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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 {$$\vec a_k$$} is a sequence of vectors in $$\mathbb{R}^n$$ and the series $$\sum_{k=1}^\infty \lVert \vec a_k \rVert$$ converges (i.e., the sequence of partial sums $$t_k = \lVert \vec a_1 \rVert + ... + \lVert \vec a_k \rVert$$ is a convergent subsequence of real numbers). Then the series $$\sum_{k=1}^\infty \vec a_k $$ of vectors converges (i.e., the sequence of partial sums $$\vec s_k = \vec a_1 + ... + \vec a_k$$ is a convergent sequence of vectors in $$\mathbb{R}^n$$).

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