The Positivity of Differential Operator with Nonlocal Boundary Conditions

We study a structure of fractional spaces E-alpha (L-p[0,1], A(x)) generated by the positive differential operator Ax defined by the formula A(x)u = - a(x)d(2)u/dx(2) + delta u with domain D(A(x)) = {u is an element of C-(2) [0,1] : u(0) = u(mu), u' (0) = u' (1), 1/2 <= mu <= 1}. Here, a(x) is a smooth function defined on the segment [0,1] and a(x) = a > 0, d > 0. It is established that for any 0 < alpha < 1/2, 1 <= p < infinity, the norms in the spaces E-alpha (L-p[0,1], A(x)) andW(p)(2 alpha) [0,1] are equivalent. The positivity of the differential operator Ax in W-p(2 alpha) [0,1], (0 <= alpha <= 1/2,1 <= p < infinity) is established. In applications, well- posedness of nonlocal boundary problems for elliptic equations is established.