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Math Help - existence of solutions using Gronwall

  1. #1
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    existence of solutions using Gronwall

    I must complete the following exercise by the end of tonight:

    Let f\in C((a,b)\times\mathbb{R}^n,\mathbb{R}^n) such that

    |f(t,x)|\leq p(t)|x|+q(t), (t,x)\in(a,b)\times\mathbb{R}^n,

    where p,q are continuous on (a,b). Show that the solutions of the IVP

    x'=f(t,x), x(t_0)=x_0

    exist on (a,b).
    It is suggested that we use the following formulation of the Gronwall inequality:

    Suppose that u(t) is a continuous solution of

    \displaystyle u(t)\leq f(t)+\int_{t_0}^t h(s)u(s)ds, t\geq t_0,

    where f,h are continuous and h(t)\geq 0 for t\geq t_0. Then

    \displaystyle u(t)\leq f(t)+\int_{t_0}^t f(s)h(s)\exp\left(\int_s^t h(\sigma)d\sigma\right)ds, t\geq t_0.
    Also, I suspect (though I am not certain) that the following theorem, part of the unit of this exercise, might be useful:

    Let D=(a,b)\times\mathbb{R}^n, -\infty\leq a<b\leq\infty, t_0\in(a,b), f\in C(D,\mathbb{R}^n), and f is bounded. Then every solution of x'=f(t,x), x(t_0)=x_0, exists on (a,b).
    Any help would be much appreciated!
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  2. #2
    Senior Member bkarpuz's Avatar
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    Quote Originally Posted by hatsoff View Post
    I must complete the following exercise by the end of tonight:
    ...
    Any help would be much appreciated!
    This may help you
    http://www.mathhelpforum.com/math-he...ons-91772.html
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