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Combine the damped free-transport evolution and the linear Boltzmann collision operator in a time-ordered Bochner integral. Its norm obeys $\Vert \tau f(t){\Vert }_2\le C{\int }_0^t\Vert f(s){\Vert }_{2}ds$. This reduces the characteristic integral equation to $(I-\tau )f=U_a(t,0)f_0$.

A linear kinetic transport equation with velocity-integral gain $K_t$ and nonnegative loss coefficient $a$. Its characteristic integral formulation combines the damped free-transport evolution with the Boltzmann Volterra operator. A uniform square-integrable velocity kernel yields bounded gain on $L_{x,v}^2$ and a globally convergent Volterra series for the linear Boltzmann equation on every finite time interval.

The damped free-transport evolution is $S(t)h(x,v)=e^{-t}h(x-tv,v)$. Substituting it into the Duhamel principle givesThe spatial shifts all follow the same backward characteristic curve ending at $(x,v)$ at time $t$; the velocity is constant.

Let $h_0=f_{1,0}-f_{2,0}\ge 0$. The damped free-transport evolution preserves nonnegativity, and a nonnegative kernel makes $\tau$ preserve it as well. Every term in ${\sum }_{n\ge 0}{\tau }^{n}F(h_0)$ is therefore nonnegative. By linearity, the difference $f_1-f_2$ solves the fixed point equation with initial value $h_0$. The assumed uniqueness identifies it with this nonnegative sum on every finite interval. Hence the order preservation for the linear Boltzmann equation givesContinuity upgrades pointwise limits in the series to the stated continuous solution; no pointwise differentiability or minimum principle is needed.

Free transport equation is the measure-preserving shear $(x,v)\mapsto (x-vs,v)$. Its composition operator preserves the $L_{x,v}^2$ norm. Since $a\ge 0$, the damping multiplier has modulus at most one:Consequently the damped free-transport evolution is contractive:This needs no upper bound on $a$. Nonnegative integrals may be interpreted in the extended sense with $e^{-\infty }=0$; local integrability along characteristics additionally gives the usual continuous initial trace.

Denote the damped free-transport evolution from time $s$ to time $t$ byBy the argument in (b), $\Vert U_a(t,s)g{\Vert }_2\le \Vert g{\Vert }_2$. The integral operator is the Boltzmann Volterra operatorThe Minkowski integral inequality and (a) give the useful stronger pointwise boundTherefore the requested estimate isFor strongly measurable bounded $L^2$-valued $f$, these are Bochner integrals; their finite norm bounds establish existence of the integrals.