On the asymptotic behavior for a certain nonlinear evolution equation (Q760893)

Let A be an accretive operator in a Banach space X which satisfies the range condition. Given \(x\in X\) and \(g:[0,\infty)\to [0,\infty)\) a nonincreasing continuous function such that g(t)\(\to 0\) as \(t\to \infty\), consider the initial value problem: \[ (E)\quad du(t)/dt+Au(t)+g(t)u(t)\ni g(t)x,\quad t\geq 0,\quad u(0)=x_ 0\in \overline{D(A)}. \] The asymptotic behavior of the (generalized) solution of (E) both at infinity and at the origin is studied by using the idea of Kohlberg and Neyman. Our results yield improvements of some results of Reich, Reich and Israel, and Miyadera. We also give a unified proof for these results.