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Consider the following sets :

$$S=\left\{(x,y)\in \mathbb R^2:x^2+y^2=1\right\}.$$

$$D=\left\{(x,y)\in \mathbb R^2:x^2+y^2\le 1\right\}.$$

$$E=\left\{(x,y)\in \mathbb R^2:2x^2+3y^2\le 1\right\}.$$

Which of the following are correct ?

(a) If $f:D\to S$ is continuous then $f$ has a fixed point.

(b) $f:S\to S$ is continuous then $f$ has a fixed point.

(c) If $f:E\to E$ is continuous then $f$ has a fixed point.

We know from Brouwer's fixed point theorem , " A continuous mapping of a closed & convex set into itself necessarily has a fixed point. "

From this theorem we have , option (c) is correct & option (b) is wrong.

But what about options (a) ?

Are there any theorem for a continuous function from one set to another set have to a fixed point?


marked as duplicate by mechanodroid, Xam, rogerl, Trevor Gunn, Nosrati Nov 5 '17 at 16:16

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  • 1
    $\begingroup$ for a) note that $S\subset D$ and $D$ is closed and convex $\endgroup$ – Surb Feb 27 '15 at 19:37
  • $\begingroup$ So what? It is NOT clear to me...Please explain.... $\endgroup$ – Empty Feb 27 '15 at 19:39
  • $\begingroup$ Is it more clear in my answer? $\endgroup$ – Surb Feb 27 '15 at 19:40
  • $\begingroup$ Here are more fixed points theorems: en.wikipedia.org/wiki/… $\endgroup$ – Surb Feb 28 '15 at 19:20

a) note that $S\subset D$ and $D$ is closed and convex. So $f(D)\subset S\subset D$ and you can apply the Theorem.

b) depends on the function, e.g. $f(x)=x$ or $f$ is a rotation around the origin.

c) you are right.

  • $\begingroup$ In statement of Brouwer's fixed point theorem the condition is necessary..So why I am wrong in (b)? $\endgroup$ – Empty Feb 27 '15 at 19:44
  • $\begingroup$ @Panja.S. no the condition is sufficient but not necessary for the existence of a fixed point. $\endgroup$ – Surb Feb 27 '15 at 19:45
  • $\begingroup$ Ohh...So (a) is also TRUE ? $\endgroup$ – Empty Feb 27 '15 at 19:50
  • 1
    $\begingroup$ @Panja.S. yes :) $\endgroup$ – Surb Feb 27 '15 at 19:50
  • $\begingroup$ I have a last question..Are there any necessary or sufficient condition for a continuous function $f:X\to Y$ to have a fixed point? where $X$ & $Y$ are any arbitrary two spaces, not necessarily $Y\subset X$ $\endgroup$ – Empty Feb 27 '15 at 19:52

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