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

In this study, the second order differential operator Ax defined by the formula Axu = -uxx(x) + δu(x), δ = 0, with domain D(Ax) = {u(x) : u(x), u'(x), u''(x) ∈ C(R¹), u(x) = u(x + 2π), x ∈ R¹, ⊏02πu(x) dx = 0} is considered. The Green function of the differential operator Ax is constructed. The estimates for the Green function are obtained. It is proved that for any α ∈ (0, ½), the norms in the spaces Eα = Eα(°C(R¹), Ax) and °C2α(R¹) are equivalent. The positivity of the operator Ax in Hölder spaces of °C2α(R¹), α ∈ (0, ½), is proved. In the applications, theorems on well-posedness of local and nonlocal boundary value problems for elliptic equations in the Hölder spaces are obtained. [ABSTRACT FROM AUTHOR]