Maximal regularity and compactness conditions for a high order system of difference equations

Abstract

In this paper we study an infinite linear system of difference equations of high even order with the righthand side from the Hilbert space of numerical sequences. Sequences formed from the coefficients of the equations of the system for the same orders of difference 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 difference 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 differences 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 difference 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 differential equations, in the case of a difference system, it is possible to remove the condition for oscillations of the coefficients of the system.