There is a nice general book by the MAA that is cheap called Fixed Point Theorems (I think). In my field they are used to prove existence and uniqueness of solutions to ordinary differential equations or to prove that iterative numerical schemes converge to what they are supposed to converge to.

Any theorem which, in some category, states, that for f:X\to X there is an x\in X with f(x) or for some f:X\to P(X) there is an x\in X with x\in f(x) would be called a fixed point theorem. More precise statements about the set of all fixed points, that is \{ x\in X | x=f(x) \} or \{x\in X | x\in f(x)\}, respectively, would be entitled to the name 'fixed point theorem' as well.

There are, broadly speakting, two types of fixed point theorems:

1. Constructive fixed point theorems (e.g. Banach fixed point theorem) which not only claim the existence of a fixed point but yield an algorithm, too (in the Banach case fixed point iteration x_{n+1}=f(x_n)). Banach's fixed point theorem is omnipresent in analysis, both for existence proofs (Picard-Lindelöf) as for numerical schemes that allow the approximation of solutions of (differential, integral,...) equations. Any equation that can be written as x=f(x) for some map f that is contracting with respect to some (complete) metric on X will provide such a fixed point iteration. When it comes to medical science, the Banach fixed point theorem will be often (implicitly or explicitly) present when some relevant biological or chemical process is modelled by equations. I you use numerical software, it probably relies in some parts on this fixed point theorem. Some of the algorithms for the reconstruction of a 3d-image from tomographic data use fixed point iterations...

2. Non-constructive fixed point theorems

Fixed point theorems like Brouwer's, Schauder's, Kakutani's, Lefschetz', Knaster-Tarski, etc. will provide criteria for the existence of a fixed point (and, for example in the case of Lefschetz, allow an estimate of the number of fixed points, if these are sufficiently regular). The existence of a fixed point, that is the existence of the solution of some equation x=f(x) or x\in f(x) is often an important piece of information, though these theorems typically do not boil down to an algorithm which finds a solution. They will contribute to a qualitative understanding of the models but can also serve as a basis for decisions about where to look for solutions of equations. As to real world applications, there are famous examples like the existence of market equilibria in economics; for medicine it is less obvious. But, as medical science is full of dynamical systems, I suppose that these theorems will be used in relevant mathematical models. (By a rapid search I found, for example, papers on the modelling of blood coagulation that use fixed point theorems, and more examples from "mathematical biology". But I really do not know 'medical sciences', so I have to refrain from any further claims.)

I'm afraid, most of this can be found in any encyclopedia. I hope it may be useful, though.

Well, here is one very quick answer: the theory is very rich, and so is the variety of applications. Here are links to some journals with a strong focus on your area of interest:

http://www.springer.com/mathematics/analysis/journal/13663

http://www.springer.com/birkhauser/mathematics/journal/11784

http://www.pphmj.com/journals/jpfpta.htm

By simply browsing the journals' home page you will find several applications, including medical science, including mathematical biology.

There are also a fair number of basic books on the subject: see, for example the site

http://www.drkhamsi.com/fpt/books.html

which lists some books appearing before 2004.

Further, one should note that fixed point theory is a natural ingredient in the support of analysis and solutions methods for variational inequality and nonlinear optimization problems appearing in applications, and for such a book, I would take a look at this one:

http://www.amazon.com/Fixed-Theory-Springer-Monographs-Mathematics/dp/0387001735

It has been praised by many.

yours

Michael

There is a nice general book by the MAA that is cheap called Fixed Point Theorems (I think). In my field they are used to prove existence and uniqueness of solutions to ordinary differential equations or to prove that iterative numerical schemes converge to what they are supposed to converge to.

In brief, fixed point theory is a powerful tool to determine uniqueness of solutions to dynamical systems and is widely used in theoretical and applied analysis. So it must be applicable to mathematical biology as well.

I can say that when you confront with a problem in mathematics which its aim is to find a unique answer you and use fixed point theory. In medicine or engineering or others

Farshid, that means of course that one would try to establish that the problem can somehow be equivalently transformed into the solution to a fixed-point problem, for which you show that the problem-defining mapping is contractive.

That a problem has a unique solution can of course be established also by other means. In the case of an optimization problem is it enough to establish that the feasible set is closed and convex and the objective function is continuous, weakly coercive (tends to plus infinity when the norm of the variable vector does), and strictly convex on the feasible set.

An answer to one of the original questions:

Applications to medical science do exist. One such application is to feasibility and inverse optimization problems arising in the context of finding good radiation dose plans for the cure of cancer. These problems can often be formulated as inverse or feasibility problems, for which fixed-point theory provides algorithms, or arguments for the convergence of various iterative methods. Here is one out of many papers on the subject (and Y. Censor has written plenty): http://faculty.uml.edu/cbyrne/cekb05.pdf

There is a nice book dedicated to fixed point theorems and their applications :

"Zeidler, Eberhard. Nonlinear functional analysis and its applications. I. Fixed-point theorems. Springer-Verlag, New York, 1986."

For applications to economics and game theory see for example "Border, Kim C. Fixed point theorems with applications to economics and game theory. Cambridge University Press, Cambridge, 1989."

... and of course that just mentioned volume by Eberhard Zeidler is accompanied by an additional four volumes, with a slightly stronger focus on applications:

2A on Linear mootone operators,

2B on Nonlinear monotone operators,

3 on Variational methods and optimization, and

4 on Applications to mathematical physics.

I know a very nice application of fixed point theory in medical sciences:

Please read the following paper which I attached.

Great--than you.You learn every day...

