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$begingroup$

Suppose we want to solve a simple ODE $$dotx=2t(1-x)=f(t,x)qquad (1)$$ with initial condition $x_0=2$ using the contraction mapping principle.

I showed that $f$ is Lipschitz continuous over $[0,1]timesmathbb R$ in $t$ with Lipschitz constant $L=max_tin[0,1]2t$. By considering a normed vector space $(C[0,delta],||cdot||_C)$, then using the contraction map with $P:C[0,delta]to C[0,delta]$ given by $$(Pphi)(t)=x_0+int_0^tf(t,phi(tau));dtau$$ I found by direct integration that the solution of $(1)$ is $$x=1+exp(-t^2).$$

But so far we have shown that there exists a unique solution over the interval $[0,delta]$, right?

Is the next step to provide an argument that shows that our solution is unique over the entire set of real numbers? I am not sure how to make this argument. I thought that maybe by iteration and increasing $delta$ one can show that the solution is unique over a bigger interval. But this doesn't seem to me very rigorous.

I'd appreciate any help/hint. Thank you.

ordinary-differential-equations nonlinear-system contraction-operator

share|cite|improve this question

asked Mar 29 at 2:36

johnny09johnny09

705624

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Hint: for large $T$, $P: C[-T,T] rightarrow C[-T,T]$ defined by, $$(Pphi)(t) = x_0 + int_0^t f(tau,phi(tau)),dtau,$$ is no longer a contraction mapping. However, for any such fixed $T$, there exists a $k$ such that $P^k$ is a contraction mapping. Try proving this and you will have your result.
  $endgroup$
  – forgottenarrow
  Mar 29 at 5:23
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  There is a result stating that if $fcolon(a,b)timesmathbbRtomathbbR$ is continuous and there are continuous functions $M_1,M_2colon(a,b)to[0,infty)$ such that $lvert f(t,x)rvertle M_1(t)+M_2(t)lvert xrvert$ for all $tin(a,b)$ and all $xinmathbbR$ then any nonextendible solution of $dotx=f(t,x)$ is defined on the whole of $(a,b)$. The proof uses Grönwall's inequality.
  $endgroup$
  – user539887
  Mar 29 at 6:28
  
  

  
  
  
  

add a comment | 

0

$begingroup$

Suppose we want to solve a simple ODE $$dotx=2t(1-x)=f(t,x)qquad (1)$$ with initial condition $x_0=2$ using the contraction mapping principle.

I showed that $f$ is Lipschitz continuous over $[0,1]timesmathbb R$ in $t$ with Lipschitz constant $L=max_tin[0,1]2t$. By considering a normed vector space $(C[0,delta],||cdot||_C)$, then using the contraction map with $P:C[0,delta]to C[0,delta]$ given by $$(Pphi)(t)=x_0+int_0^tf(t,phi(tau));dtau$$ I found by direct integration that the solution of $(1)$ is $$x=1+exp(-t^2).$$

But so far we have shown that there exists a unique solution over the interval $[0,delta]$, right?

Is the next step to provide an argument that shows that our solution is unique over the entire set of real numbers? I am not sure how to make this argument. I thought that maybe by iteration and increasing $delta$ one can show that the solution is unique over a bigger interval. But this doesn't seem to me very rigorous.

I'd appreciate any help/hint. Thank you.

ordinary-differential-equations nonlinear-system contraction-operator

share|cite|improve this question

asked Mar 29 at 2:36

johnny09johnny09

705624

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Hint: for large $T$, $P: C[-T,T] rightarrow C[-T,T]$ defined by, $$(Pphi)(t) = x_0 + int_0^t f(tau,phi(tau)),dtau,$$ is no longer a contraction mapping. However, for any such fixed $T$, there exists a $k$ such that $P^k$ is a contraction mapping. Try proving this and you will have your result.
  $endgroup$
  – forgottenarrow
  Mar 29 at 5:23
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  There is a result stating that if $fcolon(a,b)timesmathbbRtomathbbR$ is continuous and there are continuous functions $M_1,M_2colon(a,b)to[0,infty)$ such that $lvert f(t,x)rvertle M_1(t)+M_2(t)lvert xrvert$ for all $tin(a,b)$ and all $xinmathbbR$ then any nonextendible solution of $dotx=f(t,x)$ is defined on the whole of $(a,b)$. The proof uses Grönwall's inequality.
  $endgroup$
  – user539887
  Mar 29 at 6:28
  
  

  
  
  
  

add a comment | 

$begingroup$

Suppose we want to solve a simple ODE $$dotx=2t(1-x)=f(t,x)qquad (1)$$ with initial condition $x_0=2$ using the contraction mapping principle.

I showed that $f$ is Lipschitz continuous over $[0,1]timesmathbb R$ in $t$ with Lipschitz constant $L=max_tin[0,1]2t$. By considering a normed vector space $(C[0,delta],||cdot||_C)$, then using the contraction map with $P:C[0,delta]to C[0,delta]$ given by $$(Pphi)(t)=x_0+int_0^tf(t,phi(tau));dtau$$ I found by direct integration that the solution of $(1)$ is $$x=1+exp(-t^2).$$

But so far we have shown that there exists a unique solution over the interval $[0,delta]$, right?

Is the next step to provide an argument that shows that our solution is unique over the entire set of real numbers? I am not sure how to make this argument. I thought that maybe by iteration and increasing $delta$ one can show that the solution is unique over a bigger interval. But this doesn't seem to me very rigorous.

I'd appreciate any help/hint. Thank you.

ordinary-differential-equations nonlinear-system contraction-operator

share|cite|improve this question

asked Mar 29 at 2:36

johnny09johnny09

705624

$endgroup$

share|cite|improve this question

asked Mar 29 at 2:36

johnny09johnny09

705624

share|cite|improve this question

asked Mar 29 at 2:36

johnny09johnny09

705624

asked Mar 29 at 2:36

johnny09johnny09

705624

asked Mar 29 at 2:36

johnny09johnny09

705624