Estimation of the spectral radius of an operator and the solvability of inverse problems for evolution equations

In a Banach space $E$ with reproducing cone $E_+$ consider the operator $B$ defined by the formula $Bf=l(uu_t)$, where $u(t)$ is a solution of the Cauchy problem $u_t-Au=\varPhi (t)f$, $t\in [0,T]$, $u(0)=0$, and the expression $l(u)$ has one of the following forms: $l(u)=u(t_1)$, $0<t_1\leqslant T_s$, or $l(u)=\int _0^T\nu (\tau)u(\tau )\,d\tau$ with $\nu\in L_1(0,T)$, $\nu\geqslant0$ on $[0,T]$. We prove the estimate $r(B)<1$.
We obtain this estimate under the conditions that the $C_0$-semigroup generated by the operator $A$ is positive, compact, and of negative exponential type, and the operator function $\varPhi\in C^1([0,T];\mathscr L (E))$ is such that $l(\varPhi)=I$ and $\varPhi(t)\geqslant0$, $\varPhi'(t)\geqslant0$ on $[0,t]$. Correct solvability of the corresponding inverse problem follows from this estimate.