MR3308704 Jaroslav JAROŠ and KUSANO Takaŝi Jaroslav JAROŠ and KUSANO Takaŝi Funkcialaj Ekvacioj. Serio Internacia 57 2014 467--475 http://fe.math.kobe-u.ac.jp/FE/FullPapers/57-3/57_467.pdf http://www.ams.org/mathscinet-getitem?mr=MR3308704 It is shown that cyclic differential systems of the forms (A) $x_i'=-p_i(t)x_{i+1}^{\alpha_i}$, $i=1,\ldots, n$ $(x_{n+1}=x_1)$, and (B) $x_i'=-p_i(t)x_{i+1}^{-\alpha_i}$, $i=1,\ldots, n$ $(x_{n+1} = x_1)$, where $\alpha_i>0$, $i=1,\ldots, n$, are constants and $p_i(t)>0$, $i=1,\ldots, n$, are continuous functions on $[0,\infty)$ may possess singular solutions of extinct type, that is, those positive solutions $(x_1(t),\ldots,x_n(t))$ of (A) (resp. (B)) which are defined on some finite interval $[t_0,T)$, $0\leq t_0<T<\infty$, and satisfy $x_i(t)>0$, $t\in[t_0,T)$, and $\lim_{t\to T-0}x_i(t)=0$, $i=1,\ldots, n$. Cyclic systems of differential equations, Extinct solutions. 34C11. 57-467 2014 Jaroslav JAROŠ and KUSANO Takaŝi Jaroslav JAROŠ and KUSANO Takaŝi

1 Cecchi, M.; Došlá, Z.; Marini, M. Georgian Math. J. 7 2000 269-282 MR1779551

2 Chanturija, T. A. Colloq. Math. Soc. János Bolyai, Vol. 15, North-Holland, Amsterdam 1977 pp. 107-119 MR0591720

3 Jaroš, J.; Kusano, T. Fukuoka Univ. Sci. Rep. 29 1999 7-12 MR1696364

4 Jaroš, J.; Kusano, T. Funkcial. Ekvac. 43 2000 491-509 MR1815473 1815473 5 Kiguradze, I. T.; Chanturia, T. A. Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations Mathematics and its Applications (Soviet Series), 89, Kluwer Academic Publishers Group, Dordrecht 1993 MR1220223

6 Kusano, T.; Naito, M. Fukuoka Univ. Sci. Rep. 29 1999 243-252 MR1718808

7 Kusano, T.; Naito, M. J. Inequal. Appl. 5 2000 487-496 MR1800976

8 Mirzov, J. D. Asymptotic properties of solutions of the systems of nonlinear differential equations Maikop, Adygeja 1993 (in Russian)