关于某类具有复什偏差变元的泛函微分方程

本文对方程F(t,x(t),(t),x(f(t,x(t),(t))))=0提出“徽分特征映射”的概念。应用这一概念可以把它化为G (t,(t),x(t),x(x(t)))=0型的方程,同时给出一种求解这类方程的途径。从而完整地解答了1806年Poisson几何问题中提出的一类泛函微分方程,得出参数平面的区域划分与相应的解族。

ON A FUNCTIONAL DIFFERENTIAL EQUATIONS WITH COMPLICATED DEVIATING ARGUMENTS

Since latter twenty years, the theory of FDE had developped essentially and systematically whether in the application or in the theory. Nevertheless, so far there were no methods for finding solutions of some special type FDE induced earlist from geometric problems during latter two centuries. Generally, there is no method to know that which curves can, satisfy this differential equation in which the argument depends on its solution, even depends on derivative of its solutions,such as Euler's problem(1750), poisson's problem (1806)〔1〕Babbege's problem (1816),and Barba's problem (1930)〔2〕etc. Except that one special solution can be guessed for some specific equation.this problem was not investigated efficientlly all the time. In other respect, applied background of this class of equations has over- stepped the bounds of classical geometry with late total, development in the field of FDE, and its importance appears again. The paper proposed the conception of differential chiaracteristic map for equation F(t,X(t),X(t), X(f(t,X(t)X(t)))) =0 using the concep- tion, we can transform it into the form as G(t,X(t), X(t),X (X(t),) ) =0 and in detail discuss one special case---the FDE in poisson's prpb- lem,i.e. epuation X~2(t)+X~2(t) X~2(t)+bX~2(t+X(t)X(t))=a, where a, b∈R. some families of solution are given.