Overview
In this chapter we’ll study the best-known of all fixed-point theorems: the Banach Contraction-Mapping Principle, which we’ll apply to Newton’s Method, initial-value problems, and stochastic matrices.Prerequisites. Undergraduate-level real analysis and linear algebra. The basics of metric spaces: continuity and completeness.
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Notes
- 1.
These are often just called “contractions”; the terminology here is more in keeping with conventions used in (linear) operator theory.
- 2.
See [101, Theorems 7.14 and 7.15, pp. 150–151], for example.
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Shapiro, J.H. (2016). Contraction Mappings. In: A Fixed-Point Farrago. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-319-27978-7_3
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