1

In my assignment a question was given on strong contraction. I could understand the meaning and searched in wikipedia and also in books but I could find the definition of it. Can anyone give the definition of strong contraction and how it is different from normal contraction in banach's contraction principle.

Thanks!

uuuuuuuuuu
  • 993
  • 2
    I would guess that a weak contraction is a map $f$ such that $d(f(x),f(y))<d(x,y)$ for all distinct $x,y$ and a strong contraction is a map $f$ such that there exists a constant $c<1$ such that $d(f(x),f(y))\leq cd(x,y)$ for all $x,y$. – Eric Wofsey Apr 02 '16 at 17:57
  • @Eric Wofsey Is it true that in case of weak contraction there may be a number of fixed points? – uuuuuuuuuu Apr 02 '16 at 18:01
  • @Saikat: No (why not?), but a weak contraction (in this sense) on a complete metric space may have no fixed point at all. – Andrew D. Hwang Apr 02 '16 at 18:25
  • 1
    The function $f:X\to X$ which satisfies $d(f(x),f(y))<d(x,y)$ for all $x,y\in X$ distinct numbers is called as "contractive" or "strict contraction". If $X$ is a compact metric space, $f$ has a unique fixed point. http://math.stackexchange.com/questions/1397478/counterexample-of-banach-fixed-point-banachs-contraction-theorem (i guess strong contraction is same as banach contraction ) – Domates Apr 02 '16 at 18:25

0 Answers0