A convergence result on random products of mappings in metric spaces

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Khamsi, M.A. & Louhichi, I. Fixed Point Theory Appl (2012) 2012: 43. https://doi.org/10.1186/1687-1812-2012-43


Let X be a metric space and {T1, ..., T N } be a finite family of mappings defined on DX. Let r : ℕ {1,..., 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 x0∈D;andxn+1=Tr(n)(xn),foralln≥0.Open image in new window" role="presentation" style="box-sizing: border-box; display: block; line-height: normal; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: 100%; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; width: 745px; overflow: auto hidden; position: relative;">𝑥0∈𝐷;𝖺𝗇𝖽𝑥𝑛+1=𝑇𝑟(𝑛)(𝑥𝑛),𝖿𝗈𝗋𝖺𝗅𝗅𝑛≥0.Open image in new window

In particular, we extend the study of Bauschke [1] from the linear case of Hilbert spaces to metric spaces. Similarly we show that the examples of convergence hold in the absence of compactness. These type of methods have been used in areas like computerized tomography and signal processing.