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Given information: Let $f$ be a continuous function defined for all $(x,y) \in \mathbb{R}^2$. Let $f$ also satisfy a Lipschitz condition with respect to $y$. Let $f$ be periodic with respect to $x$ of period $w$, and let $y_1, y_2$ be such values that $f(x,y_1)f(x,y_2) \lt 0$ for all $x$.

Question: Show that the equation $y'=f(x,y)$ has at least one periodic solution of period $w$. Then apply this result to the equation $y'+p(x)y = q(x)$ where $p(x) \not = 0$ and $q(x)$ are continuous period functions of period $w$.

I'm given a hint that is:

Consider the map

$$p: \mathbb{R} \to \mathbb{R}, y_0 \mapsto p(y_0):=y_w$$

where $y_{w}=y(w)$ with $y$ being the solution of the initial value problem

$$y'=f(x,y), y(0)=y_0$$

The map $p$ is a Poincaré map, which maps the $y$-value $y_0$ of the solution curve through the point $(x,y)=(0,y_0)$ to the $y$-value of this solution curve $x=w$.

A periodic solution of $y'=f(x,y), $ corresponds to a fixed Poincare map, i.e. the existence of a $y^* \in \mathbb{R}$ with $p(y^{*})=y^*$

Random thoughts: I have to prove the existence of a fixed point by noting that the Poincare map is continuous as follows from a theorem.

Please help !

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Nobody has any idea? :-( –  MSKfdaswplwq Jan 14 '13 at 23:29
    
where did you find this problem? –  yiyi Jan 16 '13 at 0:56
    
In an introductionary book on differential equations –  MSKfdaswplwq Jan 16 '13 at 2:25
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2 Answers

up vote 5 down vote accepted
+100

Assume $y_1<y_2$ and $f(x,y_1)>0$, $\ f(x,y_2)<0$ for all $x$. By general principles about ODE's any solution $x\mapsto \phi_\eta(x)$ starting at a point $(0,\eta)$, $\ y_1\leq\eta\leq y_2$, will finally leave the rectangle $R:=[0,w]\times[y_1,y_2]$. But it cannot do so along the horizontal edges of this rectangle. It follows that the solution $\phi_\eta$ will pass through a point $(w,\eta')$, $\ y_1\leq\eta'\leq y_2$, on the right edge of $R$. In this way a so-called Poincaré map $$\Phi:\quad [y_1,y_2]\to[y_1,y_2],\qquad \eta\mapsto \eta'=:\Phi(\eta)$$ is defined. Again by general principles this $\Phi$ is continuous. By Brouwer's fixed point theorem (or using the intermediate value theorem) it follows that $\Phi$ has a fixed point $\eta_*\in[y_1,y_2]$. The solution $\phi_{\eta_*}$ starting at $(0,\eta_*)$ is then periodic.

If instead of the initial assumption on $f$ we have $f(x,y_1)<0$, $\ f(x,y_2)>0$ we start the argument at $x=w$ and proceed to the left.

In the example $y'+p(x) y=q(x)$ we have $$f(x,y)=p(x)\left({q(x)\over p(x)} -y\right)\ .$$ As $p$ and $q$ are periodic and continuous, and $p(x)\ne0$ for all $x$ there is an $M>0$ such that $$-M<{q(x)\over p(x)}<M\qquad\forall x\ .$$ It follows that by choosing $y_1:=-M$, $\ y_2:=M$ we can fulfill the assumptions of the "theorem".

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I don't know Brouwers fixed point theorem. Only Banachs fixed point theorem... –  MSKfdaswplwq Jan 16 '13 at 19:37
    
@Joyeuse Saint Valentin: As indicated, in the one-dimensional case the intermediate value theorem allows to prove that any continuous map $\Phi:\ [0,1]\to[0,1]$ has a fixed point. A hint: Look at the graph of $\Phi$ and at the line $y=x$. –  Christian Blatter Jan 16 '13 at 20:59
    
So this proves the whole thing? Why are we not using the Lipschitz condition? –  MSKfdaswplwq Jan 19 '13 at 13:31
1  
@Joyeuse Saint Valentin: The Lipschitz condition guarantees the existence and uniqueness of solutions as used in my argument. So it has in fact be used tacitly. –  Christian Blatter Jan 19 '13 at 16:45
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Draw a rectangle with sides $(0, w) $ and $(y_1, y_2)$ and pay close attention to the condition $ f(x, y_1)f(x,y_2)<0$ for all $x$. Let me assume WLOG that $f(x, y_1)<0$ and $y_1 < y_2$. Draw the integral curves of the vector field $ f(x,y)$. You'll notice that they all point upwards at the upper side of the rectangle and downwards at the bottom. Since integral curves cannot cross, this implies that your map contains the interval $(y_1, y_2)$ in the image and moreover that it is monotone on $p^{-1}((y_1,y_2)) \subset (y_1,y_2)$.

This is the big intuitive picture. Can you make it rigorous now?

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Thank you for your reponse. To be honest I'm not sure how to exactly formulate it. I see what you mean. Can you say something more about it? Or anybody else? :) –  MSKfdaswplwq Jan 15 '13 at 15:47
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