the system (1) from Corollary 2. Because of the limited space, we omit them here. In the sequel, we choose the function U(t, s, r) = (t s)(s r)a or U(t, s, r) = (t s)a(s r) for a > 1/2. then we have the following interesting oscillation results: Theorem 3. Assume that there exist a function f(t) 2 C1[t0, 1) and a constant a > 1/2 such that a(t)P(t) 6 In for t P t0, and for each r P t0, lim sup
t!1
Z
1 ðt rÞ
DðsÞ þ
2aþ1
k1
t 2
aðsÞðt sÞ ðs rÞ
2a
r
at ða þ 1Þs þ r a RðsÞ ds > ; ðt sÞðs rÞ ð2a 1Þð2a þ 1Þ
ð15Þ
where a(s) and D(s) are defined as in Theorem 1, then (1) is oscillatory. Proof. Suppose to the contrary that there exists a prepared solution Y(t) of (1) such that detY(t) 5 0 on [r, 1) for some r > t0. Let V(t) be defined by (8). Similar to the proof of Theorem 1, we have that (13) holds. Noting that a(t)P(t) 6 In for t P t0, we get from (13) Z
t
U2 ðt; s; rÞ½DðsÞ þ Uðt; s; rÞU0s ðt; s; rÞRðsÞ ds 6 r
Z
t
U02 s ðt; s; rÞI n ds;
r
where U(t, s, r) = (t s)(s r)a. It follows that Z
t
at ða þ 1Þs þ r RðsÞ ds aðsÞðt sÞ ðs rÞ DðsÞ þ ðt sÞðs rÞ 2
k1 r
Z
6 r
2a
t
U02 s ðt; s; rÞds:
ð16Þ
Y.G. Sun, F.W. Meng / Appl. Math. Comput. 170 (2005) 545–555
Since Z t r
U02 s ðt; s; rÞds
Z
551
t
½aðt sÞðs rÞa1 ðs rÞa 2 ds Z t Z t 2 2ða1Þ 2a1 ds 2a ðt sÞðs rÞ ds ¼ a2 ðt sÞ ðs rÞ r r Z t 2a ðs rÞ ds þ r Z t Z t 2a2 2a1 ¼ ðt sÞðs rÞ ds ðs rÞ2a ds 2a 1 r r Z t Z t a 2a 2a ðs rÞ ds ¼ ðs rÞ ds þ 2a 1 r r a ðt rÞ2a1 : ¼ ð2a 1Þð2a þ 1Þ ¼
r
Thus, from (16) and the above equalities we have Z t 1 at ða þ 1Þs þ r 2 2a RðsÞ ds k aðsÞðt sÞ ðs rÞ DðsÞ þ 1 ðt sÞðs rÞ ðt rÞ2aþ1 r a 6 ; ð2a 1Þð2a þ 1Þ which contradicts the assumption (15) when we take the sup limit in the above inequality. This completes the proof of Theorem 3. h Let U(t, s, r) = (t s)a(s r) for a > 1/2 and note that Z t Z t a a1 2 02 Us ðt; s; rÞds ¼ ½ðt sÞ aðt sÞ ðs rÞ ds r Z t Zr t 2a 2a1 ðt sÞ ds 2a ðt sÞ ðs rÞds ¼ r r Z t þ a2 ðt sÞ2ða1Þ ðs rÞ2 ds Z t Z t r 2a 2a ðt sÞ ds ðt sÞ ds ¼ r r Z t 2a2 þ ðt sÞ2a1 ðs rÞds 2a 1 r Z t a a 2a 2aþ1 ðt rÞ ðt sÞ ds ¼ : ¼ 2a 1 r ð2a 1Þð2a þ 1Þ Similar to the proof of Theorem 3, the following theorem is immediate. Theorem 4. Assume that there exist a function f(t) 2 C1[t0, 1) and a constant a > 1/2 such that a(t)P(t) 6 In for t P t0, and for each r P t0,
552
Y.G. Sun, F.W. Meng / Appl. Math. Comput. 170 (2005) 545–555
limt!1 sup
Z
1 2aþ1
k1
t
aðsÞðt sÞ2a ðs rÞ2
ðt rÞ r t ða þ 1Þs þ ar a RðsÞ ds > ; DðsÞ þ ðt sÞðs rÞ ð2a 1Þð2a þ 1Þ
ð17Þ
where a(s) and D(s) are defined as in Theorem 1, then (1) is oscillatory.
