Oscillation results for matrix differential systems with damping

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Abstract

By introducing a class of new parameter functions Φ(t, s, r), we establish some new oscillation criteria in terms of the coefficients for the matrix differential system with damping (P(t)Y′)′ + R(t)Y + 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, ∞). Our results are not contained in known ones.

Introduction

Consider the second order linear matrix differential system with damped term(P(t)Y)+R(t)Y+Q(t)Y=0,tt0,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, ∞). 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(P(t)Y)+Q(t)Y=0,tt0,which is the particular case of the matrix Hamiltonian systemU(t)=A(t)U(t)+B(t)V(t),V(t)=C(t)U(t)-A(t)V(t),with A(t)  0, B−1(t) = P(t) and C(t) = Q(t). When R(t)  0, (1) in general cannot be rewritten to system (3). Therefore, all the existing oscillation results for (2), (3) generally cannot be applied to (1).

By now, there have been many papers (see, for example, [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22] and the references quoted therein) devoted to the oscillation of systems (2), (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)  0 for at lease one t  [t0, ∞)) of (2) is said to be prepared if for t  [t0, ∞),Y(t)P(t)Y(t)(Y(t))P(t)Y(t).

A nontrivial solution (U(t),V(t)) (i.e., det U(t)  0 for at lease one t  [t0, ∞)) of (3) is said to be prepared if for t  [t0, ∞),U(t)V(t)-V(t)U(t)0.

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)  C([t0, ∞),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  [t0, ∞),Y(t)P(t)Y(t)-(Y(t))P(t)Y(t)0,Y(t)R(t)Y(t)-(Y(t))R(t)Y(t)0,i.e., Y*(t)P(t)Y′(t) and Y*(t)R(t)Y′(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, ∞).

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), (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  C(D, R+), where D = {(t, s) : t0  s  t}, which satisfies H(t, t) = 0, H(t, s) > 0 for t > s, and has partial derivative ∂H/∂s and ∂H/∂t on D such thatHt=h1(t,s)H(t,s)andHs=-h2(t,s)H(t,s),where h12,h22 are locally integrable in D.

In this paper, we define another function class Y. We say that a function Φ = Φ(t, s, r) belongs to the function class Y, denoted by Φ  Y, if Φ  C(E, R), where E = {(t, s, r) : t  s  r  t0}, which satisfies Φ(t, t, r) = 0, Φ(t, r, r) = 0 and has the partial derivative Φs on E such that (∂Φ/∂s)2 is locally integrable in E.

Remark 1

We can construct a function Φ(t, s, r)  Y in terms of two functions in X. For example, let Φ(t, s, r) = H1(t, s)H2(s, r), where H1, H2  X. It is easy to see that Φ(t, s, r)  Y.

In Sections 2 Oscillation criteria of Kamenev type, 3 Interval oscillation criteria of this paper, we will establish some new oscillation results for system (1) by using the auxiliary function Φ  Y. For the sake of convenience we define the operator T[·, ·, ·; r, t] in view of Φ  Y as the following:Ta(D,E,F;r,t)=rta(s)[Φ2(t,s,r)D(s)+Φ(t,s,r)Φs(t,s,r)E(s)-Φs2(t,s,r)F(s)]ds,where a(t) is a positive and continuously differentiable function on [t0, ∞), D(t), E(t) and F(t)  Rn×n (n × n matrices of real valued continuous functions on the interval [t0, ∞)).

Section snippets

Oscillation criteria of Kamenev type

Now, let us give the main results of this paper.

Theorem 1

If there exist Φ  Y and f  C1[t0, ) such that for each r  t0limtsupλ1[Ta(D,R,P;r,t)]>0,wherea(s)=exp-2sf(t)dt,D(s)=(M-RP-1R/4)(s)andM(s)=Q(s)+(f2P)(s)-(fP)(s)-(fR)(s),then system (1) is oscillatory.

Proof

Suppose to the contrary that there exists a prepared solution Y(t) of (1) such that det Y(t)  0 on [r, ∞) for some r > t0. LetV(t)=a(t)[(PYY-1)(t)+f(t)P(t)]fortr,then V(t) is symmetric for t  r. From (1) we haveV(t)=-2(fV)(t)+a(t)[-RYY-1-Q-P(YY-1)2+(fP)

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, ∞), rather than on the whole half-line. Therefore, our results can be applied to extreme cases such as λ1[t0Q(s)ds]=-.

Theorem 5

Suppose that for each T  t0, there exist constants b > c  T, f(t)  C1[t0, ) and Φ  Y such thatλ1[Ta(D,R,P;c,b)]>0,where a(s) and D(s) are defined as in Theorem 1, then (1) is

Examples

Example 1

Consider the following Euler matrix differential systemY+ltY+mt2Y=0,t1,where l, m are constants, P(t) = In, Q(t)=mt2InandR(t)=ltIn. 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 α > 1/2 and for each r  1, the left-hand side of (15) takes the formlimsupt1(t-r)2α+1rt(t-s)2(s-r)2αm-l2/4+l/2s2ds=m-l2/4+l/2α(2α-1)(2α+1)limt(t-r)2αt2α=m-l2/4+l/2α(2α-1)(2α+1).

Since (l  1)2 < 4m, i.e., 4m  l2/4 + l/2 > 1/4, we can choose an

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