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In the present paper the modified Crank-Nicholson difference schemes for the approximate solutions of the nonlocal boundary value problem v 0 (t) + Av(t) = f(t)(0 ≤ t ≤ 1), v(0) = v(λ) + µ, 0 < λ ≤ 1 for differential equations in an arbitrary Banach space E with the strongly positive operator A are considered. The well-posedness of these difference schemes in C β,γ τ (E) and Ce β,γ τ (E) spaces is established. In applications, the coercive stability estimates for the solutions of difference schemes of the second order of accuracy over time and of an arbitrary order of accuracy over space variables in the case of the nonlocal boundary value problem for the 2mth-order multidimensional parabolic equation are obtained

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