3. Interval oscillation criteria In this section, we give several interval criteria for the oscillation of system (1), that is, criteria given by the behavior of system (1) (or of P, Q and R) only on a sequence of subintervals of [t0, 1), rather than on the whole half-line. Therefore, our results can be applied to extreme cases such as R1 k1 ½ t0 QðsÞds ¼ 1. Theorem 5. Suppose that for each T P t0, there exist constants b > c P T, f(t) 2 C1[t0, 1) and U 2 Y such that k1 ½T a ðD; R; P ; c; bÞ > 0;
ð18Þ
where a(s) and D(s) are defined as in Theorem 1, then (1) is oscillatory. Proof. In fact, if we replace t, r by b, c, respectively, repeat the proof of Theorem 1, we can show that for every prepared solution Y(t) of (1), detY(t) has at least one zero in [c, b], i.e., detY(t) has arbitrarily large zero on [t0, 1). This completes the proof of Theorem 5. h The following oscillation result is evident. Corollary 6. Assume that for each T P t0, there exist constants b > c P T, f(t) 2 C1[t0, 1) and H 2 X such that Z b k1 aðsÞH ðb; sÞH ðs; cÞ½DðsÞ þ /ðb; s; cÞRðsÞ /2 ðb; s; cÞP ðsÞ ds > 0; c
ð19Þ where a(s) and D(s) are defined as in Theorem 1, then (1) is oscillatory. 4. Examples Example 1. Consider the following Euler matrix differential system l m Y 00 þ Y 0 þ 2 Y ¼ 0; t t
t P 1;
ð20Þ
Y.G. Sun, F.W. Meng / Appl. Math. Comput. 170 (2005) 545–555
553
where l, m are constants, P(t) = In, QðtÞ ¼ tm2 I n and RðtÞ ¼ lt I n . Using Theorem 3 of this paper, we will prove that (20) is oscillatory when (l 1)2 < 4m. In fact, let f(t) 0, then for any constant a > 1/2 and for each r P 1, the left-hand side of (15) takes the form lim sup t!1
Z
1 ðt rÞ
2aþ1
t 2
ðt sÞ ðs rÞ
r
m l2 =4 þ l=2 ds s2
2a
2
¼
2a
m l =4 þ l=2 ðt rÞ m l2 =4 þ l=2 lim : ¼ 2a að2a 1Þð2a þ 1Þ t!1 að2a 1Þð2a þ 1Þ t
Since (l 1)2 < 4m, i.e., 4m l2/4 + l/2 > 1/4, we can choose an appropriate constant a > 1/2 such that 4m l2/4 + l/2 > a2, and hence 4m l2 =4 þ l=2 a > : að2a 1Þð2a þ 1Þ ð2a 1Þð2a þ 1Þ Thus, (15) holds. By Theorem 3, we have that (20) is oscillatory when 2 (l 1)2 < 4m. On the other hand, if (l p1)ffiffiffiffiffiffiffiffiffiffiffiffi P 4m, evidently, (20) has an nonosffi cillatory prepared solution Y ðtÞ ¼ t our results are sharper.
1lþ
ðl1Þ2 4m 2
I n . In this sense, we know that
Example 2. Consider the following matrix differential system Y 00 þ Y 0 þ QðtÞY ¼ 0;
ð21Þ
t P 0;
where Q(t) = q(t)In and 8 3k 6 t 6 3k þ 1; > < cðt 3kÞ; qðtÞ ¼ cðt þ 3k þ 2Þ; 3k þ 1 < t 6 3k þ 2; > : nðt 3k 2Þð3k þ 3 tÞ; 3k þ 2 < t 6 3k þ 3; where c is a constant and k 2 K = {0, 1, 2, . . .}. For any constant T P t0, there exists k 2 K such that 3k P T. Let c = 3k, b = 3k + 1, f(t) 0 and U(t, s, r) = (t s)(s r), then we have Z 3kþ1 2 2 k1 ½T a ðD; R; P ; c; bÞ ¼ ½ð3k þ 1 sÞ ðs 3kÞ ðcðs 3kÞ 1=4Þ 3k
þ ð3k þ 1 sÞðs 3kÞð6k þ 1 2sÞ 2
ð6k þ 1 2sÞ ds: It is easy to see that condition (18) holds for sufficiently large c, and hence, the system (21) is oscillatory for sufficiently large c by Theorem 5. However, in R1 this case we have k1 ½ 0 QðtÞdt ¼ 1.
