Science topic

# Fixed Point Theory - Science topic

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. Results of this kind are amongst the most generally useful in mathematics.
Questions related to Fixed Point Theory
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If possible, give an example of a continuous function defined on a convex subset of a Banach space $X$ satisfies Kannan contraction but does not satisfy Banach contraction.
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Hi everyone
I want to start the discussion on a real-life application of metrical and Generalized Metrical(b-metric, partial metric, rectangular metric, complex-valued metric, C*-algebra valued metric, etc) Fixed Point Theory. I hope you can contribute to the discussion.
Thank you very much for your valuable views @Dr. Oluma Ararso
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I am working on geometry of banach spces and applications in metric fixed point theory , especially  my interesting is renorming of Banach spaces, Is anyone interested in collaboration
I am interested in the applications of Fixed Point Theory to differential and integrodifferential equations (fractional o generalized). Can be?
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Actually proving a new metric-like space to be metrizable is an interesting study in fixed point theory. Though I don't have so much knowledge or idea about all the metrization theorems except for Urishon metrization theorem. So I want to know more about this area. Thank you.
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.
Question
Dear researchers
As we know, recently a new type of derivatives have been introduced which depend two parameters such as fractional order and fractal dimension. These derivatives are called fractal-fractional derivatives and divided into three categories with respect to kernel: power-law type kernel, exponential decay-type kernel, generalized Mittag-Leffler type kernel.
The power and accuracy of these operators in simulations have motivated many researchers for using them in modeling of different diseases and processes.
Is there any researchers working on these operators for working on equilibrium points, sensitivity analysis and local and global stability?
Thank you very much.
Best regards
Yes I am
Question
Hi everyone,
I want to start the discussion about open problems in the field of Fixed Point Theory. I hope you can contribute to the discussion.
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
Question
This set of ODEs describes a dynamical system where a,b,c,d and e are positive real valued parameters of the system. We know at a fixed point all system dynamics are zero. Though in this special case, the set of nonlinear equations after setting all dynamics to zero, states that one of the system parameters (e) should be equal to zero wich is not the case. Does such a system (for particular this one) have a equilibrium point or this result states that it has not.
No equilibrium points.
Question
Dear Researchers
As you know, nowadays, the mathematical models of different phenomena and processes are designed by means of fractional operators in the context of various systems of boundary or initial value problems with boundary conditions. Some of such well-known mathematical models can be found in many papers like pantograph equations, Langevin equations, Jerk equations, Snap equations, etc. All of these models are practical examples of fractional boundary value problems.
Could you suggest other examples about practical applications in fractional boundary value problems? I am going to study the dynamical behaviors of these new models theoretically and numerically.
Thank you very much.
Best regards
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,…
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As question stated , maybe can also include symplectic topology.
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:
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Please let me know if such function, which does not satisfy Banach contraction principle, but the fixed points exist for these functions.
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".
Question
How we can studying the existence and uniqueness of solutions of the functional equations by fixed point theory
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.
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for the approximate solution of different
types of differential equations"
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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.
Fixed point theory solved many nonlinear problems. In the future, it will play more crucial role in many real world problems.
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There are many generalizations about Banach's contraction principle in fixed point theory.  Are there any applications these generalizations  in any applied fields?
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.
Question
What is the stability of a fixed point (non-isolated), which has zero eigenvalues? Is there any analytical procedure for finding stability of these type of fixed points. If numerical analysis can do the job, then how to do that?
I suggest you the stability of the iteration processes.
Question
I read in many articles a statement that best proximity point theory has applications in the field of Economics, game theory, biological science. But I want to see some physical problem that was mathematically modeled and then solved by using best proximity point theory or fixed point theory.
Please se my recent paper, "Approximation of fixed points and best proximity points of relatively nonexpansive mappings".
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I'm currently working on fixed point theorems on uniformly convex spaces and I will love if anyone can point my attention to spaces that are uniformly convex apart from the ones I have listed above.
I suggest you the 2-uniformly convex hyperbolic spaces.
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i have trouble in solving a problem using an implicit algorithm. I am not able to get the initial approximation and also how will i proceed to the next iteration.
Hello, try this book as an initiation : Iterative Methods for Linear
and Nonlinear Equations, C. T. Kelley, North Carolina State University, Philadelphia 1995.
