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Which of the following statements are true?
(a) let $f: (0,∞) →(0,∞) $ be such that $|f(x)-f(y)| ≤(1/2)|x-y|$ for all $x$ and $y$.then $f$ has a fixed point.
(b) let $f:[-1,1] →[-1,1]$ be continuous. Then $f$ has a fixed point.
(c) let $f: \mathbb{R}→\mathbb{R}$ be continuous and periodic with period $T<0$ . the there exist a point $x_0 \in \mathbb{R}$ such that $f(x_0)=f(x_0+T/2)$.

(a) I am not sure . I can not apply banach fixed point theorem here.
(b)true.
(c) not sure.

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a) False b) and c) True. –  Une Femme Douce Jan 19 '13 at 19:18
    
why is a) not true by the Contraction Mapping Theorem? –  mathemagician Jan 19 '13 at 19:31
1  
@mathemagician the given set is not complete –  poton Jan 19 '13 at 19:33
    
true. thank you. –  mathemagician Jan 19 '13 at 19:38

1 Answer 1

up vote 1 down vote accepted

HINTS: For (a) consider $f(x)=\frac{x}2$. For (c) you presumably want $T>0$; consider the function $$g(x)=f\left(x+\frac{T}2\right)-f(x)$$ and the intermediate value theorem.

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then I can say that (a) is false and (c) is true. am I right? –  poton Jan 19 '13 at 19:21
    
@poton: Yes, (a) is false, and (c) is true, though you should make sure that you can justify those conclusions if asked. –  Brian M. Scott Jan 19 '13 at 19:22

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