As we know, the vastness of the subject is realized by the variety of interdisciplinary subjects that belong to several mathematical domains such as classical analysis, differential and integral equations, functional analysis, operator theory, topology and algebraic topology; as well as other subjects such as Economics, Commerce etc.
Recently fixed point theory in the field of control and communication engineering has broad application. mostly by imposing distributed computing and game theory resource allocation scheme which is now growing fast to prepare the required theories and assumptions for the mentioned two disciplines. I guess the modern fixed point theorem such as algebraic topology is the most promising approach to maturate the next generation of advanced fixed point theorems for high end engineering applications.
Fixed point theory is quite useful in the existence theory of differential, integral, partial differential and functional equations. This is a basic mathematical tool used in showing the existence of solution in game theory and mathematical economics. Applications in best approximation theory, optimization problems, variational inequalities, complementarity problems and eigenvalue problems are well known. Fixed point theory is a very important tool used in nonlinear problems arising is diverse field like mathematical economics, game theory, biology, engineering and physics, and is backbone of the nonlinear analysis.
The recent applications are seen in Communication engineering and some other engineering branches.
Very interesting applications of fixed point theory are related to stability of dynamic systems of any type ( continuous- time, discrete- time, hybrid of both of them, those involving time-delayed dynamics etc.) . Of course, the same applications can be focused on for any kind of differential equations ( ordinary, functional , partial - derivative...) / or their difference/ or their hybrid counterparts. It can be noted that fixed points of a differential system or differential / difference/ hybrid equations are also (non-necesarily stable) stability points. The limit cycles of the above structures could also be studied in that way ( roughly speaking, a limit cycle is an asymptotic closed solution which can be formed by a sequence of points of a certain mapping , each one of them being a limit ( fixed point) of one of the subsequences generated by the mapping which defines the trajectory solution.
Concepts and memes are fixed points in human communication field; mutual understanding (emergence of language) is also an example of fixed points in humanity's communication field
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