Stability of an iteration in cone metric spaces (Q2912208)

Let \((X,\|\cdot\|)\) be a cone normed space with respect to a normal cone \(P\) in a real Banach space \(E\), and \((X,d)\) the corresponding cone metric space, in the sense of [\textit{L.-G. Huang} and \textit{X. Zhang}, J. Math.\ Anal.\ Appl.\ 332, No.\ 2, 1468--1476 (2007; Zbl 1118.54022)]. Let \(E\) be a nonexpansive self-map on \(X\). Consider the iteration procedure \(x_{n+1}=S_nx_n\), where \(S_n=\frac1n(I+T+\dots+T^{n-1})\). The author proves that, if there exist \(a\geq0\) and \(b\in(0,1)\) such that \(d(S_ny,q)\leq ad(S_ny_n,y_n)+bd(y_n,q)\) for all sequences \(\{y_n\}\) in \(X\) with \(d(Ty_n,y_n)=o(1/(n-1))\) and for every fixed point \(q\) of \(T\), then the given iteration is \(T\)-semistable. No examples are given.