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Example 3. Consider the following matrix differential system Y 00 þ cos tY 0 þ c sin tY ¼ 0;
t P 0;
ð22Þ
where c is a constant, R(t) = cos tIn and Q(t) = csin tIn. For any constant T P t0, there exists k 2 K such that 2kp P T. Let c = 2kp, b = (2k + 1)p. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Choose f(t) 0 and Uðt; s; rÞ ¼ j sinðt sÞ sinðs rÞj, then we have cos2 s 2 2 sin s c sin s k1 ½T a ðD; R; P ; c; bÞ ¼ þ sin scos s cos s ds 4 0 4c 2 p p ¼ : 3 3 32 2 From Theorem 5, we see that (22) is oscillatory for c > 34 17p þ 23 . 32 Z
p
2
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555
[16] Y.G. Sun, New oscillation criteria for linear matrix Hamiltonian systems, J. Math. Anal. Appl. 279 (2003) 651–658. [17] Y.G. Sun, F.W. Meng, New oscillation criteria for linear matrix Hamiltonian systems, Appl. Math. Comput. 155 (2004) 259–268. [18] Y.G. Sun, New Kamenev type oscillation criteria for linear matrix Hamiltonian systems, Appl. Math. Comput. 158 (2004) 69–78. [19] Y.G. Sun, New Kamenev type theorems for second-order linear matrix differential systems, Appl. Math. Lett. 17 (2004) 875–880. [20] I. Sowjaya Kumari, S. Umanaheswaram, Oscillation criteria for linear matrix Hamiltonian systems, J. Differen. Equat. 165 (2000) 174–198. [21] Q.R. Wang, Oscillation theorems for second order matrix differential systems, Math. Nachr. 266 (2004) 92–105. [22] X. Yang, Oscillation criteria for certain second-order matrix differential equations, J. Math. Anval. Appl. 265 (2002) 285–295.
Oscillation results for matrix differential systems with damping Yuan Gong Sun *, Fan Wei Meng Department of Mathematics, Qufu Normal University, Qufu, Shandong 273165, China
Abstract By introducing a class of new parameter functions U(t, s, r), we establish some new oscillation criteria in terms of the coefficients for the matrix differential system with damping (P(t)Y 0 ) 0 + R(t)Y 0 + Q(t)Y = 0 under the hypothesis H: P(t) = P*(t) > 0, Q(t) = Q*(t) and R(t) = R*(t) are n · n matrices of real valued continuous functions on the interval [t0, 1). Our results are not contained in known ones. 2005 Elsevier Inc. All rights reserved. Keywords: Oscillation; Matrix differential system; Damping
1. Introduction Consider the second order linear matrix differential system with damped term ðP ðtÞY 0 Þ0 þ RðtÞY 0 þ QðtÞY ¼ 0;
t P t0 ;
*
Corresponding author. E-mail address: [email protected] (Y.G. Sun).
