Periodicity and stability of linear Volterra difference systems. (English) Zbl 0796.39004

Elaydi, Saber

J. Math. Anal. Appl. 181, No. 2, 483-492 (1994).

A sufficient condition for the existence of a unique \(N\)-periodic solution of the difference equation \(x_{n+1} = A_ nx_ n + \sum^ n_{i=- \infty} B_{ni} x_ i + c_ n\), \(n \geq 0\), where \(N\) is a positive integer, \((A_ n)\), \((B_{ni})\) are \(N\)-periodic sequences of \(k\)-dimensional matrices with \(\sum^ n_{i=- \infty} | B_{ni} |<\infty\) for all \(n \geq 0\) and where \((c_ n)\) is an \(N\)-periodic sequence of \(k\)-dimensional column vectors, is given. Moreover, a sufficient condition for the uniform asymptotic stability of the zero solution of the difference equation \(x_{n+1} = A_ n x_ n + \sum^ n_{i=0} B_{ni} x_ i\), \(n \geq 0\), where \(A_ n\), \(B_{ni}\) are \(k\)-dimensional matrices, is given.

Reviewer: H.Länger (Wien)

MSC:

39A10	Additive difference equations
39A11	Stability of difference equations (MSC2000)

Keywords:

periodicity; stability; linear Volterra difference systems; resolvent matrix; variation of constants; uniform asymptotic stability