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NOn-linear equation, when has a solution?

  1. Apr 3, 2007 #1

    tpm

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    NOn-linear equation, when has a solution??

    For linear system of equations:

    [tex] A_{ij}x^{i}=b_{j} [/tex] (implicit sum over repeated indices)

    a necessary and sufficient condition to exist is that |detA| >0

    but what happens whenever you have a Non-linear equation:

    [tex] f(x_{i},x_{j} , x_{k})=b_{j} [/tex] ??

    How do you know it will have a solution or not??...

    the problem arises mainly in NOn-linear equation theory..how do you know that equation:

    [tex] \int_{0}^{\infty} K(x,y,f(x),)dy = g(x) [/tex]

    has a solution applying a 'quadrature method' ??
     
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  3. Apr 3, 2007 #2

    arildno

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    Well, for non-linear diff.eq systems, you may prove, in special cases, the existence of a unique solution function in some neighbourhood of the initial value.

    A crucial tool in deriving this, is the use of a contractive mapping procedure.

    For algebraic non-linear systems, in so far as you can prove you get a contractive mapping by iteration should also guarantee a solution.

    Perhaps there exist less crude tools, I dunno.
     
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