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We consider a second order differential operator AAxx with Samarkii-Ionkin type

conditions. We establish the positivity of the differential operator in C[0,1]. Moreover, we investigate the structure of fractional spaces Eα(C[0,1], A). Furthermore, we prove for each 0 < α < 1/2 that the topological equivalence of the normed spaces Eα(C[0,1], A) and C^2α[0,1] and hence we establish the positivity of A in °C^2α[0,1]. We also apply the theoretical results to obtain noval coercive inequalities for the solution of a nonlocal boundary value problem for parabolic equations. This is a joint work with Prof. Dr. Allaberen Ashyralyev, Bahçeşehir University.