Volterra and Functional Differential Equations

ON VOLTERRA EQUATIONS WITH LOCALLY FINITE MEASURES AND L^∞-PERTURBATIONS

Stig-Olof Londen

Mathematics Research Center University of Wisconsin-Madison Madison, Wisconsin and Institute of Mathematics Helsinki University of Technology Otaniemi, Finland

DOI: 10.1201/9781003420026-7

In this lecture we discuss the asymptotics of the scalar, real, nonlinear Volterra differential equation of convolution type

$$\begin{array}{ll}\hfill & x^{\prime }\left(t\right)+{{\displaystyle \int }}_{\left[0,t\right]}g\left(x\left(t-s\right)\right)du\left(s\right)=f\left(t\right),t\in R^{+}=\left[0,\infty \right)\hfill \\ \hfill & x\left(0\right)=x_0,\hfill \end{array}$$ (1)

Here g and f are given functions, μ is a given real Borel neasure on R⁺ and x (t) is the unknown solution. In particular, we concentrate on the case when both and f are large; that ...