On a class of second order quasilinear ordinary differential equations
On a class of second order quasilinear ordinary differential equations
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关于一类二阶拟线性常微分方程
DOI：
10.32917/hmj/1206127714
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发表时间：
1995
影响因子：
0.2
通讯作者：
T. Kusano
中科院分区：
文献类型：
--
作者：
Motohiko Kitano;T. Kusano
(B) y(to) = y<»y'(to) = yi；(a)在区间/ c [α， oo]上的解是指函数yeC^)，它具有l/ leC ^)的性质，并且在所有点tel上满足方程。如果解存在于整个区间[α， oo]上，则称其为全局解。我们将特别证明初值问题(A)-(B)对于任意给定的y0和y±，只要q(i)是正的，并且在[α， oo]上局部有界变分，具有唯一的全局解。其次，我们研究了(A)在无穷邻域中的解的振荡（和非振荡）性质。如果这样的解在无穷远处有一个无穷序列的零聚类，我们就说它是振荡的；否则我们说它是非振荡的。因此，非振荡解必须最终为正或最终为负。(A)型方程的振荡理论最早是由Mirzov[10-13]和Elbert[3,4]提出的。最近的论文对他们的理论作了大量补充[2,5 -8]。由此可见，(A)的振荡特性在很大程度上与Emden-Fowler型方程的振荡特性相同
(B) y(to) = y<» y'(t0) = yι. By a solution of (A) on an interval / c [α, oo) we mean a function yeC^) which has the property l/ leC 1 ^) and satisfies the equation at all points tel. A solution is said to be global if it exists on the whole interval [α, oo). It will be shown in particular that the initial value problem (A)-(B) has a unique global solution for any given values of y0 and y± provided q(i) is positive and locally of bounded variation on [α, oo). Secondly, we investigate the oscillatory (and nonoscillatory) behavior of solutions of (A) which are defined in a neighborhood of infinity. Such a solution is said to be oscillatory if it has an infinite sequence of zeros clustering at infinity; otherwise it is said to be nonoscillatory. Thus a nonoscillatory solution must be eventually positive or eventually negative. Oscillation theory of equations of the type (A) was first developed by Mirzov [10-13] and Elbert [3, 4]. A considerable amount of addition to their theory has been given in the recent papers [2, 5-8]. It has thus turned out that the oscillatory character of (A) is to a large extent in common with that of the Emden-Fowler type equation