Asymptotic behavior of inhomogeneous iterates for nonlinear operators in ordered Banach spaces. (Q2784681)

A sequence \(\{f_n\}\) of operators in a complete metrically convex metric space or in a positive cone of an ordered Banach space is considered. The author shows that if \(f_n \to f_0\) uniformly and certain additional assumptions are fulfilled, then \(\lim_{n\to \infty} f_n \circ f_{n-1} \circ \dots \circ f_1(x_1) = x_0\) for any \(x_1\), where \(x_0\) is the unique fixed point of the operator \(f_0\). In the case of a metrically convex space, the basic assumptions are that all the \(f_n\) are Lipschitz with the same Lipschitz constant and \(d(T_0^l x,T_0^l y) \leq \phi(d(x,y))\) for some \(l>1\), where \(\phi:[0,\infty) \to [0,\infty)\), \(\phi(t)<t\) for all \(t>0\), and \(\phi(0)=0\). In the case of an ordered Banach space, the basic assumptions are that the partial ordering is given by a closed convex cone \(P\) with a nonempty interior \(P^0\), \(f_n: P^0 \to P^0\), \(f_0^l(y) \geq \phi(t)f_0^l(x)\) for some \(l\) positive integer and all \(y \geq tx\), \(x,y \in P^0\), where \(\phi:(0,1) \to (0,1)\), \(\phi(t)>t\) for all \(t \in (0,1)\), and \(\phi(1)=1\). An application to Hammerstein integral operators is given.NEWLINENEWLINEFor the entire collection see [Zbl 0971.00016].