On difference linear periodic systems II. Non-homogeneous case. (English) Zbl 0588.39001

Zaballa, Ion; Gracia, Juan M.

Apl. Mat. 30, 403-412 (1985).

Given \(N\in {\mathbb{N}}\) let \(x(n+1)=A(n)x(n)+f(n)\), \(n\in {\mathbb{N}}\), be a non-homogeneous system of linear difference equations over \({\mathbb{C}}\) such that for the matrix of coefficients A(n) and the vector f(n) the equations \(A(n+N)=A(n)\) and \(f(n+N)=f(n)\) hold for all \(n\in {\mathbb{N}}\). Analogously to part I [ibid. 28, 241-248 (1983; Zbl 0529.39003)] the given periodic system is reduced to a linear non-homogeneous system of difference equations with constant coefficients and independent term. This allows to obtain the solutions of the given system in closed form, to study the asymptotic behaviour and to derive theorems concerning the existence and properties of N-periodic solutions.

Reviewer: D.Dorninger

MSC:

39A10	Additive difference equations
39A12	Discrete version of topics in analysis

Keywords:

reduction; explicit solution; non-homogeneous system; periodic system; asymptotic behaviour; periodic solutions

Citations:

Zbl 0529.39003

References:

[1]	R. Bellman: On the boundedness of solutions of nonlinear differential and difference equations. Trans. Amer. Math. Soc. 62 (1974), no. 3, 357-386. · Zbl 0031.39901
[2]	A. Halanay D. Wexler: Qualitative Theory of Sampled-Data Systems. (Russian translation), Mir, Moscow (1971). · Zbl 0226.34001
[3]	I. Zaballa J. M. Gracia: On difference linear periodic systems I. Homogeneous case. Apl. mat. 28 (1983) 241-248. · Zbl 0529.39003