The first and second orders of accuracy difference schemes for the approximate solutions of the nonlocal boundary value problem v'(t) + Av(t) = f(t) (0 ≤ t ≤ 1), v(0) = v(λ) + μ, 0 ＜ λ ≤ 1, for differential equation in an arbitrary Banach space E with the strongly positive operator A are considered. The well-posedness of these difference schemes in difference analogues of spaces of smooth functions is established. In applications, 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.
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