Fixed Point Theory - Science topic

Thank you very much for your valuable views @Dr. Oluma Ararso

I am interested in the applications of Fixed Point Theory to differential and integrodifferential equations (fractional o generalized). Can be?

To the best of my knowledge, there is not such a book. Maybe professor James F Peters would know better.

Although a classical result, Urysohn theorem only provides a sufficent condition for a space to be metrizable. Nagata-Smirnov theorem gives both sufficient and necessary conditions.

Another common way is to show that a uniformity generating a given topology admits a countable base, see e.g.

Also, we have the so called Nemytzki-Wilson metrization criterion, which is very useful for spaces endowed with a distance function, see e.g.

Yes I am

I can help you with two possible ways to obtain new results:

1. Extending the space, as Issam Kaddoura said. Generally, fixed point theorems are given in complete metric spaces, particularly Banach spaces or Hilbert spaces. So, it's new to study the fixed point (unique or not) problem in non-metrizable spaces. For exemple, if E is a locally convex space and T:D(T)->E is an operator( D(T) can be E itself), you can study properties of E and T so that T have fixed points(unique fixed point).

2. If X is a Banach space and T:X->X is some operator having fixed points, you can study the problem of finding operators S:X->X so that operators T+S, T-S, TS, ST have also the fixed point property.

You can visit my Researchgate page to read some of my papers about:

- Fixed Points for Perturbed Contractions, Fixed Point Theory, 2011

- On Firmly Nonexpansive Perturbations of the Identity in Hilbert Spaces, Georgian Mathematical Journal, 2014

- On the Difference of a Contraction and an Inverse Strongly Monotone Operator, Operators and Matrices, 2017

No equilibrium points.

Examples on practical applications in fractional boundary value problems;

Fractional differential equations /models are applied in different fields of studies: mathematics, physics, chemistry, biology, medicine,…

They are used to describe different physical process such as dynamics of blood flow, wave motions, elasticity,…

For more information, read attached files!

You may want to learn about Floer Lagrangian Homology. It aims to study the minimal number of intersections of two Lagrangian submanifolds in a symplectic manifold, in particular, it gives you information about the minimal number of fixed points of Hamiltonian diffeomorphisms (and hence, stable orbits in geometric dynamical systems). Also, there is an old (but extremely interesting) paper by Moser where the topic of fixed points is explored in some detail:

I provide you a ver simple example. :Let Tx=2x then d(Tx,Ty)=2|x-y|>|x-y|=d(x,y). But T has a unique fixed point "0".

You must rearrange the original functional equation in the form fixed point equation. Then using Banach or other related result to guarantee the existence of fixed point. Actually this fixed point is the solution of the original functional equation. To find this fixed point we use Picard, Mann or Ishikawa iterations.

Please download the paper "The new modified Ishikawa iteration method

for the approximate solution of different

types of differential equations"

Fixed point theory solved many nonlinear problems. In the future, it will play more crucial role in many real world problems.

There are some papers available on the research-gate and internet. However the most good generalization is the Zemfirisco Fixed Point Theorem in this direction.

I suggest you the stability of the iteration processes.

Please se my recent paper, "Approximation of fixed points and best proximity points of relatively nonexpansive mappings".

I suggest you the 2-uniformly convex hyperbolic spaces.

Hello, try this book as an initiation : Iterative Methods for Linear

and Nonlinear Equations, C. T. Kelley, North Carolina State University, Philadelphia 1995.

No. It's not true. The statements are contradictory. Instead we say If M(x,y,t) >M(p,q,t) then M(p,q,t) <=M(x,y,t)

Please don't forget about many(!) applications of partitions of unity in differential inclusions. It can be found in any book on the subject.

Excellent answers

James F Peters, Artur Sergyeyev Thank you very much Professors for your king suggestions. However, I am looking for where I can get a free e-copy of the book.

Yes. You can do this.

