# NOn-linear equation, when has a solution?

• tpm

#### tpm

NOn-linear equation, when has a solution??

For linear system of equations:

$$A_{ij}x^{i}=b_{j}$$ (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:

$$f(x_{i},x_{j} , x_{k})=b_{j}$$ ??

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:

$$\int_{0}^{\infty} K(x,y,f(x),)dy = g(x)$$

has a solution applying a 'quadrature method' ??

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.