Dear RG friends:
In two weeks time, I am all set to conduct a technical session on Analysis.
I plan to deliver a long lecture on "Fixed Point Theorems". Of course, Banach fixed point theorem is useful to establish the local existence and uniqueness of solutions of ODEs, and contraction mapping ideas are also useful to develop some simple numerical methods for solving nonlinear equations. Are there any other interesting science / engineering applications?
Kindly let me know! Thank you for the kind help.
With best wishes,
Sundar
Hi,
I think that a good book for your intention is: Eberhard Zeidler, Nonlinear Functional Analysis and its Applications, I: Fixed-Point Theorems, Springer-Verlag, 1986
Regards, Vasile
Hi,
I think that a good book for your intention is: Eberhard Zeidler, Nonlinear Functional Analysis and its Applications, I: Fixed-Point Theorems, Springer-Verlag, 1986
Regards, Vasile
See my paper "Existence of Solutions of the Hartree Equations for N Electrons
An Application of the Schauder-Tychonoff Theorem"
The Schauder-Tychonoff Theorem is a fixed point theorem. In general fixed point theorems give existence of solutions of equations but not uniquiness.
We're often using spectral method approaches for solving mechanical equilibrium in volume elements subjected to periodic boundary conditions.
The initial method is based on a fixed point iteration
http://hera.polytechnique.fr/users/bornert/MSE_multiech08/Articles/Moulinec_Suquet.pdf
I agree with the RG friends that typically fixed point theorems are used to show the existence of solutions of some equations. In my early research, I used Brouwer's fixed point theorem to establish that a certain mapping is surjective.
Brouwer's fixed-point theorem is a fixed-point theorem in topology, which states that for any continuous function f mapping a compact convex set into itself there is a point x0 such that f(x0) = x0.
Best wishes, Sundar
A generalization of Brouwer's fixed point theorem is the famous Schauder fixed point theorem.
Schauder fixed point theorem asserts the following:
"Every continuous function from a convex compact subset K of a Banach space to K itself has a fixed point."
Recent text-books also call Schauder fixed point theorem as Schauder-Tychonoff fixed point theorem.
Best wishes, Sundar
As applications : logic programming and fuzzy logic programming
Nash equilibrium and game theory.
I also have some experience of apllications of Browder fixed point theorem (that is a generalization of the Schauder one) to the existence of periodic solutions of some impulsive systems . It works in some special cases, when Brouwer theorem fails. (The formulation of Browder theorem can be found in Duke Math. J., 26(2):291-303, 1959.)
Perhaps you should take a look at the website
http://signal.ece.utexas.edu/seminars/dsp_seminars/02spring/danilo.html
I would recommend you the monograph
V.G.Angelov "A Method for Analysis of Transmission Lines Terminated by
Nonlinear Loads"Nova Science, New York,2014
No one seems to have mentioned one of the earliest applications; namely implicit function theorems, particularly in steering so more general than $\mathbb{R}^n$.
I recommend you the monograph
V.G.Angelov "A Method for Analysis of Transmission Lines Terminated by Nonlinear Loads",Nova Science , New York, 2014
Implicit function theorem and inverse function theorem are very fundamental results in differential geometry. I have used inverse function theorem quite a lot in my control papers!
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