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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 $$\lim_{x\to\infty} \varphi(x)= \int_0^\infty g(x) dx /\int_0^\infty xv(x) dx$$.

##### MSC:
 45D05 Volterra integral equations 45M05 Asymptotics of solutions to integral equations
##### Keywords:
renewal equation; resolvent; asymptotic behaviour