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Fuzzy metric spaces
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)
Question
I would like to collect the applications of partition of unity theorem in math, for example manifold, topology, fixed point theory, differential forms, differential geometry, vector analysis, algebraic topology, differential topology , and any other related fields.
Please don't forget about many(!) applications of partitions of unity in differential inclusions. It can be found in any book on the subject.
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suggest me any good materilas on fixed point theory and dynamic programing,and fuzzy metric space
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Please I would like your help on getting an e-copy of the book entitled "The theory of fixed point classes" by Tsai-Han Kiang.
Best regards,
M.S. Abdullahi
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.
Question
Can we use fixed point theory to study the inventory control problems? If yes then please share any literature.
Yes. You can do this.
Question
How we can use Fixed point theory to find the fluid layer stability or instability?
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.
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Whenever we try to introduce Fixed Point Theory, we can use some real life examples. I want to know some of such examples. Like the map meets the floor in one and only one fixed point. If anybody can present more of such type?
Question
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?
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.
Question
How we can use Fixed point theory to find stability condition of some Flow problems?
Please look at the attached fille. Perhaps it will be helpful. The published article can be find at
Question
I am trying to prove the existence of positive solutions to:
-y''=f(y), 0<x<1,
y(0)=y(1)=0.
where f(y)=k sin(y)-sin(2y) "k~ arbitrary constant". Any help with known result or useful technique is much appreciated.
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)
Question
For example, when we xerox a document, the printer takes the A4 page to the same. Are there other better examples?
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.
Question
What are the applications of implicit iterative algorithms? Moreover, is there any nonlinear nonexpansive mapping example satisfying some result (any research paper in literature with implicit iteration) having implicit iteration process?
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.
Question
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.
Question
i am looking for a software tools set that let me setup a multi-point to multi-point VPN which is called VPLS over IP (not MPLS) . does any one have any suggestion how can  I get in to this?
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:
Question
MRI which uses NMR enjoy the resonance feature of hydrogen nucleus for imaging. this technique is completely different from reflection feature of objects used in radar scanning. i would like to know why this technique is not used for radar scanning to detect other material that used in aircraft or devices that have special materials .
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.
Question
Dear All,
Why most mathematicians choose kkm-mapping to solve the problems in variational inequality ?  while the problems can been solve by another methods such as fixed point theorem, critical point theorems and so on.
best regards
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.
Question
We know that from Shukla's paper (Shukla, S: Partial b-metric spaces and fixed point theorems. Mediterr. J. Math. (to appear).doi:10.1007/s00009-013-0327-4)
If (X,p) be a partial metric space and q≥1, then (X,p_{b}) is a partial b-metric space with the coefficient s=2^{q-1} , where p_{b} is defined by p_{b}(x;y)=(p(x,y))^{q}. Whether this statement is true according to a modified definition of partial b-metric in the paper of Z. Mustafa and et al.(Z. Mustafa, J.R. Roshan, V. Parvaneh and Z. Kadelburg, Some common fixed point results in ordered partial b-metric spaces, Journal of Inequalities and Applications, 2013, 2013:562.)
This is a good question.
In considering partial b-metric spaces, a good place to start is
See, also:
Question
Let X be a complete metric space (C*-algebra valued) and CB(X) be nonempty closed and bounded subset of X. Can we define a mapping T:X------->CB(X), i.e. Can we have a C*-algebra valued multi-valued mapping? If Yes, please can someone provide me with an example of such a mapping?
Because I know if X=R(set of real numbers), any interval I be it open, closed, half-open or half-closed then I is a subset of R and we can define such a mapping T by for instance Tx = [0, x/2), thus we can be able to get a fixed point depending on some conditions either on T, X or both.
I hope my question is well constructed and understandable.
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
That is a perfect set valued mapping.
:-P
Cheers
p.s.: The composition was wrong, I corrected the mistake
Question
when I studied some delay systems, the fixed points of these systems have zero eigenvalue. I don't how can study its stability?
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.
Question
Such as the differential equations or more generally, evolution equations are the fields where the Banach's Contraction Principle in Fixed Point Theory is recommended., Book of "A. Pazy, Semigroups of linear operators and Applications to Partial Differential Equations".
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.
Question
What are the additional conditions as needed in order to establish the
existence of a unique fixed point satisfying the condition for any α ∈ (0, 1] there exists
β ∈ (0, α) such that dα(x, z) ≤ s[dβ(x, y) + dβ(y, z)] in Gbq-family (X, dα).
