Oscillatory and asymptotic behaviours of first order functional differential equations (Q1064454)

The author discusses the following first order functional differential equations: (1) \(x'(t)+\int^{b}_{a}p(t,\xi)x[g(t,\xi)]d\sigma (\xi)=0,\) (2) \(x'(t)+\int^{b}_{a}f(t,\xi,x[g(t,\xi)])d\sigma (\xi)=0.\) Some sufficient conditions of oscillation and nonoscillation are obtained, and two asymptotic properties and their criteria are given. These criteria can be used in the following equations: (3) \(x'(t)+\sum^{n}_{i=1}p_ i(t)x[g_ i(t)]=0,\) (4) \(x'(t)+\sum^{n}_{i=1}f_ i(t,x[g_ i(t)]=0.\)