Classes of stable three-layer schemes

The problem statement. In this section we establish sufficient stability conditions and a priori estimates for three-layer schemes on the basis of their canonical form 

Here and below, j/q Vi arbitrary given vectors of a finitedimensional real space H, (p t) is a given arbitrary abstract Jf-valued function of the variable t . A, B and R are linear operators in the space H. The dependence of y t) = yhr t), = Phrit), (0 = hr, B, R, y and 2/j on h and r is not explicitly indicated. We proceed as usual. This amounts to introducing more compact notations  

All the tricks and turns remain unchanged by analogy with Section 2 first, a solution of problem (1) can be arranged as a sum y = y + y, where y is a solution of the homogeneous equation  

Second, an alternative form of writing appears useful  

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Further development of the three-layer scheme (3) is connected with the functional known as the compound-norm  

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