Agree with George Stocia, 

Its nice and useful generalization is in fuzzy setup and its nice application in fuzzy filters, image processing, solution of functional,ntegral,fuzzy differential equation etc...

Dear Colleague,

Take a closer look at the monograph

V.G.Angelov "A Method for Analysis of Transmission Lines Terminated by Nonlinear Loads" Nova Science, New York, 2014.

www.nova.publishers.com

Dear Colleague,

Take a closer look at the Book

Fixed-Point Algorithms for Inverse Problems in Science and Engineering

Editors: Heinz H. Bauschke, Regina S. Burachik, Patrick L. Combettes, Veit Elser, D. Russell Luke, Henry Wolkowicz

ISBN: 978-1-4419-9568-1 (Print) 978-1-4419-9569-8 (Online)

Fixed point theorems have proved to be very powerful method to prove existence and under certain conditions in many areas of analysis from dynamic systems (Poincre-Birkhoff theory) to non-linear and/or functional ordinary and partial differential equations and integral equations. They have also provided a powerful tool in topology and global analysis (Lefschetz fixed point theorem and later the Atiyah-Bott fixed point theorem applied to general Elliptic Complexes on a smooth manifold) . They have even found their way into algebra (Knaster-Tarski fixed-point theorem. ).

Between the Contraction Principle and particularly the Tychonoff fixed-point theorem, there have been many applications to non-linear and functional differential equations, both ordinary and partial that arise in applied sciences. Although the Contraction Principle gives a constructive proof of a solution to such problems, the Tychonoff fixed-point theorem gives more general results.  

In his wonderful book, "Ordinary Differential Equations,"  Phil Hartman has two chapters devoted to the functional analytic aspects of ordinary differential equations and the used of fixed-point theory as a powerful tool. That would be a recommended start.  

Applications of Banach's Contraction Principle and its generalizations in Applied Fields---engineering (image and signal reconstruction and decompression problems), computer tomography and radiation treatment planning (convex feasibility problems), astronomy (adaptive optics), crystallography (molecular structure reconstruction), computational chemistry (molecular structure simulation) , . ., antenna optimization, location problems

Dear colleagues,

Thank you very much for these informations, I think that the Banach principle has various kinds of generalizations and utilizations. Recently Burton gave a new method based on fixed point technique, to study the stability of integral or differential equations with delay, which replaced the classical method of Lyponov function, also Amini Herandi applied Caristi type fixed point in the optimization, and there is a group in Naresuan University (Thailand) worked in this direction ( applications of fixed point approach in optimization theory) as Poom Kumam and others.

As prof  Truman Prevatt    mention "' Between the Contraction Principle and particularly the Tychonoff fixed-point theorem, there have been many applications to non-linear and functional differential equations, both ordinary and partial that arise in applied sciences''.

A closed ball in finite dimensional space has FPP (fixed point property)and it is also compact.

Every set which homeomorphic to a closed ball in finite dimensional space has FPP.

Then every bounded  closed convex set  in finite dimensional space has FPP

What is generalization  in infinite dimensional space. It seems natural to consider  a convex set K  in a Banach space and continuous mapping  T of K intoself.

But we need  we need to suppose that set or its image is compact .

Here we will shortly mention  the Schauder fixed point theorem, which is applicable  to topological vector spaces.

The Schauder fixed point theorem is an extension of the Brouwer fixed point theorem to topological vector spaces, which may be of infinite dimension. It asserts that if  {\displaystyle K} is a nonempty convex subset of a Hausdorff topological vector space   {\displaystyle V} and  {\displaystyle T} is a continuous mapping of   {\displaystyle K} into itself such that   {\displaystyle T(K)} is contained in a compact subset of {\displaystyle K} , then  {\displaystyle T} has a fixed point.

A consequence, called Schaefer's fixed point theorem, is particularly useful for proving existence of solutions to nonlinear partial differential equations. Schaefer's theorem is in fact a special case of the far reaching Leray–Schauder theorem which was discovered earlier by Juliusz Schauder and Jean Leray. The statement is as follows:

Let T be a continuous and compact mapping of a Banach space  {\displaystyle X} into itself, such that the set

{ x ∈ X : x = λ T x }  for some  0 ≤ λ ≤ 1

is bounded. Then  {\displaystyle T} has a fixed point.   For applications of FPT  in PDE  see  for example

Elliptic Partial Differential Equations of Second Order.

by   Gilbarg, David, Trudinger, Neil S.

See also Browder-Kirk theorem in which we do not suppose compactness hypothesis.

At this moment we will illustrate   applications  of FPT to  the Existence and Uniqueness Theorem for first-order ordinary differential equations.

''One of the most important theorems in Ordinary Differential Equations is Picard’s

Existence and Uniqueness Theorem for first-order ordinary differential equations. Why is

Picard’s Theorem so important? One reason is it can be generalized to establish existence

and uniqueness results for higher-order ordinary differential equations and for systems of

differential equations. Another is that it is a good introduction to the broad class of existence

and uniqueness theorems that are based on fixed points.''