Linear stability analysis is applied to the problem of a density-stratified fluid contained in an inclined slot being subjected to a lateral temperature gradient. Stability equations are solved using the Galerkin technique with 12 terms in the truncated expansion series. Within the range of 8 considered, < 75", critical instability was found to be of the stationary type. Results of critical thermal Rayleigh numbers and

wavenumbers at all inclination angles are in good agreement with the experimental results obtained earlier (Paliwal & Chen 1980). Contrary to intuition, these results show that the system is more stable when the lower wall is heated. This is shown to be the result of the increased vertical solute gradient in the steady state prior to the onset

of instabilities when the heating is from below.

see the following references

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.

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.

Please look at the attached fille. Perhaps it will be helpful. The published article can be find at

Joachim> I have got into some blackout state,

I assumed you were just temporarily misled by your "constant restoring force" (V = |y|) analysis :-)

Actually, the problem of how the return time varies with (kinetic) energy is quite interesting and complicated, and varied. Which is why I hid behind the intuition phrase. It is easy to construct potentials where almost any behaviour can occur. So, I just believe that the specific differential equation under discussion leads to the behaviour I conjectured; other cases will very likely be different.

The equations analysed in the AMS paper can be interpreted as one-dimensional motion of a point particle influenced by a time and position dependent force. I am rather unimpressed by the results in that paper; they look like mechanical trivialities dressed in abstract mathematical jargon, decorated with complicating concepts and notation. (I think am allowed to say that, since I graduated is from a department of mathematics :-D)

One of the most practical areas of Mathematics is the theory of differential equations.  it plays a vital role in Physics and Engineering.  Fixed point theory guarantees that these equations have solutions.  This is one very practical result that should be noted.

I guess Newton's method for optimization qualifies: it is based on the solution of a system of linear equations, and a line search, both of which in general have multiple solutions. Even the very basic algorithm called "steepest descent" is an implicit such method, as it relies on a line search which may have more than one solution. In effect, most known methodologies within nonlinear optimization - at least - are implicit.

1. You must describe the mapp which has the fixed point - at least where this mapp is defined.

2. The topological and /or the metric properties of this mapp.

If you clear this particulars, you can receive an useful answer.

The answer is yes, you can do this, BUT the techniques are still sort of vendor-specific. For instance, Juniper and Cisco have their solutions, which can build the VPLS using BGP or MPLS (respectively RFC 4761 and RFC 4762).

This will give you a single, virtual layer 2 network, like a (potentially) giant single Ethernet. So the same considerations should apply as they would with a non-virtual large single LAN. Meaning, it's not necessarily a great idea!

Although the VPLS is built on an IP infrastructure, traffic within the VPLS does not need to be IP at all. Same as in any other layer 2 network.

Looks like Nokia offers a 1-week course on this subject, various dates, various places:

Radars are used to detect the moving objects that are faraway from the radar. That is to say it teledetects the objects. It loos lie seeing the objects by the eye of an observer.These objects are normally in air and their case is made from metal. So, the best way to detect them is by using  radio waves. in order to resolve the objects geometrically one uses microwaves radiators which emits in a relatively narrow angles and therefore., the transmitter radiates microwave, which propagates till meeting the object where it bounces back to the radar station where the receiver reside. So, in the radar station both the transmitter and receiver are present side by side. By receiving the bounced wave one can measure the delay time between the transmitted and the revived reflected wave , from which one can measure the distance between the radar station and the object.

If one exploits the absorption of waves such as in NMR, one would put the receiver behind the object to measure the transmittance for teledetection this would be practically impossible. 

This may shed some light on answering your question.

How are you sure that all VI problems are solved by KKM? have you read the book by Facchinei and pang? i have seen several usages of fixed point theorem to the existance and uniqueness of the solutions.

This is a good question.

In considering partial b-metric spaces, a good place to start is

See, also:

If I am not mistaken, you gave an example yourself:

Take ℂ as Hilbert space and define for every ω ε ℝ the continuous linear mulitplication operators A_ω :ℂ →  ℂ, z → A_ω(z)  = ωz, and set X:= {A_ω | ω ε ℝ}. Then X ⊂ L(ℂ) is a C*-Algebra and is isometrically canonically isomorphic to ℝ via isomorphisms φ : X → ℝ , A_ω → ω and ψ : ℝ → X, ω → A_ω, that is ψ = φ^{-1}. X is the algebra of real 1x1 matrices. Then, the mapping  Π :=ψ∘T∘φ : X → CB(X), where for I ⊂ ℝ φ(I):= {φ(x) | x ε I} and T is defined like in your question is an example for the sought for type of mapping.

