一类三维微分系统非振动解的一种渐近性

The Asymptotic Behavior of the Set of Nonoscillatory Solutions of Three-Dimensional Differential Systems

研究三维微分系统:{u′1=a1(t)|u2|λ1sgnu2u′2=a2(t)|u3|λ2sgnu3u′3=-a3(t)|u1|λ3sgnu1(S)假设λi(i=1,2,3)是正的常数,ai(t)(i=1,2,3)在区间[0,∞)上是正的连续函数,根据u的分量ui(ui(t)>0(i=1,3),u2(t)<0))的特殊渐近条件,应用Schauder-Tyehnoff不动点定理给出了一种特殊类型非振动解存在的充分必要条件。系统满足条件∫∞0ai(t)dt=∞,i=1,2。

The three-dimensional differential system{u′1=aq（t）｜u2｜λ1sgnu2 u′2=a2（t）｜u3｜λ2sgnu3 u′3=-a3（t）｜u1｜λ3sgnu1is considered under the assumptions that λi （i= 1,2,3 ） are positive constants andai （t） （i= 1,2,3） are positive continuous functions on [0, ∞）, one class of nonoscillatory solutions u of （S）（ui （t）〉0 （i = 1, 3）, u2 （t）〈0） having specific asymptotic growths as t→∞ are established, and applicate of Schauder-Tyehnoff fixed point Theorem proving in the nessary and suffcient integral conditions for the existence of eventually positive solution of（S） having specific asymptotic growths as t→∞. The system satisfies ∫0^∞=ai（t）dt=∞,i=1,2.