The author consider an abstract parabolic equation

${v}^{\text{'}}\left(t\right)+Av\left(t\right)=f\left(t\right)$ where the initial condition is replaced by the nonlocal condition

$v\left(0\right)=v\left(\lambda \right)+\mu $. All variables and constants takes values in a Hilbert space

$E$ and

$A$ is a linear and possible unbounded operator on this space. Under the assumption that the operator

$-A$ generates an analytic semigroup

${\{exp(-At)\}}_{t\ge 0}$ with exponential decay, it is shown that the solutions to the nonlocal parabolic equation satify a coercivity estimate in terms of

$f$ and

$\mu $ with the implication that the problem is well-posed. In addition, first and second order difference schemes are given and so called almost coercive inequalities are established for these (the multiplier in the inequality contains the factor

$min\{1/\tau ,|ln\parallel A{\parallel}_{E\to E}\left|\right\}$, where

$\tau $ is the time step).