A note on fractional spaces generated by the positive operator with periodic conditions and applications

In this study, the second order differential operator A(x) defined by the formula A(x)u = -u(xx)(x) + delta u(x), delta >= 0, with domain D(A(x)) = {u(x) : u(x), u'(x), u ''(x) is an element of C( R-1), u(x) = u(x + 2 pi), x is an element of R-1, integral(2 pi)(0) u(x) dx = 0} is considered. The Green function of the differential operator A(x) is constructed. The estimates for the Green function are obtained. It is proved that for any alpha is an element of (0, 1/2), the norms in the spaces E-alpha = E-alpha (C degrees(R-1), A(x)) and C-2 alpha (R-1) are equivalent. The positivity of the operator A(x) in Hlder spaces of C degrees(2 alpha) (R-1), alpha is an element of (0, 1/2), is proved. In the applications, theorems on well-posedness of local and nonlocal boundary value problems for elliptic equations in the Hlder spaces are obtained.