A PRIORI ESTIMATES FOR SOME CLASSES OF ONE-DIMENSIONAL INITIAL AND BOUNDARY-VALUE DIFFERENCE PROBLEMS

A new approach to the investigation of the correctness of difference schemes for abstract linear parabolic equations is proposed, which uses weaker assumptions about the initial difference operator of the problem than the property of uniform positiveness (semi-positiveness). Using this method in the norms of mesh spaces L(ph), 1 less-than-or-equal-to p less-than-or-equal-to infinity (the mesh analogues of Lebesgue spaces), a priori estimates are constructed of difference schemes which approximate a one-dimensional initial and boundary-value problem on non-uniform space meshes.