Well-posedness of the difference schemes for elliptic equations in spaces

The second order of accuracy difference scheme for the approximate solutions of the nonlocal boundary-value problem − v ″ ( t ) + A v ( t ) = f ( t ) ( 0 ≤ t ≤ 1 ) , v ( 0 ) = v ( 1 ) , v ′ ( 0 ) = v ′ ( 1 ) for differential equations in an arbitrary Banach space E with a strongly positive operator A is considered. The well-posedness of this difference scheme in C τ β , γ ( E ) spaces is established. In applications, a series of coercivity inequalities in difference analogues of various Hölder norms for the solutions of difference schemes of the second order of accuracy over one variable for the approximate solutions of the nonlocal boundary value problem for elliptic equations are obtained.