Differential and difference equations carry many common features.
A strong tendency may be observed of transferring pieces of the
long experience with the former to the recently increasingly
popular domain of the latter. A part of the motivation lies in the
growth of the role played by the computers. That's why the author
moved temporarily from her traditional readership (J. Math. Anal.
Appl.) to a more computer-oriented public. Her compact and
comparatively closed review as well as new rigorous results (see
the title) in a problem with huge applicability. The text offers a
satisfactory reading about the discrete version of the
inhomogeneous linear Sturm Liouville problem and about its
solution via Green's functions. After a thorough review of her own
recent very interesting results - with coauthors - on Green's
(difference) function, she concentrates on the problems of
stability and absolute stability and proves two theorems giving
necessary and sufficient conditions in cases where the
''potential" remains non-negative. In all the theorems and proofs
the key role is played by the auxiliary sequences of Bulabaev and
Shuster. A sample of applications of the theory illustrates, e.g.,
the relevance of the criteria for the inversion problem (cf.
Theorem 3.4), for the tracing of parallels with differential
equations (cf. Theorem 3.4) and a few characteristic and specific
differences (cf. examples in the last section).




MR2000588 Chernyavskaya, N. A. Stability and absolute stability of
a three-point difference scheme. Advances in difference equations,
IV. Comput. Math. Appl. 45 (2003), no. 6-9, 1181--1194.