一类二维微分差分方程具有多个周期的周期解的一个条件

Existence of Manyperiodic Solutions to the Two-dimensional Differential Difference Equations

给出了二维微分差分方程(E)具有周期为4/2n+1,4/2n-1, 4/2n-3,…,4/7,4/5,4/3,4的周期解的一个条件,并在定理的证明过程中给出了如何求出其相应周期解的方法.

Using for the most pare geometric methods,many periodic solutions to the two-dimensional differential difference equations are as followsdxdt=f(x(t-1),y(t-1)), dydt=g(x,y,x(t-1),y(t-1)).(E)Authors obtained main results as follows: Let f:R 2R and g:R 4R be real variable function. If there exists a function y=y(x),x∈R,definition the invariane curve of equation (E) by y=y(x).Use y(x),y(x(t-1)) for y,y(t-1) in equation (E).Then equation (E) can be turn into as following from:dxdt=f(x(t-1),y(x(t-1))=defF(x-(t-1)), dydt=f(x,y(x),x(t-1),y(x(t-1)).(E 0)Next let a l be real numbers and l>0.Let F:RR be the l-Periodical bounded function: (ⅰ) F(a)=0, F(x±l)=-F(x),x∈R; (ⅱ) F is uniformly continuons in (a,a+l) and F(y)=F(x)>0,y=2a+l-x,x∈(a,a+l); (ⅲ) 0<d=∫ a+l adxF(x)≤12n+1(n=0,1,2,...). Then the equation (E) has 42n+1,42n-1,...,47,45,43,4-periodic solutions.