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A convergence result on random products of mappings in metric trees. (English) Zbl 1346.54018

Summary: Let \(X \) be a metric space and \(\{T_1,\dots, T_N\}\) be a finite family of mappings defined on \(D \subset X\). Let \(r : \mathbb N \to \{1,\dots, N\}\) be a map that assumes every value infinitely often. The purpose of this article is to establish the convergence of the sequence \((x_n)\) defined by \[ x_0 \in D; \;{\text{and}} \;x_{n+1} = T _{r(n)} (x_n), \quad \text{for all} \;n \geq 0. \] In particular we prove Amemiya and Ando’s theorem [I. Amemiya and T. Ando [Acta Sci. Math. 26, 239–244 (1965; Zbl 0143.16202)] in metric trees without compactness assumption. This is the first attempt done in metric spaces. These type of methods have been used in areas like computerized tomography and signal processing.

MSC:

54H25 Fixed-point and coincidence theorems (topological aspects)
06F30 Ordered topological structures
47J25 Iterative procedures involving nonlinear operators

Citations:

Zbl 0143.16202
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References:

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