Maximal regularity and compactness conditions for a high order system of di˙erence equations

Abstract

In this paper we study an infinite linear system of di˙erence equations of high even order with the right-hand side from the Hilbert space of numerical sequences. Sequences formed from the coe􀀁cients of the equations of the system for the same orders of di˙erence can be unlimited, and their growth may not be subject to the growth of the potential. The previously developed methods, which essentially use the dominant potential growth in the di˙erence systems of Sturm-Liouville type equations, do not pass here, since In the case under consideration, the potential may turn out to be zero, or not having a definite sign by a sequence. We give conditions for the correct solvability of the system, as well as optimal estimates of the norms of the solution and its di˙erences up to the highest order. Conditions for the compactness of the resolvent of the corresponding system of a degenerate operator are obtained. We prove some di˙erence weight inequalities of Hardy type having independent scientific interest. They are used in the proof of the main results of the paper. It is shown that, in comparison with degenerate di˙erential equations, in the case of a di˙erence system, it is possible to remove the condition for oscillations of the coe􀀁cients of the system.