On coercive solvability of parabolic equations with variable operator

In a Banach space $E$, the Cauchy problem
$$v'(t)+A(t)v(t)=f(t)\quad (0\leq t\leq1),\qquad v(0)=v_0$$
is considered for a differential equation with linear strongly positive operator $A(t)$ such that its domain $D=D(A(t))$ is everywhere dense in $E$ independently off $t$ and $A(t)$ generates an analytic semigroup $\exp\{-sA(t)\}$ ($s\geq0$). Under some natural assumptions on $A(t)$, we establish coercive solvability of the Cauchy problem in the Banach space $C_0^{\beta,\gamma}(E)$. We prove a stronger estimate of the solution compared to estimates known earlier, using weaker restrictions on $f(t)$ and $v_0$.