| 摘要: | 擴展帶有停留( Retarded)特質的高階常微分方程式( * ) ，LnX( t ) +(-1)n+1 Σi=klPi (t)Φ(X( t-Zi)=0,此處, LkX(t)=ak(t)(Lk-1 X(t))，k=1,2………………………n，( = d/ dt ),且 a0(t) = an(t) = 1 , LoX(t) = X(t)，對每一 k,k= 1,2……………………n-1，ak(t) C (〔0,∞)，(0，∞))，另外對每一 i =1,2…………k，Pi(t) C (〔0,∞)，(0，∞))，Zi為正常數，且Φ滿足，當y≠0，則 Φ(y)/y>0，利用積分及有界函數的極限性質，證明出下列任一條件成立①，某一 i,i = 1,2…………k，∫t-zi 1/a1(Sn-1) ∫Sn-1 1/a2(Sn-2) ∫Sn-2…………∫S Pi(S)dSdS1…………dSn-1>Φ,② ∫t-z 1/a1(Sn-1) ∫Sn-1 1/a2 (Sn-2)…………∫S Σi=1 Pi(S)dS dS1…………dSn-1 >Φ，則可推出(*)之任何有界解必振盪(Oscillatory)以上，Φ = max｛lim y→∞ sup y / Φ(y)，lim y→-∞ Tnfy/Φ(y)｝，Z = min ｛Z1,Z2,Z3…………Zk｝進而討論高階前進(Advanced)微分方程式(* *)，LnX(t) - Σj=1qj(t) A(x(t+δj)) = 0 及混合型(***)Ln X(t) + (-1)Σi=1 Pi(t)Φ(X(t-2i))+(-1) Σj=1 qj(t)A(X(k+δj)) = 0，類似的振盪定理，這裏A，qj(t)，δj與Φ，Pi(t)，Zi有相同的性質。
Our goal in this paper is to discuss ; using integration and the limit property of bounded, the oscillatory behavior of retarded differential equation form ( * )， LnX( t ) +(-1)n+1 Σi=klPi (t)Φ(X( t-Zi)=0,where, LkX(t)=ak(t)(Lk-1 X(t))，for k=1,2………………………n，( = d/ dt ) and Assume a0(t) = an(t) = 1 , LoX(t) = X(t)，also for k= 1,2……………………n-1，ak(t) C (〔0,∞)，(0，∞)) and i =1,2…………k，Zi is opsitive constant, Pi(t) C (〔0,∞)，(0，∞)) and Assume y≠0，imply Φ (y)/y>0，we prove that each of the following condition (1) for same i = 1,2…………k，∫t-zi 1/a1(Sn-1) ∫Sn-1 1/a2(Sn-2) ∫Sn-2…………∫S Pi(S)dSdS1…………dSn-1> ,2 ∫t-z 1/a1(Sn-1) ∫Sn-1 1/a2 (Sn-2)…………∫S Σi=1 Pi(S)dS dS1…………dSn-1 > ，where Z = min ｛Z1…………Zk｝Φ = max｛lim y→∞ sup y / Φ(y)，lim y→-∞ Tnfy/Φ(y)｝imply all bounded solution of (*) oscillatory our result also extend the oscillatory behavior of advanced differenbal equation(* *)，LnX(t) - Σj=1qj(t) A(x(t+δj)) = 0 及 mix type equation (***)Ln X(t) + (-1)Σi=1 Pi(t)Φ(X(t-2i))+(-1) Σj=1 qj(t)A(X(k+δj)) = 0，where A and qj(t)，δj，being the same property with Φ and Pi(t),Zi.# |
