方程(dx/dt)=x^n+sum from i=0 to (n-1) a_i(t)x^2的闭解的个数及存在的条件  

THE CONDITIONS OF EXISTENCE AND THE NUMBER FOR THE CLOSED SOLUTIONS OF THE EQUATION ■

作　　者：李今明[1] 蔡燧林[1] 

机构地区：[1]浙江大学应用数学系,杭州310027

出　　处：《应用数学学报》1997年第1期86-92,共7页Acta Mathematicae Applicatae Sinica

基　　金：国家自然科学基金

摘　　要：本文给出方程n＝3时分别正好存在1个闭解，3个闭解，以及至少存在2个闭解的充分条件，并研究了这些闭解的稳定性．当n＝4且ai（t）（i＝0，1，2，3）为t的P次多项式时，文[1]曾猜想其时闭解重数的上界为max｛4，p＋3｝．本文举例指出，即使p＝3，闭解重数的上界也可以大于7．In this paper, we obtain the sufficient conditions for the equationhaving exactly one closed solution, three closed solutions and at most two closed solutionsrespectively where ai(t) (i = 0, 1,''', n - 1) is continuous on [O,1]. The stability of thoseclosed solutions are also considered. The paper [1] conjecture that the super bound of themultiplicity of closed solution is max {4,p + 3} when n = 4 and ai(t) (i = 0, 1, 2, 3) is thepolynomial of defree p of t. We point out that, the super bound of the multiplicity is greaterthen 7 when p = 3. So the conjecture in [1] is wrong.

关 键 词：ABEL方程 闭解 RICCATI方程 稳定性 

分 类 号：O175.1[理学—数学] O231[理学—基础数学]