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Thread: Continuous dependence on the initial point

  1. #1
    Senior Member bkarpuz's Avatar
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    Exclamation Continuous dependence on the initial point

    Dear MHF members,

    I need the proof of the following result but I don't know where can I get it.

    Theorem. Let $\displaystyle A\in\mathrm{C}([0,\infty),\mathbb{R}_{n}^{n})$, $\displaystyle F\in\mathrm{C}([0,\infty),\mathbb{R}^{n})$ and $\displaystyle X_{0}\in\mathbb{R}^{n}$.
    Then the unique solution $\displaystyle \varphi(\cdot,s)$ of the initial value problem
    $\displaystyle \begin{cases}X^{\prime}=A(t)X+F,\ r\geq t\geq s\geq0\\ X(s)=X_{0}\end{cases}$,
    where $\displaystyle r>0$ is fixed, is continuous in $\displaystyle s$.

    I would be very glad if you can help me in this direction.
    Thanks.
    bkarpuz
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  2. #2
    Senior Member bkarpuz's Avatar
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    Re: Continuous dependence on the initial point

    Okay, I got the proof. Please let me know if you think that it is wrong.
    Proof. Set $\displaystyle Y(t):=\varphi(t,s_{2})-\varphi(t,s_{1})$ for $\displaystyle t\geq s_{2}\geq s_{1}\geq0$. Then, for all $\displaystyle t\geq s_{2}$, we have
    $\displaystyle Y(t)=\bigg(X_{0}+\int_{s_{2}}^{t}A(u)\varphi(u,s_{ 2}) \mathrm{d}u\bigg)-\bigg(X_{0}+\int_{s_{1}}^{t}A(u)\varphi(u,s_{1}) \mathrm{d}u\bigg)$
    ......_$\displaystyle =\int_{s_{2}}^{t}A(u)Y(u) \mathrm{d}u-\int_{s_{1}}^{s_{2}}A(u)\varphi(u,s_{1})\mathrm{d} u,$
    which yields
    $\displaystyle \|Y(t)\|\leq\int_{s_{2}}^{t}\|A(u)\|\|Y(u)\| \mathrm{d}u+\int_{s_{1}}^{s_{2}}\|A(u)\|\|\varphi( u,s_{1})\| \mathrm{d}u$
    ......___$\displaystyle \leq\bigg(\int_{s_{1}}^{s_{2}}\|A(u)\|\|\varphi(u, s_{1})\| \mathrm{d}u\bigg)\exp\bigg\{\int_{s_{2}}^{t}\|A(u) \| \mathrm{d}u\bigg\},$
    where we have applied the Gronwall's inequality in the last line.
    This shows that picking $\displaystyle s_{2}$ sufficiently close to $\displaystyle s_{1}$, the right-hand side of the above inequality can be made sufficiently small. This proves continuity of $\displaystyle \varphi(\cdot,s)$ in $\displaystyle s$ follows.
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  3. #3
    A Plied Mathematician
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    Re: Continuous dependence on the initial point

    I don't have the book with me, but this sort of thing is in Coddington and Levinson. Check that out.
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