Necessary and sufficient conditions for the existence of periodic solutions of Riccati's equation with periodic coefficients

In this paper, the concept of generalizedω-periodic solution is given for Riccati's equationy'=a(t)y2+b(t)y+c(t) with perriodic coefficients, the relation between generalizedω-periodic solutions and the characteristic numbers of systemx'1=c(t)x2,x'2=−a(t)x1-b(t)x2 is indicated, and several necessary and sufficient conditions are given using the coefficients. Moreover, in the case ofa(t) without zero, the relation between the number of continuousω-periodic solutions ofy'=a(t)y2+b(t)y+c(t)+δ and the parameter δ is given; thus the problem on the existence of continuousω-periodie solutions is basically solved.