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 $$ \|K^nx − \bar x\| ≤ \sum_{j=n}^\infty θ_j\cdot \|Kx − x\|,\quad x∈ C. $$
Weissinger's Theorem. How to prove?
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fixed-point-theorems
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0What is $K_n$? Do you know how to use Tex in this site? – 2012-10-15
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0@DavideGiraudo: Perhaps it is to be read as $K^n$? – 2012-10-15
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0Davide, 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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0@Klara Please check if I did the TeXification correclty. – 2012-10-15
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0@ Martini you did great! – 2012-10-15