Consider  the equation y'(x)= f(x,y), y(x_0)=y_0.   Suppose that  

I f(t,y_1) -f(t,y_2) I\leq L |y_1-y_2|  on some rectangle  [x_0 -e,x_0 +e]x [y_0 -d,y_0 =d,] around (x_0,y_0).

Idea of proof is  to consider  y(x)=y_0+\int_{x_0}^x f(t,y(t) dt  and the operator  T[y](x):= y_0+\int_{x_0}^x f(t,y(t) dt .

It is fascinating  that  there is small neigh  [a,b]  of x_0  that  T is a contraction on C[a,b]  with max norm (if f is Lipschitz)! Here we can understand the power of  Banach contraction principle    and also that the space  C[a,b] with max norm can have an important  role (it is not only mathematical imagination)

For applications of  The Schauder fixed point  see Gilbarg-Trudinger book.

Recall Czeslaw Bessaga \cite{bessaga}, from 1959:

\begin{thm}

(i)Let $f : X \rightarrow X$ be a map of an abstract set such that each

iterate $f^n$ has a unique fixed point x_0. Let $q \in (0, 1)$, then

(ii) there

exists a complete metric on $X$

such that $f$ is contractive, and $q$ is the contraction constant.

\end{thm}

Example. Let f(z)= e^{i \alpha}$, where \alpha is an irational multiple of \pi and X =U the unit disk.

Bessaga (i) implies (ii). But now (ii) implies (iii) each

iterate $f^n(x)$ converges to x_0.

As far as I can see my example satisfies (i), but it does not satisfy

(iii).

I discussed the subject with prof. Pasiki.

Perhaps all circle C_r= $z: |z| = r $, 0

There is an application of FPT to economy and in particular to

Game theory.

We consider a game with

n≥2 players, under the assumption that the players do

not cooperate among themselves. Each players pursue a

strategy, in dependence of

the strategies of the other players. Denote the set of all possible strategies of the

kth  player by   K_k, and set

K=K_1×···×K_n. An element   x∈K  is called a  strategy   profile. For each k, let  f_k: K→R

be the   loss function  of the

k-th player. If

\sum_{k=1}^n f_k(x) = 0,∀x∈K (1)

the game is said to be of

zero-sum. The aim of each player is to minimize his loss,

or, equivalently, to maximize his gain.

Definition

A Nash equilibrium

is a strategy profile with the property that no

player can benefit by changing his strategy, while all other players keep their strategies unchanged.

We need of course further hypotheses on the sets K_k

and on the maps f_k. It is reasonable to assume that, with all the other strategies fixed, the loss function f_k has a small variation in correspondence of a small variation of    x_k. Also, loosely speaking, it is assumed that the average of losses corresponding to two different strategies of the k^th

player is grater than the loss corresponding to the “average”

strategy. Convexity can suitably translate this issue.

The fundamental result of game theory is

here that there is a Nash equilibrium.

[von Neumann] The hypotheses can be weakened if we consider a two-player zero-sum game (sometimes called a duel).

one of the important application of Banach's Contraction Principle is in integral equations. It is considered as a source of existence and uniqueness theorems in integral equations.  It is also used in solving systems of linear algebraic equations, that may arise in vibration problems , by iteration methods 

Ali Turab  asked a question  related to  the paper:

VASILE BERINDE, and ABDUL RAHIM KHAN, On a functional equation arising in mathematical biology  and theory of learning, CREAT. MATH. INFORM. 24 (2015), No. 1, 9 - 16.

It seems that it is possible to extend the main result of this paper.

Let $E$ be Lipschitz space of functions $f$ on $[0,1]$ with $f(0)=0$ and $f(1)=1$.

We consider the operator $T : E \rightarrow C[0; 1]$, defined by

$(T\varphi)(x) = x\varphi(f(x)) + (1 -x)\varphi(g(x))$, $x \in [0; 1]$ (2.16)

where $f, g : [0; 1] \rightarrow [0; 1]$ are contractions with the contraction coefficients $\alpha$ and $\beta$ satisfying

the ''one side'' boundary conditions (1.6). Obviously, $T$ maps $E$ into itself and the

functional equation (1.7) is equivalent to the fixed point operator equation

$T\varphi = \varphi$.

At this moment it seems to me that it is useful to use that Lipschitz space can describe by f is AC on [0,1] and $f'$ is esentially bounded on [0,1].

Then the proof that $Tf$ is a contraction on E with Lipschitz norm is easier.

Idea: $(T\varphi)'(x)= I_1 + I_2 =\varphi(f(x))-\varphi(g(x)+ x \varphi(f(x))' + (1 -x) \varphi(g(x))' $ , where

$I_1=h_1- h_2$, $I_2= x (h_1)'+ (1 -x) (h_2)'$

$h_1=\varphi\circ f$, $h_2=\varphi\circ g$.