Usually qestions of the type: „Can I define a map from one set to another?“ can usually be answered with a yes: As long as a mapping has no requirements to satisfy (other than being a mapping) you can define anything. For example I can define a map M that maps yourself to the set of all the questions you have asked on researchgate, that is

Muhammad →M(Muhammad) := {question | question was asked by Muhammad on RG}

That is a perfect set valued mapping.

:-P

Cheers

p.s.: The composition was wrong, I corrected the mistake

Another way would be to use the center manifold theorem. To simplify, we assume 0 is the only imaginary root of the characteristic equation (other cases can be dealt with in a similar way but the calculations can be too difficult) and that all the other eigenvalues have strictly negative real part (otherwise the fixed point would be unstable).

The center manifold theorem consists of reducing the local stability analysis of any DDE to that of a system ODEs (with dimension being equal to the multiplicity of zero as a root of your characteristic equation).

The calculations of the center manifold can be tedious but doable. The resulting system ODEs may be difficult to analyse though.

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.

What is the map here please clarify your question...

see links below

www.ncbi.nlm.nih.gov › NCBI › Literature › PubMed Central (PMC)

Hello, yes there is...it is Cauchy's integral. See attached paper for further details.

Ricci flow is:

μ ∂G_μν/∂μ=α’ R_μν

This equation describes how the metric changes with energy scale. Idea of Ricci flow was used by Perelman to prove Poncare conjecture. See http://arxiv.org/abs/math/0307245, and references in this paper.

The remarkable relationship between a diagonalized matrix, eigenvalues, and eigenvectors follows from the beautiful mathematical identity (the eigen decom position) that a square matrix A can be decomposed into the very special form

              A = PDP^-1                                                                                        (1)

where P is a matrix composed of the eigenvectors of A , D is the diagonal matrix constructed from the corresponding eigenvalues, and P^-1  is the matrix inverse of P . According to the eigen decomposition theorem, an initial matrix equation

             AX =Y                                                                                                (2)

can always be written

             PDP^-1 X=Y                                                                                        (3)                                                                                                                                                                 

(at least as long as P is a square matrix), and premultiplying both sides by P^-1    gives

            DP^-1 X=P^-1 Y                                                                                     (4)

Since the same linear transformation P^-1 is being applied to both X and Y , solving the original system is equivalent to solving the transformed system

            DX^' =Y^',                                                                                              (5)

where   X^'=P^-1 X   and  Y^' = P^-1 Y . This provides a way to canonicalize a system into the simplest possible form, reduce the number of parameters from nxn for an arbitrary matrix to n for a diagonal matrix, and obtain the characteristic properties of the initial matrix. This approach arises frequently in physics and engineering, where the technique is oft used and extremely powerful.

                                                                                           From Wolfram Math Worl

To the difference between Lobachevsky's geometry and Riemann's: the former is homogeneous, that is, you can transport a geometric figure of finite size without deforming it, from one place to the next. In other words, the curvature is constant, and, as turns out, negative. The other possibilities are constant positive curvature, corresponding to a sphere, and vanishing curvature, corresponding to a plane. Riemannian geometry, on the other hand, is a generalisation of the geometry of arbitrary curved surfaces, with (generally) non constant curvature, so that there are no congruences. It is thus very different from both Euclidean and non-Euclidean geometry.

I see, very nice. So the interval [0,1] is such a variety, isn't it? (take a function f : [0,1] ---> [0,1] and consider the function f(x) - x. If this function changes the sign, it must have a fixpoint by the intermeiate value property. If it does not changes the sign, then f(0) = 0 or f(1) = 1 is a fixpoint.) 

So on one hand, the fix point property for [0,1] has a proof which is quite similar to your problem, but for a special family of submanifolds N. On the other hand [0,1] is a variety with bord, and this helps the following construction: Take N to be the union of two closed straight line intervals of the plane, which do not intersect the diagonal of the square. One of them connects the point (1,0) with the point (1/3, 1/3 - epsilon), the other one connects the point (0,1) with the point (2/3, 2/3 + epsilon). Of course N projects surjectively on M, has dimension 1 as a manifold with bord (as M also) but does not intersect the diagonal.

