NOn-linear equation, when has a solution?

[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' ??

 

[arildno]

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.

 

[tacman]

In general, there is no easy way to determine if a non-linear equation has a solution. Unlike linear equations, there is no simple condition like |detA| > 0 to check for existence of a solution. Non-linear equations can have multiple solutions, no solutions, or even an infinite number of solutions.

One approach to solving non-linear equations is through numerical methods such as the quadrature method mentioned. These methods involve approximating the solution by breaking the equation into smaller, simpler pieces and iteratively finding a solution. However, these methods may not always work and may require some initial guess or approximation of the solution.

Another approach is to use analytical techniques such as substitution or elimination to simplify the equation and potentially find a solution. However, these techniques may not always be possible or may result in a very complex solution.

Ultimately, the best way to determine if a non-linear equation has a solution is to try different methods and see if a solution can be found. If a solution cannot be found, it may indicate that the equation does not have a solution or that more advanced techniques are needed to find one.