As a direct application of the fixed point theorem (successive approximation method), I recall approximating the solutions of some scalar or operatorial equations which cannot be solved exactly. Here it should be mentioned that in some cases, the iteration from the contraction principle is the same as that of the Newton's method. As "theoretical applications", I could mention the proof of the existence and uniqueness of the solution of the Cauchy problem for differential equations (and systems of equations), as well as the proof of the implicit function theorem.

Most of the physical problems of applied sciences and engineering are usually formulated in the form of fixed point equations. This field containing  the solutions of some scalar or operational equations which cannot be solved exactly specially random fixed point theory. Also using the iteration schemes to approximate the fixed point which it consider the solution of  differential equations, fractional differential equations, nonlinear integral equation and many others........................

The fixed point theorem based on the contraction principle is applied usually in proving the existence and uniqueness of the solutions of some scalar and mainly functional equations (for example differential and integral equations). Other fixed point principles are applied in solving moment problems (for example, see the paper of Christian Berg and A. J. Durán, "The fixed point for a transformation of Hausdorff moment sequences and iteration of a rational function". Math. Scand. 103 (2008), pp. 11-39.

Disregard the introduction and continue to Section 2, in the attached file. ..find an application of the logistic map, which shows that there is less chaos when the dimensions of the variables in the iteration scheme are kept.

https://www.researchgate.net/publication/312053900_Concepts_of_nco_related_to_cosmos_states_Feynman-collapse_and_bifurcations_in_the_logistic_map

If you haven't already browsed the page below, please do - it is filled with nice applications in medicine of various mathematical optimization approaches:

http://math.haifa.ac.il/YAIR/censor-imrt.html

I would like to know what applications fixed point theory has in financial science, economic and banking. I need them for a new research field

The excellent discussion of the role of a fixed point theorem in economy is presented in the book by McLennan

http://cupid.economics.uq.edu.au/mclennan/Advanced/advanced_fp.pdf

The book is written in the excellent style and it shows multiple connection between the various fields of mathematics. I'm sure that this text can suggest also many new subjects of research in the field of economy.

Whenever we have to find a unique solution of any process then fixed point theorems can be applied easily. For example if we want to converge a sequence to a fixed number we can apply approximations with the help of fixed point. Whenever we want to find a unique solution for a process of integration and differentiation we can apply it whenever we want to find an optimum solution in a game (saddle point) we can apply fixed point theorems and many more examples are there.

One of the most common applications of fixed-point theory is to demonstrate the existence of solutions of some differential equations by achieving certain conditions.

A book which contains an entire chapter devoted to fixed points theorems, including applications, is: "Functional Analysis", by L. V. Kantorovich and G. P. Akilov. I do not know whether a translation from Russian  to English has been published. A more recent paper on the moment problem and related problems, which uses a basic fixed point theorem is: "The fixed point for a transformation of Hausdorff moment sequences and iteration of a rational function", Math. Scand. 103 (2008) 11-39, by Christian Berg and Antonio  J. Duran. For applications of ordered vector spaces theory to fixed point  and further related results see: "Ordered Vector Spaces and Linear Operators", Academiei, Bucharest and Abacus Press, Tunbridge Wells, Kent (1976), by Romulus Cristescu. Observe that a differential equation is usually equivalent to an integral equation. Fixed point theorems have applications to such equations, but also to some other equations or systems of equations.

Hello Sachin, Please refer any "Functional Analysis Book" for Fixed point theory. This theory is most useful for the Data Mining, Wavelet Analysis, Data filtering, etc......

I am not sure about your second question, but here I'll let you two applications of fixed point theorems.

1. By a fix point theorem argument, one show that it is possible to find at any time two diametrically opposed points on Earth having simultaneously the same temperature and pressure.

2. The a version of a fix point theorem is also used to prove the existence of mixed Nash equilibriums in multiplayer games.

You can also check this link: https://fixedpointtheoryandapplications.springeropen.com/

Fixed point theorems provide sufficient conditions under which there exists a fixed point for a given function, and thus allow us to guarantee the existence of a solution of the original problem.

Many problems of physical world are modelled in the form of operator equations. The fixed point equation Tx= xis one among them. Fixed point theorems are very important tools for proving the existence and uniqueness of solutions to various mathematical models, differential, integral, partial differential equations and variational inequalities etc. representing phenomena arising in different fields, such as steady state temperature distribution, chemical equations, neutron transport theory, economics theories and flow of fluids. They are also used to study the problems of optimal control related to these systems. Fixed point theorems of ordered Banach spaces provide us exact or approximate solutions of boundary value problems.

Fixed point theorems have numerous applications in mathematics. They are also used in new areas of mathematical applications equation in mathematical economics, game theory, communication network space etc. In economics fixed point is used in formal role of predicting how the game will he played and it explain the strategy, which produces the most favourable outcomes for player. This advancement in fixed point theory diversified the applications of various fixed point theorems in different areas such as the existence theory of differential and integral equations, dynamic programming, fractal and chaos theory, discrete dynamics, population dynamics, differential inclusions, system analysis, interval arithmetic, optimization and game theory, variational inequalities and control theory, elasticity and plasticity theory, mathematical economics, engineering, physics, biology, chemistry, partial differential equation, approximation theory and other diverse disciplines of mathematical sciences. The method can be applied not just to numerical equation but also to equations involving vector or function.

In mathematics, a fixed-point theorem is a result saying that a function F will have at least one fixed point (a point x for which F(x) = x), under some conditions on F that can be stated in general terms. One of the remarkable application of Fixed point theorem in Numerical Analysis, is the Fixed point algorithm for solving nonlinear equation (system).

There are fixed point medical applications. you can see the following link

https://arxiv.org/pdf/1709.00810.pdf

have applied in chaios theory