Stability analysis of nonlinear retarded difference equations in Banach spaces (Q1827156)

Consider the nonlinear difference equation \[ x(k+1)=f(k,x(k),x(k-1),\dots,x(k-r)),\quad k\in Z^{+},\;x(k)\in X,\quad r\geq 1,\tag{*} \] where \((X,\left\| .\right\| )\,\) is an infinite-dimensional Banach space with the norm \(\left\| .\right\|,\;Z^{+}=\{0,1,2,\dots\}\) and \(f(k,0,\dots,0)=0,\forall k\in Z^{+}.\) The authors prove the following theorem that gives sufficient conditions for the asymptotic stability of (*). Theorem: Assume that \(\exists \alpha >0\), \(p_i\geq 0\), \(i=0,1,\dots,r\) such that \(\left\| f(k,y_0,y_{1},\dots,y_k)\,\right\| \leq \alpha \prod_{i=0}^r\left\| y_i\right\| ^{p_i} \forall y_i\in X\), \(k\in Z_{k_0}\) and let \(P=p_0+p_1+\cdots +p_r\). Then the system (*) is asymptotically stable if one of the following two conditions holds : (i) \(P>1\), \(\alpha >0\); (ii) \(P=1\), \(0<\alpha <1.\) Another theorem for the instability of (*) is proved. Several examples are given to show the power of their results. Reviewer's comment : It has to be noticed that the proofs of the two theorems are elegant. In reference [6] ``S. Elaydi'' should be replaced by ''W. G. Kelley''.