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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Mathematics, a fixed point (sometimes shortened to fixpoint) of a function is a point that is mapped to itself by the function Results of this kind are amongst the most generally useful in mathematics.
The Banach fixed point theorem gives a general criterion guaranteeing that, if it is satisfied, the procedure of iterating a function yields a fixed point. The Banach Fixed point theorem (also known as the contraction mapping theorem or contraction mapping principle) is an important tool in the theory of Iteration means the act of repeating Mathematics Iteration in mathematics may refer to the process of iterating a function, or to the techniques used
By contrast, the Brouwer fixed point theorem is a non-constructive result: it says that any continuous function from the closed unit ball in n-dimensional Euclidean space to itself must have a fixed point, but it doesn't describe how to find the fixed point (See also Sperner's lemma). In Mathematics, the Brouwer fixed point theorem is an important Fixed point theorem that applies to finite-dimensional spaces and which forms the basis for several In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output In Mathematics, a unit Sphere is the set of points of Distance 1 from a fixed central point where a generalized concept of distance may be used a closed You may be looking for Sperner's theorem on set families In Mathematics, Sperner's lemma is a combinatorial analog
For example, the cosine function is continuous in [-1,1] and maps it into [-1, 1], and thus must have a fixed point. This is clear when examining a sketched graph of the cosine function; the fixed point occurs where the cosine curve y = cos(x) intersects the line y = x. Numerically, the fixed point is approximately x = 0. 73908513321516 (thus x = cos(x)).
The Lefschetz fixed-point theorem (and the Nielsen fixed-point theorem) from algebraic topology is notable because it gives, in some sense, a way to count fixed points. In Mathematics, the Lefschetz fixed-point theorem is a formula that counts the number of fixed points of a continuous mapping from a compact Nielsen theory is a branch of mathematical research with its origins in Topological Fixed point theory. Algebraic topology is a branch of Mathematics which uses tools from Abstract algebra to study Topological spaces The basic goal is to find algebraic
There are a number of generalisations to Banach spaces and further; these are applied in PDE theory. In Mathematics, Banach spaces (ˈbanax named after Polish Mathematician Stefan Banach) are one of the central objects of study in Functional analysis In Mathematics, partial differential equations ( PDE) are a type of Differential equation, i See fixed point theorems in infinite-dimensional spaces. In Mathematics, a number of fixed point theorems in infinite-dimensional spaces generalise the Brouwer fixed point theorem.
The Knaster-Tarski theorem is somewhat removed from analysis and does not deal with continuous functions. In the mathematical areas of order and Lattice theory, the Knaster–Tarski theorem, named after Bronisław Knaster and Alfred Tarski It states that any order-preserving function on a complete lattice has a fixed point, and indeed a smallest fixed point. In Mathematics, a complete lattice is a Partially ordered set in which all subsets have both a Supremum (join and an Infimum (meet See also Bourbaki-Witt theorem. In Mathematics, the Bourbaki–Witt theorem in Order theory, named after Nicolas Bourbaki and Ernst Witt, is a basic Fixed-point theorem
A common theme in lambda calculus is to find fixed points of given lambda expressions. In Mathematical logic and Computer science, lambda calculus, also written as λ-calculus, is a Formal system designed to investigate function Every lambda expression has a fixed point, and a fixed point combinator is a "function" which takes as input a lambda expression and produces as output a fixed point of that expression. An important fixed point combinator is the Y combinator used to give recursive definitions. Recursion in computer science is a way of thinking about and solving problems
The above technique of iterating a function to find a fixed point can also be used in set theory; the fixed-point lemma for normal functions states that any continuous strictly increasing function from ordinals to ordinals has one (and indeed many) fixed points. The fixed-point lemma for normal functions is a basic result in Axiomatic set theory stating that any Normal function has arbitrarily large fixed points In Set theory, an ordinal number, or just ordinal, is the Order type of a Well-ordered set.
Every closure operator on a poset has many fixed points; these are the "closed elements" with respect to the closure operator, and they are the main reason the closure operator was defined in the first place. A closure operator on a set S is a function cl P ( S) → P ( S) from the Power set of S In Mathematics, especially Order theory, a partially ordered set (or poset) formalizes the intuitive concept of an ordering sequencing or arrangement
The collage theorem in fractal compression proves that, for many images, there exists a relatively small description of a function that, when iteratively applied to any starting image, rapidly converges on the desired image. Fractal compression is a lossy image compression method using Fractals to achieve high levels of compression
See also
- Atiyah–Bott fixed-point theorem
- Borel fixed-point theorem
- Brouwer fixed point theorem
- Caristi fixed point theorem
- Diagonal lemma
- Fixed point property
- Injective metric space
- Kakutani fixed-point theorem
- Kleene fixpoint theorem
- Woods Hole fixed-point theorem
References
- Agarwal, Ravi P. In Mathematics, the Atiyah–Bott fixed-point theorem, proven by Michael Atiyah and Raoul Bott in the 1960s is a general form of the Lefschetz fixed-point In Mathematics, the Borel fixed-point theorem is a Fixed-point theorem in Algebraic geometry. In Mathematics, the Brouwer fixed point theorem is an important Fixed point theorem that applies to finite-dimensional spaces and which forms the basis for several In Mathematics, the Caristi fixed point theorem (also known as the Caristi-Kirk fixed point theorem) generalizes the Banach fixed point theorem for maps In Mathematical logic, the diagonal lemma or fixed point theorem establishes the existence of Self-referential sentences in formal theories of A mathematical object X has the fixed point property if every suitably well-behaved mapping from X to itself has a fixed point. In Metric geometry, an injective metric space, or equivalently a hyperconvex metric space, is a Metric space with certain properties generalizing those In Mathematical analysis, the Kakutani fixed point theorem is a Fixed-point theorem for Set-valued functions It provides sufficient conditions for a set-valued In the mathematical areas of order and Lattice theory, the Kleene fixed-point theorem, named after American mathematician Stephen Cole Kleene ; Meehan, Maria; O'Regan, Donal (2001). Fixed Point Theory and Applications. Cambridge University Press. ISBN 0-521-80250-4.
- Border, Kim C. (1989). Fixed Point Theorems with Applications to Economics and Game Theory. Cambridge University Press. ISBN 0-521-38808-2.
- Brown, R. F. (Ed. ) (1988). Fixed Point Theory and Its Applications. American Mathematical Society. ISBN 0-8218-5080-6.
- Dugundji, James; Granas, Andrzej (2003). Fixed Point Theory. Springer-Verlag. ISBN 0-387-00173-5.
- Kirk, William A. ; Goebel, Kazimierz (1990). Topics in Metric Fixed Point Theory. Cambridge University Press. ISBN 0-521-38289-0.
- Kirk, William A. ; Khamsi, Mohamed A. (2001). An Introduction to Metric Spaces and Fixed Point Theory. John Wiley, New York. . ISBN 978-0-471-41825-2.
- Kirk, William A. ; Sims, Brailey (2001). Handbook of Metric Fixed Point Theory. Springer-Verlag. ISBN 0-7923-7073-2.
- Šaškin, Jurij A; Minachin, Viktor; Mackey, George W. (1991). Fixed Points. American Mathematical Society. ISBN 0-8218-9000-X.
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