Fixed Point Theory: Lipschitz or Contraction?

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SUMMARY

The discussion centers on the relationship between Lipschitz continuous functions and contraction mappings in fixed point theory. It is established that contraction mappings are a specific subset of Lipschitz continuous functions, characterized by a Lipschitz constant \( K \) that satisfies \( 0 \leq K < 1 \). Therefore, Lipschitz mappings are indeed more general than contraction mappings. This distinction is crucial for understanding the broader applications of fixed point theory.

PREREQUISITES
  • Understanding of fixed point theory
  • Familiarity with Lipschitz continuity
  • Knowledge of contraction mappings
  • Basic mathematical analysis concepts
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  • Research the properties of Lipschitz continuous functions
  • Explore the implications of contraction mappings in fixed point theorems
  • Study examples of Lipschitz mappings in various mathematical contexts
  • Learn about applications of fixed point theory in optimization problems
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Mathematicians, students of analysis, and researchers interested in fixed point theory and its applications in various fields such as optimization and numerical methods.

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 :)
 

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