= Solution
On each finite interval use the <Banach space> $X_T=C([0,T];E)$ with the supremum <norm>. The free term $g(t)=U_tf_0$ belongs to $X_T$, and the time-<integral> operator $\tau$ maps $X_T$ to itself with <norm> at most $2T$. The <strongly continuous semigroup> property and boundedness of $B$ justify continuity of the <Bochner integral>.
Define
$$
\boxed{f=\sum_{j=0}^\infty\tau^jg.}
$$
The <factorial bound for a Volterra iterate> gives $\|\tau^jg\|_{X_T}\leq(2T)^j\|f_0\|_1/j!$. The series therefore converges absolutely in $X_T$. Since $\tau$ is a bounded <linear operator> on $X_T$, it can be passed through the convergent sum, giving
$$
\tau f=\sum_{j=1}^\infty\tau^jg=f-g.
$$
Thus $f=g+\tau f$, the required <integral> formulation, and $f(0)=f_0$. Each term on a larger interval restricts to the identical term on a smaller interval, so these constructions define a single global solution without having to restart at successive times. This is an <integrable Volterra solution for normalized velocity relaxation>. It is a <mild solution of an abstract Cauchy problem> in $L^1$ and hence an $L^1$ weak solution in the paper's <integral>-formulation sense. No smallness condition such as $2T<1$ is needed.
Back to article page
