High-accuracy Stable Difference Schemes for Well-posed NBVP

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The single step difference schemes of the high order of accuracy for the approximate solution of the nonlocal boundary value problem (NBVP) $$ v'\left( t \right) + Av\left( t \right) = f\left( t \right)\left( {0 \leqslant t \leqslant 1} \right),v\left( 0 \right) = v\left( \lambda \right) + \mu , 0 < \lambda \leqslant 1 $$ for the differential equation in an arbitrary Banach space E with the strongly positive operator A are presented. The construction of these difference schemes is based on the Padé difference schemes for the solutions of the initial-value problem for the abstract parabolic equation and the high order approximation formula for \( v\left( 0 \right) = v\left( \lambda \right) + \mu \) . The stability, the almost coercive stability and coercive stability of these difference schemes are established.