So the property might be true, but possibly needs more conditions:  that M and N have no singularities (so both must be smooth) and possibly that both are connected. 

S^2 has not the fixpoint property because there is an antipodal map which is continuous and has no fixpoint. Also the torus S^1 x S^1 and the circle S^1 do not have the fixpoint property, because they have various rotations. Which example of variety with fixpoint property do you know excepting [0,1]? (it must be a smooth variety with local dimension = n for all points, unlike [0,1]...) For smooth surfaces of genus g \geq 2 I can also imagine kind of symmetries (similar to antipodal mappings) which have no fixpoint.

I second Peter's answer: it is often rather easy to transform a problem into a fixed-point one, BUT the properties of the mapping F in the fixed-point problem "F(x) = x" may have weird properties - in particular it may be far from contractive. So I would most often NOT force the original problem to be a fixed-point problem, but to treat it more like what it is when you first formulated in, be it a system of equations, a variational inequality, or indeed an optimization problem with or without constraints.

Note that our work is in progress  and  that  at this point   we give less or more heuristic arguments.

It seems  that  we can interpret  N²    as vertices of the  corresponding graph (nets)  and get a fixed point theorem.

 

First recall.  Suppose that

g maps N² into 1/2N², which satisfies 

(*)  |g(n+1,m) -g(n,m) | <= 1/2,    |g(n,m+1) -g(n,m) | <= 1/2,

   Then g has a fixed point.

For any topological space X the (Alexandroff) one-point compactification αX of X is obtained by adding one extra point ∞ (often called a point at infinity) and defining the open sets of the new space to be the open sets of X together with the sets of the form G ∪ {∞}, where G is an open subset of X such that X \ G is closed and compact.

Suppose  that N² ∪ {∞}      is   one-point compactification  of  N²  and

g maps N2 ∪ {∞},   into N² ∪ {∞}, which satisfies   (*)   on  N² .  Then g has a fixed point.

We can interpret  N²    as vertices of the  corresponding graph (nets).

When two vertices of a graph are connected by an edge, these vertices are said to be adjacent, and the edge is said to join them.

Every planar graph can be embedded on a sphere.

 

Using stereographic projection p  we can  embed    N² ∪ {∞} as subset  of the Riemann sphere S² .   Denote by M, M_1,  M* and   M*_1  the graphs  which are   p(N^2), p(1/2N^2),   p(N² ∪ {∞})   and    p( 1/2N² ∪ {∞}). If  g maps M* into M₁*   and has properties 

(X) if    two  vertices  of  M   are  adjacent   then  they have the same image or  their   images by g  are  adjacent  in  M₁.  Then  g has at least one fixed point in M*  .

Dear Arpit,

Due to some numerical problems (such as near zero elements), matrices could appear nearly singular in MATLAB preventing their exact inversion. To get around this problem,

pseudoinverse (which is a type of generalized inverse) could be helpful. MATLAB does this through the "pinv" command which calculates the Moore-Penrose pseudo inverse of a singular or even a non-square matrix A. It is based on singular value decomposition (SVD).

Its use is very simple:

B=pinv(A);

as opposed to exact inversion

B=inv(A).

A stable equilibrium point is when the state of the system ( often expressed as an energy functional, expressed say as f(x)) does not change as the system variables are changed. i.e. , the energy functional is at a minimum, df/dx<0. Of course an equilibrium point may not be stable, i.e. df/dx>0. When the dynamics of a system can  be expressed by a differential equation  say dh/dt=g(x), then the steady state of the system is when g(x)=0 (this may also represent a minimum in some appropriate energy functional). Often a fixed point iteration system is used to find the steady state. That is the steady state condition g(x)=0 is rewritten as G(x_k)=x_k+1,  so that at the fixed point G(x_n)=x_n gives the root of g(x_n)=0. This is how Newton's method works. To solve g(x)=0 you write x_k+1=x_k-g(x_k)/(dg/dx)_k=G(x_k). There are many ways to create a G(x) so that you can do fixed point iteration, but the fixed point may not be stable. If dg/dx is not zero at the fixed point, then Newton's method will converge ( if your initial guesx x_0 is close enough), and do so quadratically (very fast convergence). That is why it is so popular.