0096-3003/$ - see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2004.12.012
ð1Þ
546
Y.G. Sun, F.W. Meng / Appl. Math. Comput. 170 (2005) 545–555
where P(t) = P*(t) > 0 (i.e., P(t) is positive definite), Q(t) = Q*(t) and R(t) = R*(t) are n · n matrices of real valued continuous functions on the interval [t0, 1). By M* we mean the transpose of the matrix M. In the absence of damping, i.e., R(t) 0, (1) reduces to the following matrix differential system 0
ðP ðtÞY 0 Þ þ QðtÞY ¼ 0;
t P t0 ;
which is the particular case of the matrix Hamiltonian system 0 U ðtÞ ¼ AðtÞU ðtÞ þ BðtÞV ðtÞ; V 0 ðtÞ ¼ CðtÞU ðtÞ A ðtÞV ðtÞ;
ð2Þ
ð3Þ
with A(t) 0, B1(t) = P(t) and C(t) = Q(t). When R(t) f 0, (1) in general cannot be rewritten to system (3). Therefore, all the existing oscillation results for (2) and (3) generally cannot be applied to (1). By now, there have been many papers (see, for example, [1–22] and the references quoted therein) devoted to the oscillation of systems (2) and (3). It is well known that a successful oscillation theory for matrix differential system can be carried out only for the class of prepared solutions. As usual, a nontrivial solution Y(t) (i.e., detY(t) 5 0 for at lease one t 2 [t0, 1)) of (2) is said to be prepared if for t 2 [t0, 1), 0
Y ðtÞP ðtÞY 0 ðtÞ ðY ðtÞÞ P ðtÞY ðtÞ:
ðaÞ
A nontrivial solution (U(t),V(t)) (i.e., det U(t) 5 0 for at lease one t 2 [t0, 1)) of (3) is said to be prepared if for t 2 [t0, 1), U ðtÞV ðtÞ V ðtÞU ðtÞ 0:
ðbÞ
To best of our knowledge, it seems to us that little has been known about the oscillation of (1) except [21]. In [21], the author studied the particular case of (1) with R(t) = r(t)P(t), where r(t) 2 C([t0, 1),R). The purpose of this paper is to deal with the oscillation of the more general system (1). In this paper, we say a nontrivial solution Y(t) of (1) is prepared if for t 2 [t0, 1), ( Y ðtÞP ðtÞY 0 ðtÞ ðY ðtÞÞ0 P ðtÞY ðtÞ 0; ð4Þ 0 Y ðtÞRðtÞY 0 ðtÞ ðY ðtÞÞ RðtÞY ðtÞ 0; i.e., Y*(t)P(t)Y 0 (t) and Y*(t)R(t)Y 0 (t) are symmetric. A prepared solution Y(t) of (1) is said to be oscillatory, if det Y(t) has arbitrarily large zeros on [t0, 1). As we can see, the important tool in the study of oscillatory behavior of solutions for the scalar equation (as a special case of (1) or (2)) is the averaging technique, which involves a function class X. Say a function H = H(t, s) belongs to the function class X, if H 2 C(D, R+), where D = {(t, s) : t0 6 s 6 t}, which satisfies H(t, t) = 0, H(t, s) > 0 for t > s, and has partial derivative oH/os and oH/ot on D such that
Y.G. Sun, F.W. Meng / Appl. Math. Comput. 170 (2005) 545–555
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi oH ¼ h1 ðt; sÞ H ðt; sÞ and ot
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi oH ¼ h2 ðt; sÞ H ðt; sÞ; os
547
ð5Þ
where h21 ; h22 are locally integrable in D. In this paper, we define another function class Y. We say that a function U = U(t, s, r) belongs to the function class Y, denoted by U 2 Y, if U 2 C(E, R), where E = {(t, s, r) : t P s P r P t0}, which satisfies U(t, t, r) = 0, U(t, r, r) = 0 and has the partial derivative oU on E such that (oU/os)2 is locally integrable os in E. Remark 1. We can construct a function U(t, s, r) 2 Y in terms of two functions in X. For example, let U(t, s, r) = H1(t, s)H2(s, r), where H1, H2 2 X. It is easy to see that U(t, s, r) 2 Y. In Sections 2 and 3 of this paper, we will establish some new oscillation results for system (1) by using the auxiliary function U 2 Y. For the sake of convenience we define the operator T[Æ, Æ, Æ; r, t] in view of U 2 Y as the following: Z t T a ðD; E; F ; r; tÞ ¼ aðsÞ½U2 ðt; s; rÞDðsÞ þ Uðt; s; rÞU0s ðt; s; rÞEðsÞ r
U02 s ðt; s; rÞF ðsÞ ds;
ð6Þ
where a(t) is a positive and continuously differentiable function on [t0, 1), D(t), E(t) and F(t) 2 Rn·n (n · n matrices of real valued continuous functions on the interval [t0, 1)).
