Comparison theorems for first order retarded functional differential equations and their applications (Q810729)

Let \(C=\{\phi: \phi: (-\infty,0]\to R\) is continuous, \(\lim_{s\to - \infty}\phi(s)\) exists and is finite\}. The author establishes some comparison theorems for the retarded functional differential inequalities \(x(t)(x'(t)+f(t,x_ t))\leq 0,\quad t\geq t_ 0,\) and \(x(t)(x'(t)+f(t,x_ t))\geq 0,\quad t\geq t_ 0,\) where \(f: [t_ 0,\infty)\times C\to R\) is continuous, \(x: (-\infty,A)\to R\) is continuous, \(t_ 0<A\leq \infty\), and \(x_ t(s)=x(t+s)\) for \(s\leq 0\). By using the comparison theorems obtained, the author also establishes some results about the asymptotic behavior and oscillation for the retarded functional differential equation \(x'(t)+f(t,x_ t)=0,\quad t\geq t_ 0.\)