The positivity of the second order difference operator with periodic conditions in H?lder spaces and its applications

摘要

In this study, the second order of approximation of the difference operator A_h~x approximates the second order differential operator A~x defined by the formula with domain is considered. The Green's function of the difference operator A_h~x is constructed. The estimates for the Green's function are obtained. The positivity of operator A_h~x in the Banach space C(R_(1h)) of periodic mesh functions defined on R_(1h) is established. Here, R_(1h) = {x_k = kh, k = 0,?1,?2,...}. It is proved that for any α ∈ (0, 1/2), the norms in spaces Eα = Eα (C(R_(1h)),A_h~x) and C~(2α) (R_(1h)) are equivalent uniformly with respect to h. The positivity of the operator A_h~x in H?lder spaces of °C ~(2α) (R_(1h)),α ∈ (0, 1/2) is proved.We discuss its applications to theory of difference schemes for the approximate solution of local and nonlocal boundary value problems for elliptic differential equations.