On quasilinear Volterra integrodifferential equation in a Banach space (Q1821294)

The authors study the existence, uniqueness, continuation, boundedness, asymptotic behaviour and the growth of solutions of the equation \[ du/dt+A(t,u(t))u(t)=g(t,u(t),\int^{t}_{0}h(t,s,u(s))ds),\quad t\geq 0, \] with initial condition \(u(0)=u_ 0\) in a Banach space B. The main tools are the Schauder fixed-point theorem and some integral inequalities established by the second author. The assumptions include for example the following conditions: \[ \| [A(t,A_ 0^{-\alpha}\nu -A(\tau,A_ 0^{-\alpha}\omega)]A^{-1}(s,A^{-1},\omega)\| \leq C(N)(| t- \tau |^{\sigma}+\| \nu -\omega \|^{\rho}), \] where \(A_ 0=A(0,u_ 0)\) is a closed operator with dense domain that satisfies \(\| (\lambda I-A_ 0)^{-1}\| \leq C/(1+| \lambda |)\) for all \({\mathfrak R}\lambda \leq 0\).