zbMATH — the first resource for mathematics

New theorems for the renewal integral equation. (English. Russian original) Zbl 0894.45003
J. Contemp. Math. Anal., Armen. Acad. Sci. 32, No. 1, 2-16 (1997); translation from Izv. Nats. Akad. Nauk Armen., Mat. 32, No. 1, 5-20 (1997).
The authors show that if the kernel \(v\) is nonnegative, integrable, and satisfies \(\int_0^\infty v(x) dx= 1\), then the renewal resolvent \(\Phi\) that is the solution to the equation \[ \Phi(x)= v(x)+ \int_0^x v(x-t)\Phi(t) dt, \] can be written in the form \[ \Phi(x)= \left (\int_0^\infty xv(x) dx\right)^{-1} +\Phi_1(x) +\Phi_2(x), \] where \(\Phi_1\) is continuous with \(\lim_{x\to\infty} \Phi_1(x)= 0\) and \(\Phi_2\in L^1\). From this follows that if \(g\) is integrable, bounded, and tends to zero, then the solution \(\varphi\) of the renewal equation \[ \varphi(x)= g(x)+ \int_0^x v(x-t)g(t) dt, \] satisfies \(\lim_{x\to\infty} \varphi(x)= \int_0^\infty g(x) dx /\int_0^\infty xv(x) dx\).

45D05 Volterra integral equations
45M05 Asymptotics of solutions to integral equations