Abstract
Let K be a bounded, closed, and convex subset of a Banach space. For a Lipschitzian self‐mapping A of K, we denote by Lip(A) its Lipschitz constant. In this paper, we establish a convergence property of infinite products of Lipschitzian self‐mappings of K. We consider the set of all sequences of such self‐mappings with the property limsupt→∞Lip(At ) ≤ 1. Endowing it with an appropriate topology, we establish a weak ergodic theorem for the infinite products corresponding to generic sequences in this space.
Highlights
The asymptotic behavior of infinite products of operators finds applications in many areas of mathematics
Given a bounded, closed, and convex subset K of a Banach space and a sequence A = {At}∞t=1 of self-mappings of K, we are interested in the convergence properties of the sequence of products {An · · · A1x}∞n=1, where x ∈ K
In the special case of a constant sequence A, we are led to study the asymptotic behavior of a single operator
Summary
Let K be a bounded, closed, and convex subset of a Banach space. For a Lipschitzian self-mapping A of K, we denote by Lip(A) its Lipschitz constant. We establish a convergence property of infinite products of Lipschitzian self-mappings of K. We consider the set of all sequences {At}∞t=1 of such selfmappings with the property lim supt→∞ Lip(At) ≤ 1. Endowing it with an appropriate topology, we establish a weak ergodic theorem for the infinite products corresponding to generic sequences in this space
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