On the estimate of solutions of perturbed linear difference equations (Q748771)

The author gives bounds for the norm of solutions of the vector difference equation \(y(k+1)=A(k)y(k)+f(k,y(k)),\) \(y\in {\mathbb{R}}^ n\), by means of the solutions of the scalar difference equation \(z(k)=z(p)+\sum^{t}_{i=s}(\gamma /\alpha)[F(i,z(i))+B(i)\alpha^{- i}],\) where \(t=k-1\), \(s=p\) or \(t=p-1\), \(s=k\) in the case \(k\in {\mathbb{N}}^+:=\{0,1,2,...\}\) (forward difference equation) or \(k\in {\mathbb{N}}^-:=\{0,-1,-2,...\}\) (backward difference equation) respectively; \(\alpha\),\(\gamma\) are any constants which majorise the growth of solutions of the vector linear equation \(x(k+1)=A(k)x(k),\) and \(\| f(m,u)\| \leq F(m,\| u\|)+B(m)\) for all \((m,u)\in {\mathbb{N}}^+\times {\mathbb{R}}^ n\) or \((m,u)\in {\mathbb{N}}^-\times {\mathbb{R}}^ n\) respectively (F,A,B satisfy some additional assumptions). Sufficient conditions for the zero solutions of the vector equation to be uniformly asymptotically stable are also given (in the case \(k\in {\mathbb{N}}^+)\). Applying the proved theorems, some earlier results of \textit{B. Aulbach} [J. Math. Anal. Appl. 112, 317-327 (1985; Zbl 0595.34060)] are deduced.