Positive age-renewal evolution and L1 data stability

For nonnegative initial data, sums and nonnegative scalar multiples of the solutions in [F2] satisfy the corresponding characteristic formulas and renewal equations in [F1]; uniqueness in [F2] therefore makes $S(t)$ additive and nonnegatively homogeneous on $L_{+}^1$. For arbitrary real $f\in L^1$, put $f^{+}=\max \{f,0\}$, $f^{-}=\max \{-f,0\}$ and define $S(t)f=S(t)f^{+}-S(t)f^{-}$. If also $f=p-q$ with $p,q\ge 0$, then $p+f^{-}=q+f^{+}$, so cone additivity gives $S(t)p-S(t)q=S(t)f^{+}-S(t)f^{-}$. This independence of decomposition gives additivity on $L^1$, and nonnegative homogeneity together with $S(t)(-f)=-S(t)f$ gives real linearity. The extension obeys the same characteristic formulas and renewal relation by subtraction; it agrees with [F2] on $L_{+}^1$ and is positive there.

Integrating the two characteristic branches in [F2] gives $N(t)={\int }_0^{\mathrm{\infty }}{n}_0(x)S(x,t)dx+{\int }_0^{t}B(t-a)S(0,a)da$. Bounded $\mu$ and continuous $B$ permit differentiation under these integrals; using ${\mathrm{\partial }}_{t}S(x,t)=-\mu (x+t)S(x,t)$ and ${\mathrm{\partial }}_{a}S(0,a)=-\mu (a)S(0,a)$ yields $N^{\mathrm{\prime }}(t)=B(t)-{\int }_0^{\mathrm{\infty }}\mu (a)n(t,a)da$ for almost every $t$. Since $0\le B(t)\le {\beta }_{\mathrm{\infty }}N(t)$ and $\mu n\ge 0$, we have $N^{\mathrm{\prime }}(t)\le {\beta }_{\mathrm{\infty }}N(t)$.

Integrating the differential inequality in step 1.2 gives $N(t)\le N(0)+{\int }_0^t{\beta }_{\mathrm{\infty }}N(s)ds$. Apply [F3] with $a(t)=N(0)$ and $b={\beta }_{\mathrm{\infty }}$ to obtain the asserted bound for nonnegative data. For arbitrary real $f\in L^1$, positivity and linearity from step 1.1 give $\mathrm{\mid }S(t)f\mathrm{\mid }\le S(t)f^{+}+S(t)f^{-}=S(t)\mathrm{\mid }f\mathrm{\mid }$ almost everywhere. Integrating and applying the nonnegative-data bound proves $\mathrm{\parallel }S(t)f{\mathrm{\parallel }}_1\le e^{{\beta }_{\mathrm{\infty }}t}\mathrm{\parallel }f{\mathrm{\parallel }}_1$.

Subtract the two transport equations and write $w=n-\tilde{n}$. Along characteristics, its interior forcing is $(\tilde{\mu }-\mu )\tilde{n}$ and its inflow forcing is $\int \beta w+\int (\beta -\tilde{\beta })\tilde{n}$. Taking absolute values in the characteristic integral formula, integrating over age, and using nonnegative survival factors gives $E(t)\le E(0)+{\int }_0^t[{\beta }_{\mathrm{\infty }}E(s)+\delta \tilde{N}(s)]ds$, where $E=\mathrm{\parallel }w{\mathrm{\parallel }}_1$ and $\tilde{N}=\mathrm{\parallel }\tilde{n}{\mathrm{\parallel }}_1$.

By step 2.1 applied to $\tilde{n}$, $\tilde{N}(s)\le e^{{\tilde{\beta }}_{\mathrm{\infty }}s}\mathrm{\parallel }{\tilde{n}}_0{\mathrm{\parallel }}_1$. Thus the nondecreasing function $a(t)=E(0)+\delta t{e}^{{\tilde{\beta }}_{\mathrm{\infty }}t}\mathrm{\parallel }{\tilde{n}}_0{\mathrm{\parallel }}_1$ bounds the forcing integral in step 2.2, and [F3] yields the claimed coefficient-and-initial-data stability estimate.