Well-posedness of the modified Crank-Nicholson difference schemes in Bochner spaces

The nonlocal boundary value problem $v$’$(t)+Av(t)=f(t)(0\leq t\leq 1),v(0)=v(\lambda )+\mu ,0<\lambda \leq 1$ for differential equations in an arbitrary Banach space $E$ with the strongly positive operator $A$ is considered. The well-posedness of the modified Crank-Nicholson difference schemes of the second order of accuracy for the approximate solutions of this problem in Bochner spaces is established. In applications, the almost coercive stability and the coercive stability estimates for the solutions of difference schemes for the approximate solutions of the nonlocal boundary value problem for parabolic equation are obtained.