The Banach fixed-point theorem (1922) gives a general criterion guaranteeing that, if it is satisfied, the procedure of iterating a function yields a fixed point. ^[2]

By contrast, the Brouwer fixed-point theorem (1911) 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, ^[3] but it doesn't describe how to find the fixed point (see also Sperner's lemma ).

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 (known as the Dottie number ) is approximately x = 0.73908513321516 (thus x = cos( x ) for this value of x ).

The Lefschetz fixed-point theorem ^[4] (and the Nielsen fixed-point theorem ) ^[5] from algebraic topology is notable because it gives, in some sense, a way to count fixed points.

There are a number of generalisations to Banach fixed-point theorem and further; these are applied in PDE theory. See fixed-point theorems in infinite-dimensional spaces .

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. ^[6]

The Knaster?Tarski theorem states that any order-preserving function on a complete lattice has a fixed point, and indeed a smallest fixed point. ^[7] See also Bourbaki?Witt theorem .

The theorem has applications in abstract interpretation , a form of static program analysis .

A common theme in lambda calculus is to find fixed points of given lambda expressions. 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. ^[8] An important fixed-point combinator is the Y combinator used to give recursive definitions.

In denotational semantics of programming languages, a special case of the Knaster?Tarski theorem is used to establish the semantics of recursive definitions. While the fixed-point theorem is applied to the "same" function (from a logical point of view), the development of the theory is quite different.

The same definition of recursive function can be given, in computability theory , by applying Kleene's recursion theorem . ^[9] These results are not equivalent theorems; the Knaster?Tarski theorem is a much stronger result than what is used in denotational semantics. ^[10] However, in light of the Church?Turing thesis their intuitive meaning is the same: a recursive function can be described as the least fixed point of a certain functional, mapping functions to functions.

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.

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.

Every involution on a finite set with an odd number of elements has a fixed point; more generally, for every involution on a finite set of elements, the number of elements and the number of fixed points have the same parity . Don Zagier used these observations to give a one-sentence proof of Fermat's theorem on sums of two squares , by describing two involutions on the same set of triples of integers, one of which can easily be shown to have only one fixed point and the other of which has a fixed point for each representation of a given prime (congruent to 1 mod 4) as a sum of two squares. Since the first involution has an odd number of fixed points, so does the second, and therefore there always exists a representation of the desired form. ^[11]

• Trace formula

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• Berinde, Vasile (2005). Iterative Approximation of Fixed Point . Springer Verlag. ISBN  978-3-540-72233-5 .
• Border, Kim C. (1989). Fixed Point Theorems with Applications to Economics and Game Theory . Cambridge University Press. ISBN  0-521-38808-2 .
• 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 .
• ?a?kin, Jurij A; Minachin, Viktor; Mackey, George W. (1991). Fixed Points . American Mathematical Society. ISBN  0-8218-9000-X .

• Fixed Point Method