一类n阶积分微分方程的解的渐近性态

Asymptotic Behaviour of Certain Arbitrary Order Integrodifferential Equations

讨论了ｎ阶积分微分方程的解的渐近性态，其中Ｄｎ（ｔ，ｘ）：＝ｘ（ｎ）＋ａ１（ｔ）ｘ（ｎ－１）＋…＋ａｎ－１（ｔ）ｘ’＋ａｎ（ｔ）ｘ，ｔ∈［０，＋∞）．证明了在适当条件下此积分微分方程的解可由微分方程Ｄｎ（ｔ，ｘ）＝０的基本解组Ｘ１（ｔ），ｘ２（ｔ），…，ｘｎ（ｔ）表示成如下形式：Ｘ（ｔ）＝Ｃ１（ｔ）Ｘ１（ｔ）＋…＋Ｃｎ（ｔ）ｘｎ（ｔ），这里ｃｉ（ｔ）∈Ｃ（Ｒ＋，Ｒ）是有界函数且当ｔ→∞时，Ｃｉ（ｔ）存在极限，ｉ＝１，２，…，ｎ．

It is proved in the paper that, under suitable conditions on the functions f g and ai(t),every solution x(t) of the integrodifferential equation:can be expressed in the form x(t)=c1(t)x1(t) + … +cn(t)xn(t), where {x,(t), …, xn(t)} is any set of linearly independent solutions of the corresponding linear differential equation Dn(t, x)=0, and ci(t)(i= 1, 2, …, n) are suitable functions of the class C(R+, R) having finite limit as t →∞.