It is said that fixed point theory has lot of applications not only in the field of mathematics but also in various disciplines. Which one is the most important?
Please see links
http://www.springer.com/mathematics/analysis/journal/13663
http://www.springer.com/birkhauser/mathematics/journal/11784
http://www.pphmj.com/journals/jpfpta.htm
The importance of the fixed point theory lies mainly in the fact that most of the equations arising in the various physical formulations may be transformed to fixed point equations or inclusions. The theorems concerning the properties and existence of fixed points are known as
fixed point theorems.
I think may be the application to differential equations or integral equations.
Applications to non linear differential and integro-differential equations - many.
Applications to algeria geometer - Borel Fixed Point theorem.
Applications to Game Theory - Browder Fixed Point Theorem.
Applications to non-zero sum game theory and particularly the Nash equilibrium in economics - Kakutani Fixed Point Theorem.
Applications to geometry and topology of manifolds - Atiyah-Bott Fixed Point Theorem.
These are just a few of the many types of fixed point theorems and applications.
For a more complete list see.
https://en.wikipedia.org/wiki/Fixed-point_theorem
The analysis of nonlinear relationships in systems that arise in the sciences, engineering and even the social sciences (e.g., economics) often end up being expressed in the terms of nonlinear equations and/or mappings and the solution is a fixed point of such a mapping. My paper - available on Research Gate - on asymptotic integration of a large class of non-linear functional differential equations - is a good example of the power of fixed point theorems in a addressing complex not linear problems.
In physics, chemistry, and biology, there are many interesting problems that lead to a differential equations. Such differential equations sometimes can be reduced to the existence of a fixed point for a function satisfying certain properties.
In addition, there are particular real-life problems, whose statements are fairly easy to understand, that can be argued using some version of the Fixed-Point Theorem. Here is one of my favorites: at any time, there exist two diametrically opposed places on our planet having exactly the same temperature. Actually, we can find two diametrically opposed places in our planet having the same temperature and the same pressure simultaneously (see Borsuk–Ulam theorem).
On the other hand, the (Brouwer) Fixed-Point Theorem is also used to prove the existence of mixed Nash equilibriums in multiplayer games.
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