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Chapter 6: Solving Nonlinear Problems

Section 1: The Contraction Mapping Principle

Proposition 1.1 (If the sum of the lengths of a series of vectors converges, the series of vectors converges.)

Suppose {${\displaystyle {\vec {a}}_{k}}$} is a sequence of vectors in ${\displaystyle \mathbb {R} ^{n}}$ and the series ${\displaystyle \sum _{k=1}^{\infty }\lVert {\vec {a}}_{k}\rVert }$ converges (i.e., the sequence of partial sums ${\displaystyle t_{k}=\lVert {\vec {a}}_{1}\rVert +...+\lVert {\vec {a}}_{k}\rVert }$ is a convergent subsequence of real numbers). Then the series ${\displaystyle \sum _{k=1}^{\infty }{\vec {a}}_{k}}$ of vectors converges (i.e., the sequence of partial sums ${\displaystyle {\vec {s}}_{k}={\vec {a}}_{1}+...+{\vec {a}}_{k}}$ is a convergent sequence of vectors in ${\displaystyle \mathbb {R} ^{n}}$).

Defintion Let X be a subset of ${\displaystyle \mathbb {R} ^{n}}$. A function ${\displaystyle {\vec {f}}}$: X → X is called a contraction mapping if there is a constant c, 0 < c < 1, so that ${\displaystyle \lVert {\vec {f}}({\vec {x}})-{\vec {f}}({\vec {y}})\rVert \leq c\lVert {\vec {x}}-{\vec {y}}\rVert }$