Fractional spaces generated by the positive differential and difference operators in a Banach space

The structure of the fractional spaces E α,q,(L q[0, 1], A x) generated by the positive differential operator A x defined by the formula A x u = −a(x) d 2 u/dx 2 + δu, with domain D(A x) = {u ∈ C (2)[0, 1] : u(0) = u(1), u′(0) = u′(1)} is investigated. It is established that for any 0 < α < 1/2 the norms in the spaces E α,q(L q[0, 1],A x) and W q 2α [0, 1] are equivalent. The positivity of the differential operator A x in W q 2α [0, 1](0 ≤ α < 1/2) is established. The discrete analogy of these results for the positive difference operator A h x a second order of approximation of the differential operator A x, defined by the formula $$ A_h^x u^h = \left\{ { - a\left( {x_k } \right)\frac{{u_{k + 1} - 2u_k + u_{k - 1} }} {{h^2 }} + \delta u_k } \right\}_1^{M - 1} ,u_h = \left\{ {u_k } \right\}_0^M ,Mh = 1 $$ with u 0 = u M and −u 2 + 4u 1 − 3u 0 = u M−2 − 4u M−1 + 3u M is established. In applications, the coercive inequalities for the solutions of the nonlocal boundary-value problem for two-dimensional elliptic equation and of the second order of accuracy difference schemes for the numerical solution of this problem are obtained.