Fixed Point Theorem & Contractive Maps

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SUMMARY

The discussion focuses on contractive maps and their fixed points, specifically highlighting the constant real function 1 and the linear map defined by $x \mapsto kx$ for $0 \leq k < 1$. The latter map has a clear fixed point at zero. These examples illustrate fundamental concepts in fixed point theory, particularly within the context of real-valued functions.

PREREQUISITES
  • Understanding of fixed point theorems
  • Familiarity with contractive mappings
  • Basic knowledge of real analysis
  • Concept of continuity in functions
NEXT STEPS
  • Study Banach's Fixed Point Theorem
  • Explore examples of contractive maps in metric spaces
  • Learn about applications of fixed point theory in mathematical analysis
  • Investigate the implications of contractive mappings in iterative methods
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Mathematicians, students of real analysis, and anyone interested in fixed point theory and contractive mappings.

ozkan12
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Please give an example of contractive map which have fixed point...I search but I didnt find
 
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Hi,

A trivial example is the constant real function 1.
 
Another trivial example is the map $x \mapsto kx$ for $0 \leq k < 1$ over, say, the reals. It has an obvious fixed point: zero.
 

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