On Some Integral Equations with Locally Finite Measures and $L^\infty $-Perturbations

Abstract

Let $g \in C(R),f \in L_{{\text{loc}}}^\infty (R^ + )$ and let $\mu $ be a real locally finite positive definite Borel measure on $R^ + $. We investigate a relation between the solution of the nonlinear scalar Volterra equation \[ x'(t) + \int_{[0,t]} {g(x(t - s))} d\mu (s) = f(t),\quad t \in R^ + ,\quad x(0) = x_0 ,\] and the solution of the linear equation with the same data \[ z'(t) + \int_{[0,t]} {z(t - s)} d\mu (s) = f(t),\quad t \in R^ + ,\quad z(0) = x_0 .\] This relation, when combined with results (established in this paper) on the set of bounded solutions of certain limit equations \[ y(t) + \int_{R^ + } {g(y(t - s))} a(s)ds = 0,\quad t \in R,\] allows us to obtain new asymptotic results for $x(t)$ in the case when both $\mu $ and f are large in a precise sense.