Determinacy criteria for implicit discrete nonautonomous systems (Q2768852)

This paper deals with an implicit difference equation NEWLINE\[NEWLINET_nx_{n+1}+x_n=f_n\tag{1}NEWLINE\]NEWLINE in \(m\)-dimensional normed space \(X\) with degenerate linear operators \(T_n:X \to X\). The authors generalize their earlier result concerning the case where the index of each \(T_n\) equals 1 [Dopov. Nats. Akad. Nauk Ukr., Mat. Pryr. Tekh. Nauky 1998, No. 7, 11-15 (1998; Zbl 0926.39002)]. Now it is assumed that the indices of the operators \(T_n\) and the dimensions of the root spaces \(Z_n\) in the zero point of the spectrum do not depend on \(n\): \(\text{ind} T_n=r \text{dim} Z_n=q, n=0,1,2,\ldots \). Then the projectors \(P_n:X \to E_r:=\text{im}T_n^r, Q_n:X \to Z_n:=\ker T_n^r\) satisfying \(P_nQ_n=0, P_n+Q_n=\text{Id}_{X}\) are of constant rank. NEWLINENEWLINENEWLINEThe authors show that for any \(e_0\in E_0\) there exists a unique solution \(x_n\) to (1) such that \(P_0x_0=e_0\), provided that the following conditions holdsNEWLINE\[NEWLINE\begin{aligned} T_{n}Q_n\cdots T_{n+k-1}Q_{n+k-1}P_{n+k}&=0,\quad k=1,2,\ldots,r;\\ T_{n}Q_n\cdots T_{n+k-1}Q_{n+k-1}Q_{n+r}&= 0.\end{aligned} NEWLINE\]NEWLINE Some generalizations of this result are also obtained.