Dear Naceri Mostepha,

I agree with George Stoica. Furthermore, a numerical method may help you to find approximate solution(s) to investigate workability depends on which method you use for. Fixed point results provide conditions under which maps have solutions. However, the computation time and efficiency of the method must be proven by some illustratives since it is quite important in computation field.

 Dear Naceri Mostepha,

I think numerical methods are used to find the approximate solutions of a certain differential equations. But on the other hand fixed point theory provides some conditions on which a certain map have a unique solution or still have at least one solution.

Also numerical methods approximate to the exact solutions. But fixed point theory does not tell us about the exact solutions of a problem. It only gives us information about the existence of solutions.  On behalf of a numerical scheme we can tell that what kind of a certain differential equation have exact solution.

For example

If a differential equations have a solution of a parabolic type then on the basis of a numerical scheme using approximate data we can easily judge the shape of the exact solution.

But fixed point theory only tells us about the existence and it is silent to say anything on the approximate shape of an exact solution.

Kind regards

Imran Talib

Existence of a fixed point is therefore of paramount importance in many areas of Mathematics and other Sciences. Fixed point results provide conditions under which maps have solutions. The theory itself is a beautiful mixture of "applied and pure analysis", topology and geometry. Numerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations.