Sufficient conditions for the existence of positive periodic solutions of a class of neutral delay models.
Here, the authors discuss the following periodic Lotka-Volterra-type neutral delay model of single-species population growth \[ N^{\prime}(t)=N(t) \Biggl( a(t)-\beta(t)N(t)-\sum_{i=1}^{n}b_i(t)N(t-\tau_i(t))- \sum_{i=1}^{n}c_i(t)N^{\prime}(t-\gamma_i(t))\Biggr), \] where \(a(t), \beta(t), b_i(t), c_i(t), \tau_i(t)\), and \(\gamma_i(t)\) are nonnegative continuous \(T\)-periodic functions, \(i=1,\cdots,n\). By applying the continuation theorem in \textit{R. E. Gaines} and \textit{J. L. Mawhin} [Coincidence degree, and nonlinear differential equations. Lecture Notes in Mathematics. 568. Berlin etc.: Springer-Verlag (1977; Zbl 0399.47031)], sufficient conditions are derived for the existence of positive periodic solutions to (1). The results of this paper give an affirmative answer to the open problem 9.1 due to \textit{Y. Kuang} [Delay differential equations: with applications in population dynamics. Mathematics in Science and Engineering. 191. Boston, MA: Academic Press Inc. (1993; Zbl 0777.34002)]. Some existing results due to \textit{K. Gopalsamy}, \textit{X. He} and \textit{L. Wen} [Glasg. Math. J. 33, 281--286 (1991; Zbl 0737.34050)], \textit{H. Fang} and \textit{J. Li} [J. Math. Anal. Appl. 259, 8--17 (2001; Zbl 0995.34073)] and \textit{Li Yong Kun} [J. Math. Anal. Appl. 214, 11--21 (1997; Zbl 0894.34075)] are extended and improved.
scientific article; zbMATH DE number 1957142