On the Asymptotic Behavior of Volterra Integral Equations
James L. Kaplan · SIAM Journal on Mathematical Analysis · 1972
Suppose $y(t) = f(t) - \int_0^t {a(t,s)y(s)ds} $ is a system of Volterra integral equations, and let $r(t,s)$ be the resolvent kernel corresponding to this system. If $f(t)$ is continuous and $\omega $-periodic, it is shown that under suitable restrictions on $r(t,s)$, the solution $y(t)$ is asymptotically $\omega $-periodic. These conditions generalize a previous result of Miller, Nohel and Wong. For the perturbed system $x(t) = f(t) - \int_0^t {a(t,s)} \{ x(s) + g(s,x(s))\} ds$, if the resolvent kernel is “sufficiently close” to an $L^1$-function, then $| {x(t) - y(t)} | \to 0$ as $t \to \infty $ for a suitable class of perturbation terms $g(t,x)$. If the resolvent is of convolution type, this generalizes a theorem of A. Strauss. Finally, it is shown that if the resolvent kernel is of convolution type, and is in $L^1 [0,\infty )$, then the Cesaro integral mean of $| {x(t) - y(t)} |$ converges to zero, for perturbations which are bounded and diminishing.