The Asymptotic Behavior of a Volterra-Renewal Equation

Peter Ney
1977 Transactions of the American Mathematical Society  
Theorem. Assume that the functions x(-), h(-), G(-) satisfy: (i) 0 < x(t), t G [0, oo); x{t) -» 0 as t -» oo; x bounded, measurable; (ii) 0 < h(s); h(s) Lipschitz continuous for s £ /, where I is a closed interval containing the range of x; A(0) = 0, h'(0 +) = 1, A"(0 +) < 0; (iii) G a probability distribution on (0, oo) having nontrivial absolutely continuous component and finite second moment. Let Hx(t) = /"' h[x(t -y)]dG(y). //0 < (x -Hx)(t) = o(r2), with strict inequality on the left on a
more » ... on the left on a set of positive measure, then x(t) ~ y/t, t -+ oo, where y is a constant depending only on h and G. The condition o(f~2) is close to best possible, and cannot, e.g., be replaced by o(r2).
doi:10.2307/1998523 fatcat:bmgdv46jijhxrjgbwhub3t34bm