2. Oscillation criteria of Kamenev type Now, let us give the main results of this paper. Theorem 1. If there exist U 2 Y and f 2 C1[t0, 1) such that for each r P t0 lim sup k1 ½T a ðD; R; P ; r; tÞ > 0;
t!1
where Z s aðsÞ ¼ exp 2 f ðtÞdt ;
DðsÞ ¼ ðM RP 1 R=4ÞðsÞ
and 0
MðsÞ ¼ QðsÞ þ ðf 2 P ÞðsÞ ðfP Þ ðsÞ ðfRÞðsÞ; then system (1) is oscillatory.
ð7Þ
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Y.G. Sun, F.W. Meng / Appl. Math. Comput. 170 (2005) 545–555
Proof. Suppose to the contrary that there exists a prepared solution Y(t) of (1) such that det Y(t) 5 0 on [r, 1) for some r > t0. Let V ðtÞ ¼ aðtÞ½ðPY 0 Y 1 ÞðtÞ þ f ðtÞP ðtÞ
ð8Þ
for t P r;
then V(t) is symmetric for t P r. From (1) we have 2
0
V 0 ðtÞ ¼ 2ðfV ÞðtÞ þ aðtÞ½RY 0 Y 1 Q P ðY 0 Y 1 Þ þ ðfP Þ ðtÞ ¼ 2ðfV ÞðtÞ ðaRP 1 ÞðtÞ½ða1 V ÞðtÞ ðfP ÞðtÞ aðtÞ½ða1 V ÞðtÞ ðfP ÞðtÞ P 1 ðtÞ½ða1 V ÞðtÞ ðfP ÞðtÞ ðaQÞðtÞ ¼ ðRP 1 V ÞðtÞ ða1 VP 1 V ÞðtÞ aðtÞMðtÞ;
ð9Þ
where MðtÞ ¼ QðtÞ þ ðf 2 P ÞðtÞ ðfP Þ0 ðtÞ ðfRÞðtÞ: From (4) and noting that R(t) = R*(t) we get ðRP 1 V ÞðtÞ ¼ ½aRY 0 Y 1 ðtÞ þ ðafRÞðtÞ
¼ ½aðY 1 Þ ðY 0 Þ R ðtÞ þ ðafRÞðtÞ ¼ ðV P 1 RÞðtÞ ¼ ðVP 1 RÞðtÞ:
ð10Þ
1
Set A(t) = (P R)(t)/2. From (9) and (10) we have V 0 ðtÞ ¼ ðA V þ VAÞðtÞ ða1 VP 1 V ÞðtÞ aðtÞMðtÞ ¼ a1 ðtÞ½ðV þ aPAÞ P 1 ðV þ aPAÞ ðtÞ þ ðaA PAÞðtÞ aðtÞMðtÞ ¼ a1 ðtÞ½ðV þ aPAÞ P 1 ðV þ aPAÞ ðtÞ þ ðaRP 1 R=4ÞðtÞ aðtÞMðtÞ ¼ a1 ðtÞ½ðV þ aPAÞ P 1 ðV þ aPAÞ ðtÞ aðtÞDðtÞ;
ð11Þ
where DðtÞ ¼ ðM A PAÞðtÞ ¼ ðM RP 1 R=4ÞðtÞ: Multiplying (11), with t replaced by s, by U2(t, s, r) and integrating it from r to t, we obtain Z t aðsÞU2 ðt; s; rÞDðsÞds r Z t U2 ðt; s; rÞV 0 ðsÞds ¼ r Z t a1 ðsÞU2 ðt; s; rÞ½ðV þ aPAÞ P 1 ðV þ aPAÞ ðsÞds Z rt ¼2 Uðt; s; rÞU0s ðt; s; rÞV ðsÞds Zr t a1 ðsÞU2 ðt; s; rÞ½ðV þ aPAÞ P 1 ðV þ aPAÞ ðsÞds: ð12Þ r
Y.G. Sun, F.W. Meng / Appl. Math. Comput. 170 (2005) 545–555
549
According to the direct computation, we see that
2Uðt; s; rÞU0s ðt; s; rÞV ðsÞ a1 ðsÞU2 ðt; s; rÞ½ðV þ aPAÞ P 1 ðV þ aPAÞ ðsÞ 1 0 ¼ P 1 1 ðsÞW ðt; s; rÞW ðt; s; rÞP 1 ðsÞ Uðt; s; rÞUs ðt; s; rÞEðsÞ