Yes, there are plenty of continuous Jachymski functions.  Any continuous function $\Phi:[0,\infty[\to [0,\infty[$ with $\Phi(0)=0$ and $\Phi(t)<t$ for all $t>0$ would do:  Then, for every $\epsilon>0$ we have $\Phi(\epsilon)<\epsilon$, hence by continuity there is a $\delta>0$  such that $\Phi(t) <\epsilon$ for all $\epsilon<t<\epsilon+\delta$.

Your question is not completely clear. Do you mean you are looking for a set-valued map that is not the subdifferential of a function, the word "subdifferential" being taken in one of the existing meanings given by a list of properties?

The application you have in view seem to orient to another interpretation.

You can refer to the book :

A.L. Dontchev and R.T. Rockafellar, Implicit functions and solutions mappings. A view from variational analysis, 2nd Edition, Springer, 2014.

It is not clear to me what is meant by the "method of upper and lower solution". Has this anything to do with the Tarski-Knaster fixed point theorem for complete lattices? And what is a "couple solution of a system"? (which system?)

One of the reason (probably the main reason) for the name 'fixed point Iteration(FPI)' for the method is as follows.

Consider the map(of a country e.g.) having two versions, one is the distorted (possibly bigger) of the other. Place one(smaller) map on the other. In the maps, each town in the country will be at different point(on the table), EXCEPT only one town which will be on the same point(called fixed point) on the table.

To find the fixed point town, follow the algorithm below, which is (in one  dimensional version ) the fixed point algorithm.

Assume that the distortion is represented by a function g(x,y). In 1D version, it is g(x).

1. Choose any town in the smaller map and get its point on table. ( get an initial x_0)

2. Locate the town in bigger map for that point ( x_1 = x_0 )

3. Find that town in smaller map ( x_0 = g(x_1) )

4. Go to step 2 ( x_1 = x_0 ... )

After some iterations, you will be nearing the fixed point town on both maps.

I use this example to explain the origin of FPI using(drawing) two maps of Sri Lanka in  numerical methods classes.

Hope this might help.

It does not hold when you drop the assumption of convexity. The first planar counter-example was found by Klee:

V. L. Klee, Some topological properties of convex sets. Trans. Amer. Math. Soc. 78, (1955). 30–45.

see also

V. L. Klee, An example related to the fixed-point property. Nieuw Arch. Wisk. (3) 8 1960 81–82.

Connell found other examples:

E. H. Connell, Properties of fixed point spaces. Proc. Amer. Math. Soc. 10 1959 974–979.

In particular, he showed that FPP is not closed under finite products or taking closures.

However, for convex sets in Rⁿ your question has an affirmative answer. See p. 35 in Klee's paper from TAMS and p. 958 in Connell's paper from PAMS.

yes; Just try to write another  proof of Brouwer fixed point theorem.

i give some ans, others can suggest their innovative ways and easy ways

1. Verilog real data type is only for simulation hence convert all into a BIGGG sized integer which hold the full value number with enough value precision ,as integer data type is synthesizble. (actually look for the lowest value , find how much 10's should be multiplied to make it integer , multiply this number to all the variables and constant in the algo)

2.VHDl offers fixed and float point package which is synthesizable , we use that for algo in VHDL , i think this is extendable for verilog also

3.if u know C u can write in C verify functionality in any digital tool or matlab and use xilinx HLS to make a HDL conversion that is synthesizable.

4.algo can be written straight forwardy as a set of equations in MATLAB , after verification use ,HDL tool ,Fixed point tool , simulink HDL coder to generate , HDLsome more ways ,

will discuss later , hope to receive new ways/ideas and possibilities from other company tools to do the above

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.

You also have the full-text http://cupid.economics.uq.edu.au/mclennan/Advanced/advanced_fp.pdf) by Andrew McLennan. I have not checked the quality; it is quite recent (November 2012).