Almost automorphic solutions to abstract Volterra equations on the line (Q540265)

The authors prove a number of results about the existence of a unique almost automorphic solution of the nonlinear Volterra integral equation \[ u(t) = \int_{-\infty}^t a(t-s)[Au(s) + f(s,u(s))]ds. \] Here the values of \(x\) lie in a Banach space \(X\), \(a\) is a scalar kernel in \(L^1(0,\infty)\), \(A\) is an operator in \(X\), and the equation is supposed to have an integrable resolvent family, i.e., an integrable family \(S(\cdot)\) of bounded linear operators which commute with \(A\) and satisfy \[ S(t) x = a(t) x + \int_0^t a(t-s) AS(s) x \, ds, \qquad x \in D(A), \;t \geq 0. \] The function \(f\) is almost automorphic (i.e., almost periodic in a certain generalized sense) with respect to its first argument and globally Lipshitz continuous with respect to its second argument, and the proofs are based on the Banach fixed point theorem.