þ aðsÞU02 s ðt; s; rÞP ðsÞ; where 1=2 1=2 P ðsÞ; P 1 1 ðsÞ ¼ a
EðsÞ ¼ aðsÞðA P þ PAÞðsÞ ¼ aðsÞRðsÞ;
and
W ðt; s; rÞ ¼ Uðt; s; rÞ½P 1 ðV þ aPAÞP 1 ðsÞ U0s ðt; s; rÞI n
ðan n n matrixÞ:
Thus, from (12) and the above computation, we have Z
t 2
aðsÞU ðt; s; rÞDðsÞds ¼ r
Z
t 1 P 1 1 ðsÞW ðt; s; rÞW ðt; s; rÞP 1 ðsÞds
r
Z
t
aðsÞ½Uðt; s; rÞU0s ðt; s; rÞRðsÞ
r U02 s ðt; s; rÞP ðsÞ ds:
Therefore, Z t aðsÞ½U2 ðt; s; rÞDðsÞ þ Uðt; s; rÞU0s ðt; s; rÞRðsÞ U02 s ðt; s; rÞP ðsÞ ds 6 0; r
ð13Þ i.e., T a ðD; R; P ; r; tÞ 6 0: This implies that lim sup k1 ½T a ðD; R; P ; r; tÞ 6 0;
t!1
which contradicts the assumption (7). This completes the proof of Theorem 1. h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi If we choose Uðt; s; rÞ ¼ H ðt; sÞH ðs; rÞ, where H 2 X, then we have pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi h1 ðs; rÞ H ðt; sÞ h2 ðt; sÞ H ðs; rÞ 0 ; Us ðt; s; rÞ ¼ 2 2 where h1, h2 are defined by (5). From Theorem 1, we can easily obtain the following corollary:
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Corollary 2. If there exist H 2 X and f 2 C1[t0, 1) such that for each r P t0, Z limt!1 sup k1
t
aðsÞH ðt; sÞH ðs; rÞ½DðsÞ þ /ðt; s; rÞRðsÞ /2 ðt; s; rÞP ðsÞ ds
r
ð14Þ
> 0;
h1 ðs;rÞ h2 ðt;sÞ ffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffi, a(s) and D(s) are defined as in Theorem 1, where /ðt; s; rÞ ¼ p H ðs;rÞ
2
H ðt;sÞ
2
then (1) is oscillatory. Remark 2. Under the appropriate choice of the functions U(t, s, r) and f(t), we can derive many new oscillation criteria for the system (1) from Corollary 2. Because of the limited space, we omit them here. In the sequel, we choose the function U(t, s, r) = (t s)(s r)a or U(t, s, r) = (t s)a(s r) for a > 1/2. then we have the following interesting oscillation results: Theorem 3. Assume that there exist a function f(t) 2 C1[t0, 1) and a constant a > 1/2 such that a(t)P(t) 6 In for t P t0, and for each r P t0, lim sup
t!1
Z
1 ðt rÞ
DðsÞ þ
2aþ1
k1
t 2
aðsÞðt sÞ ðs rÞ
2a
r
at ða þ 1Þs þ r a RðsÞ ds > ; ðt sÞðs rÞ ð2a 1Þð2a þ 1Þ
