某些特定非线性时滞微分方程的问题

Results of Certain Types of Nonlinear Delay Differential Equation

我们证明了若如下具有有理系数a(z),a_(i)(z),b_(j)(z)的时滞微分方程[w(z+1)w(z)-1][w(z)w(z-1)-1]+a(z)(w’(z))/(w(z))=(∑^(p)_(i=0)a_(i)(z)w^(i))/(∑^(q)_(j=0)b_(j)(z)w^(j))存在有限多个极点的超越亚纯函数解w且其超级小于1,则方程退化为一类形式更为简单的方程,改进了Liu和Song的结论.进一步,我们也研究了一类Tumura-Clunie型的时滞微分方程,并得到了其超越亚纯解的一些性质.

We show that if the following delay differential equation[w(z+1)w(z)-1][w(z)w(z-1)-1]+a(z)(w’(z))/(w(z))=(∑^(p)_(i=0)a_(i)(z)w^(i))/(∑^(q)_(j=0)b_(j)(z)w^(j))with rational coefficients a(z),a_(i)(z),b_(j)(z),admits a transcendental meromorphic solution w of finite many poles with hyper-order less than one,then it reduces into a more simple delay differential equation,which improves some known theorems obtained most recently by Liu and Song.Moreover,we also study the delay differential equations of Tumura-Clunie type and obtain some quantitative properties of transcendental meromorphic solutions.