Qualitative properties of solutions of Volterra equations in Banach spaces (Q1122100)

The authors study the qualitative behaviour of solutions of the abstract integral equation \[ u(t)+\int^{t}_{0}b(t-s)Au(s)ds\ni u_ 0+\int^{t}_{0}b(t-s)g(s)ds,\quad t\geq 0, \] in a real Banach space X. Here A is an m-accretive operator and b is a kernel of completely positive type. The authors give a thorough review of kernels of completely positive type, and establish a number of monotonicity and invariance properties of the solution. By using the method of upper and lower solutions they obtain results on the asymptotic behaviour.