运用$k$-集压缩算子的拓扑度抽象连续定理, 研究了如下一类高阶微分方程的周期解存在性问题$x^{(m)}(t)=r(t)-a(t)x(t-\sigma )-b(t)x^{(m)}(t-\tau ).$
并将其应用于研究一类中立型对数种群模型$\frac{\mathrm{d}N}{\mathrm{d}t}=N(t)[r(t)-a(t)\ln N(t-\sigma )-b(t)\frac{\mathrm{d}}{\mathrm{d}t}\ln N(t-\tau )]$
的正周期解存在性问题. 得到了正周期解存在性的新结果, 而且估计先验界的方法也是新的.

Using an abstract continuous theorem of $k$-set contractive operator, the existence of periodic solutions for a kind of high-order differential equation as follows $$x^{(m)}(t)=r(t)-a(t)x(t-\sigma )-b(t)x^{(m)}(t-\tau )$$  is   studied. The result is applied to the following neutral delay logarithmic population model
$$\frac{\mathrm{d}N}{\mathrm{d}t}=N(t)[r(t)-a(t)\ln N(t-\sigma )-b(t)\frac{\mathrm{d}}{\mathrm{d}t}\ln N(t-\tau )].$$ Some new result on the existence of positive periodic solutions for the above model is obtained. And the  method to estimate a priori bounds  of periodic solutions is new.