Related information can be found in below paper.
Kumari, Panda S., and Dinesh Panthi. "Cyclic contractions and fixed point theorems on various generating spaces." Fixed Point Theory and Applications 2015.1 (2015): 153.
Question
Distribution of nearest neighbors versus separation distance (r) is plotted for a point pattern. If the function exhibits three peaks at three different r, then what would be the interpretation of such a graph?
Question
I can see the transform whose kernel is log function
Hello, yes there is...it is Cauchy's integral. See attached paper for further details.
Question
Poincare and Einstein
Ricci flow is:
μ ∂Gμν/∂μ=α’ Rμν
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.
Question
Every group of prime order is cyclic.
Cyclic implies abelian.
Every subgroup of an abelian group is normal.
Every group of Prime order is simple.
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
Question
Manifold concept was first introduced by Riemann.
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.
Question
Assume that M is  a  compact manifold with fixed point property. Let N be  a compact submanifold of M x M, with dim (N)=dim (M). Assume that $\pi_{1}:N \to M$ is a surjective map, where $\pi_{1}$ is the projection on the first component.
Is it true that N has non empty intersection with the diagonal {(x,x)\in M x M}?
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.
Question
If there is no extranous fixed points for any iterative methods, what does mean?
Such methods, possible to diverge at any points in complex plane. ?
If there is many extranous fixed points, then what it means?
For example, one optimal fourth order modified newton's method gives 30 extranous fixed points for solving p(z)=z^3-1. for this results what say's.
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.
Question
I have a function f
f: Z^2 -> Z^2
such that
(x', y') = f(x, y)
and
x'-y' <= x-y
I need a reference paper or book that shows (perhaps using a more general statement) that iterative applications of f converge to a fixed point (x_0, y_0) IN A FINITE NUMBER OF ITERATIONS.
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  N 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*  .
Question
I have implemented beamformer(MVDR/LCMV) for speech enhancement in matlab, now I am trying to convert it to fixed point implementation. during the weight calculation step correlation matrix inversion is required, but in fixed point implementation  correlation matrix inversion is not giving correct results(may be due to very small values in correlation matrix).
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).
Question
I have always thought that fixed points and equilibria refer to the same thing until I started studying stability of fixed points where I noticed that to find an equilibrium of a function f we need to solve f(x)=0 and not f(x)=x as we do for fixed points !!!
Aren't they the same thing or am I just missing some points?
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.
Question
The definition of  Jachymski function  can be found  in the paper
''C. Alegre, J. Marn and S. Romaguera, A ﬁxed point theorem in generalized contraction involving W-distances on complete quesi-metric spaces, Fixed theory and applications, 40,1-8,(2014).
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 $\epsilon<t<\epsilon+\delta$.
Question
I'm looking for optimizing multivalued  vector valued function.
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.
Question
I am wondering if there are some theorems in nonlinear analysis that address the following properties of implicit function (set valued function) that the implicit function is f(x) : X-> P(Y) where X , Y \subset R^n , and P(Y) denotes the power set of Y. and y \in f(x) specified by F(x,y)=0. The properties are
1- Existence of y \in f(x) for any x \in Y
2- hemi-continuity ( or upper and lower hemi-continuity) of the set valued function
3- differentiability class of f(x) that is the maximum order of differentiability f(x) -( we can assume f(x) is single value if required)
4- f(x) is convex single valued or set valued function
5- f(x) is single valued
6- the set valued function f(x) is connected for any x \in X
any theorem that can address one of those properties is also useful
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.
Question
We have some important fixed point theorem likes the three celebrated fixed point
theorems: The Banach, Brouwer & Schauder theorems. My question is to use the method of
upper and lower solution and add some conditions to give new Theorem for the
existence of couple solution (u,v) for system.
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?)
Question
What type of convergence characteristics does it possess?
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.
Question
i.e.,  Is a convex set in R^n with the fixed point property always compact ?
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.
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.
Question
As you know fixed point theory has many applications in all sciences.
yes; Just try to write another  proof of Brouwer fixed point theorem.
Question
Fixed and floating point date type synthesizable HDL code what are the diff ways/tricks/new& diff ideas that a fixed or floating point algorithm can be implemented in FPGA using Verilog and the same for VHDL
For the betterment of student and professionals:SInce this info is not fully/widely available in book only engineers who were working in FPGA embedded system application development knows better ways and possibilities with current technical advancements.
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
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