Continuous Parameter Dependence in a Class of Volterra Equations

Abstract

Conditions are found under which the solution of the Volterra integral equation $x'(t) + \int_0^t {a(t - s,\lambda )x(s)ds} = k,\quad x(0) = x_0 ,$ is continuous in $\lambda $, uniformly in $\{ 0 \leqq t < \infty \} $, when $a(t,\lambda )$ is nonnegative, nonincreasing, and convex as a function of t, for each $\lambda $. The main theorem concerns the case where the kernel has a special piecewise linear form and solutions are asymptotic $(t \to \infty )$ to nondegenerate periodic functions. This is the case excluded in similar earlier results of the author.The significance of these results for certain related Volterra equations in Hilbert space is summarized.