Coercive solvability of nonlocal boundary-value problems for parabolic equations

In a Banach space $E$ we consider nonlocal problem
\begin{align*} &v'(t)+A(t)v(t)=f(t)\quad(0\leq t\leq1),\\ &v(0)=v(\lambda)+\mu\quad(0<\lambda\leq1) \end{align*}
for abstract parabolic equation with linear unbounded strongly positive operator $A(t)$ with independent of $t$, everywhere dense in $E$ domain $D=D(A(t))$. This operator generates analytic semigroup $\exp\{-sA(t)\}$ ($s\geq0$).
We prove the coercive solvability of the problem in the Banach space $C_0^{\alpha,\alpha}([0,1],E)$ $(0<\alpha<1)$ with the weight $(t+\tau)^\alpha$. This result was previously known only for a constant operator. We consider applications in the class of parabolic functional differential equations with transformation of spatial variables and in the class of parabolic equations with nonlocal conditions on the boundary of domain. Thus, this describes parabolic equations with nonlocal conditions both in time and in spatial variables.