| Abstract |
In this study, the second order of approximation of difference operator A_h^x approximates the second order differential operator A^x defined by the formula
A^x=-u_xx (x)+δu(x),δ>0
with domain
D(A^x)={u(x):u(x),u'(x),u''(x)∈C(R^1), u(x)=u(x+2π),x∈R^1,∫_0^2π (u(x))dx=0}
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 E_α=E_α (C(R_1h ),A_h^x) and C^2α (R_1h ) are equivalent. The positivity of the operator A_h^x in the Hölder spaces of C^2α (R_1h ), α∈(0,1/2) is proved.
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