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Theorem (Weissinger). Let $C$ be a (nonempty) closed subset of a Banach space $X$. Suppose $K : C → C$ satisfies $$\|K^nx − K^ny\| ≤ θ_n\|x − y\|, \quad x,y∈ C $$ with $\sum_n θ_n < ∞$. Then $K$ has a unique fixed point $\bar x$ such that $$ \|K^nx − \bar x\| ≤ \sum_{j=n}^\infty θ_j\cdot \|Kx − x\|,\quad x∈ C. $$

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    What is $K_n$? Do you know how to use Tex in this site?2012-10-15
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    @DavideGiraudo: Perhaps it is to be read as $K^n$?2012-10-15
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    Davide, No I do not know how to use Tex yet. Kn is K^n the nth iteration of K.2012-10-15
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    @Klara Please check if I did the TeXification correclty.2012-10-15
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    @ Martini you did great!2012-10-15

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