ð15Þ
where a(s) and D(s) are defined as in Theorem 1, then (1) is oscillatory. Proof. Suppose to the contrary that there exists a prepared solution Y(t) of (1) such that detY(t) 5 0 on [r, 1) for some r > t0. Let V(t) be defined by (8). Similar to the proof of Theorem 1, we have that (13) holds. Noting that a(t)P(t) 6 In for t P t0, we get from (13) Z
t
U2 ðt; s; rÞ½DðsÞ þ Uðt; s; rÞU0s ðt; s; rÞRðsÞ ds 6 r
Z
t
U02 s ðt; s; rÞI n ds;
r
where U(t, s, r) = (t s)(s r)a. It follows that Z
t
at ða þ 1Þs þ r RðsÞ ds aðsÞðt sÞ ðs rÞ DðsÞ þ ðt sÞðs rÞ 2
k1 r
Z
6 r
2a
t
U02 s ðt; s; rÞds:
ð16Þ
Y.G. Sun, F.W. Meng / Appl. Math. Comput. 170 (2005) 545–555
Since Z t r
U02 s ðt; s; rÞds
Z
551
t
½aðt sÞðs rÞa1 ðs rÞa 2 ds Z t Z t 2 2ða1Þ 2a1 ds 2a ðt sÞðs rÞ ds ¼ a2 ðt sÞ ðs rÞ r r Z t 2a ðs rÞ ds þ r Z t Z t 2a2 2a1 ¼ ðt sÞðs rÞ ds ðs rÞ2a ds 2a 1 r r Z t Z t a 2a 2a ðs rÞ ds ¼ ðs rÞ ds þ 2a 1 r r a ðt rÞ2a1 : ¼ ð2a 1Þð2a þ 1Þ ¼
r
Thus, from (16) and the above equalities we have Z t 1 at ða þ 1Þs þ r 2 2a RðsÞ ds k aðsÞðt sÞ ðs rÞ DðsÞ þ 1 ðt sÞðs rÞ ðt rÞ2aþ1 r a 6 ; ð2a 1Þð2a þ 1Þ which contradicts the assumption (15) when we take the sup limit in the above inequality. This completes the proof of Theorem 3. h Let U(t, s, r) = (t s)a(s r) for a > 1/2 and note that Z t Z t a a1 2 02 Us ðt; s; rÞds ¼ ½ðt sÞ aðt sÞ ðs rÞ ds r Z t Zr t 2a 2a1 ðt sÞ ds 2a ðt sÞ ðs rÞds ¼ r r Z t þ a2 ðt sÞ2ða1Þ ðs rÞ2 ds Z t Z t r 2a 2a ðt sÞ ds ðt sÞ ds ¼ r r Z t 2a2 þ ðt sÞ2a1 ðs rÞds 2a 1 r Z t a a 2a 2aþ1 ðt rÞ ðt sÞ ds ¼ : ¼ 2a 1 r ð2a 1Þð2a þ 1Þ Similar to the proof of Theorem 3, the following theorem is immediate. Theorem 4. Assume that there exist a function f(t) 2 C1[t0, 1) and a constant a > 1/2 such that a(t)P(t) 6 In for t P t0, and for each r P t0,
552
Y.G. Sun, F.W. Meng / Appl. Math. Comput. 170 (2005) 545–555
limt!1 sup
Z
1 2aþ1
k1
t
aðsÞðt sÞ2a ðs rÞ2
ðt rÞ r t ða þ 1Þs þ ar a RðsÞ ds > ; DðsÞ þ ðt sÞðs rÞ ð2a 1Þð2a þ 1Þ
ð17Þ
where a(s) and D(s) are defined as in Theorem 1, then (1) is oscillatory.
