I'm studying some book about Ordinary Differential Equations and I have some problem with the proof of this corollary: The solution $x(t, t_o, x_o)$ is a continuous function of $(t, x )$.
The solution $x(t, t_o, x_o)$ is a continuous function of $(t, x )$.
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ordinary-differential-equations
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0You mean "of $(t_0,x_0)$"? Study the Grönwall lemma, it answers most questions about perturbations of ODE and initial conditions. – 2017-01-13
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0@LutzL I can't understand the relation between this lemma and the corollary! Excuse me because of the lackness of my information.Because I have started this,just a few days! – 2017-01-13
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0Please add at least the theorem that your claim is the corollary to. And summarize the parts of the proof that you understand and the point where you do not understand it further. – 2017-01-13
1 Answers
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You have a solution $x_0(t)=x(t,t_0,x_0)$ and $x_1(t)=x(t,t_1,x_1)$ where $(t_1,x_1)$ is close in some to be quantified manner to $(t_0,x_0)$. Now the difference between the solutions can be decomposed using the Grönwall lemma as \begin{align} \|x_1(t)-x_0(t)\| &\le \exp(|A(t)-A(t_1)|)·\|x_1(t_1)-x_0(t_1)\| \\ &\le \exp(|A(t)-A(t_1)|)·(\|x_1-x_0\|+\|x_0-x_0(t_1)\|) \end{align} where $L(t)=A'(t)$ is a local Lipschitz constant for $f$ around $(t,x_0(t))$ in the sense of the Lipschitz condition, that is $$ \|f(t,y)-f(t,z)\|\le L(t)·\|y-z\| $$ Thus the continuity wrt. the initial condition is reduced to the local Lipschitz condition and the continuity of the reference solution $x_0$ which follows from the solvability of the ODE, i.e., the continuity of $f$.