Asymptotically exponential solutions in nonlinear integral and differential equations (Q994295)

The authors investigate the asymptotic behaviour of solutions of the nonlinear Volterra abstract integral equation of the form \[ x(t)= y(t;\varphi)+ \int^t_{t_0} T(t- s)f(s,x(\cdot))\,ds,\quad t\geq 0,\tag{1} \] under the assumption that an asymptotic formula for \(y(t;\varphi)\) is known. Moreover, it is assumed that the operator \(f\) involved in (1) is of Volterra type and \(T(t)\) is a strongly continuous semigroup of bounded operators in a Banach space \(X\). Equation (1) is considered together with the initial condition \[ x(s)= \varphi(s)\quad\text{for }s\in [t_{-1},t_0].\tag{2} \] In the main result of the paper there are given sufficient conditions implying that the asymptotic behaviour of the ``linear part'' of \(y(t;\varphi)\) is preserved for solutions of the problem (1)--(2).