3. Interval oscillation criteria In this section, we give several interval criteria for the oscillation of system (1), that is, criteria given by the behavior of system (1) (or of P, Q and R) only on a sequence of subintervals of [t0, 1), rather than on the whole half-line. Therefore, our results can be applied to extreme cases such as R1 k1 ½ t0 QðsÞds ¼ 1. Theorem 5. Suppose that for each T P t0, there exist constants b > c P T, f(t) 2 C1[t0, 1) and U 2 Y such that k1 ½T a ðD; R; P ; c; bÞ > 0;
ð18Þ
where a(s) and D(s) are defined as in Theorem 1, then (1) is oscillatory. Proof. In fact, if we replace t, r by b, c, respectively, repeat the proof of Theorem 1, we can show that for every prepared solution Y(t) of (1), detY(t) has at least one zero in [c, b], i.e., detY(t) has arbitrarily large zero on [t0, 1). This completes the proof of Theorem 5. h The following oscillation result is evident. Corollary 6. Assume that for each T P t0, there exist constants b > c P T, f(t) 2 C1[t0, 1) and H 2 X such that Z b k1 aðsÞH ðb; sÞH ðs; cÞ½DðsÞ þ /ðb; s; cÞRðsÞ /2 ðb; s; cÞP ðsÞ ds > 0; c
ð19Þ where a(s) and D(s) are defined as in Theorem 1, then (1) is oscillatory. 4. Examples Example 1. Consider the following Euler matrix differential system l m Y 00 þ Y 0 þ 2 Y ¼ 0; t t
t P 1;
ð20Þ
Y.G. Sun, F.W. Meng / Appl. Math. Comput. 170 (2005) 545–555
553
where l, m are constants, P(t) = In, QðtÞ ¼ tm2 I n and RðtÞ ¼ lt I n . Using Theorem 3 of this paper, we will prove that (20) is oscillatory when (l 1)2 < 4m. In fact, let f(t) 0, then for any constant a > 1/2 and for each r P 1, the left-hand side of (15) takes the form lim sup t!1
Z
1 ðt rÞ
2aþ1
t 2
ðt sÞ ðs rÞ
r
m l2 =4 þ l=2 ds s2
2a
2
¼
2a
m l =4 þ l=2 ðt rÞ m l2 =4 þ l=2 lim : ¼ 2a að2a 1Þð2a þ 1Þ t!1 að2a 1Þð2a þ 1Þ t
Since (l 1)2 < 4m, i.e., 4m l2/4 + l/2 > 1/4, we can choose an appropriate constant a > 1/2 such that 4m l2/4 + l/2 > a2, and hence 4m l2 =4 þ l=2 a > : að2a 1Þð2a þ 1Þ ð2a 1Þð2a þ 1Þ Thus, (15) holds. By Theorem 3, we have that (20) is oscillatory when 2 (l 1)2 < 4m. On the other hand, if (l p1)ffiffiffiffiffiffiffiffiffiffiffiffi P 4m, evidently, (20) has an nonosffi cillatory prepared solution Y ðtÞ ¼ t our results are sharper.
1lþ
ðl1Þ2 4m 2
I n . In this sense, we know that
Example 2. Consider the following matrix differential system Y 00 þ Y 0 þ QðtÞY ¼ 0;
ð21Þ
t P 0;
where Q(t) = q(t)In and 8 3k 6 t 6 3k þ 1; > < cðt 3kÞ; qðtÞ ¼ cðt þ 3k þ 2Þ; 3k þ 1 < t 6 3k þ 2; > : nðt 3k 2Þð3k þ 3 tÞ; 3k þ 2 < t 6 3k þ 3; where c is a constant and k 2 K = {0, 1, 2, . . .}. For any constant T P t0, there exists k 2 K such that 3k P T. Let c = 3k, b = 3k + 1, f(t) 0 and U(t, s, r) = (t s)(s r), then we have Z 3kþ1 2 2 k1 ½T a ðD; R; P ; c; bÞ ¼ ½ð3k þ 1 sÞ ðs 3kÞ ðcðs 3kÞ 1=4Þ 3k
þ ð3k þ 1 sÞðs 3kÞð6k þ 1 2sÞ 2
ð6k þ 1 2sÞ ds: It is easy to see that condition (18) holds for sufficiently large c, and hence, the system (21) is oscillatory for sufficiently large c by Theorem 5. However, in R1 this case we have k1 ½ 0 QðtÞdt ¼ 1.
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Y.G. Sun, F.W. Meng / Appl. Math. Comput. 170 (2005) 545–555
Example 3. Consider the following matrix differential system Y 00 þ cos tY 0 þ c sin tY ¼ 0;
t P 0;
ð22Þ
where c is a constant, R(t) = cos tIn and Q(t) = csin tIn. For any constant T P t0, there exists k 2 K such that 2kp P T. Let c = 2kp, b = (2k + 1)p. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Choose f(t) 0 and Uðt; s; rÞ ¼ j sinðt sÞ sinðs rÞj, then we have cos2 s 2 2 sin s c sin s k1 ½T a ðD; R; P ; c; bÞ ¼ þ sin scos s cos s ds 4 0 4c 2 p p ¼ : 3 3 32 2 From Theorem 5, we see that (22) is oscillatory for c > 34 17p þ 23 . 32 Z
p
2
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