Nonlocal boundary-value problems for abstract parabolic equations: well-posedness in Bochner spaces

The well-posedness of the nonlocal boundary-value problem for abstract parabolic differential equations in Bochner spaces is established. The first and second order of accuracy difference schemes for the approximate solutions of this problem are considered. The coercive inequalities for the solutions of these difference schemes are established. In applications, the almost coercive stability and coercive stability estimates for the solutions of difference schemes for the approximate solutions of the nonlocal boundary-value problem for parabolic equation are obtained.