NOn-linear equation, when has a solution?

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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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  • #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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