The positivity of the differential operator with periodic conditions.

The second order differential operator Ax defined by the formula A x u = -a(x)u xx (x)+δu(x) , with domain D(A x ) = u(x):u(x),u′(x)u"(x)∈C,u(x) = u(x+2π), 0 2π u(x)dx = 0 is considered. Here, a(x) ≥ a 0 is the continuously differentiable function. δ is the sufficiently large number. The Green's function of the differential operator Ax is constructed. The estimates for the Green's function are obtained. The positivity of the operator Ax in the Banach space C[0,2π] is established. It is proved that for any α∈(0,1/2), the norms in spaces E α = E α (C[0,2π],A x ) and C º 2α [0,2π] are equivalent. The positivity of the operator Ax in the Ho¨lder spaces of C º 2α [0,2π], α∈(0,1/2) is proved. [ABSTRACT FROM AUTHOR]

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