MHB Fixed Point Theory: Lipschitz or Contraction?

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Contraction mappings are a specific subset of Lipschitz continuous functions, characterized by a Lipschitz constant less than one. This implies that all contraction mappings are Lipschitz, but not all Lipschitz mappings are contractions. Therefore, Lipschitz mappings represent a broader class than contraction mappings. The discussion confirms that Lipschitz continuity encompasses a wider range of functions. Understanding this distinction is crucial in fixed point theory.
ozkan12
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I see that if a mapping is contraction then it is contractive then it is nonexpensive and then it is lipschtiz...so, which class of mapping is general ? lipschitz or contraction ? which one ? thank you for your attention :)
 
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Contraction is a special case of Lipschitz continuous functions, namely, with Lipschitz constant $K$ satisfying $0\le K<1$.
 
so, lipschitz mappings are more general than contraction ?
 
ozkan12 said:
so, lipschitz mappings are more general than contraction ?
Yes.
 
ok. thanks a lot :)
 

Thread 'About the definition of topological manifold using closed sets'

We all know the definition of n-dimensional topological manifold uses open sets and homeomorphisms onto the image as open set in ##\mathbb R^n##. It should be possible to reformulate the definition of n-dimensional topological manifold using closed sets on the manifold's topology and on ##\mathbb R^n## ? I'm positive for this. Perhaps the definition of smooth manifold would be problematic, though.
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