Fixed-point theorem

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.

Well-known fixed-point theorems include:

• Atiyah–Bott fixed-point theorem
• Banach fixed-point theorem, for contraction mappings
• Borel fixed-point theorem
• Brouwer fixed-point theorem, for continuous functions from the closed unit ball in n-dimensional Euclidean space to itself
• Caristi fixed point theorem
• Fixed-point lemma for normal functions, for continuous strictly increasing functions from ordinals to ordinals
• Fixed point theorems in infinite-dimensional spaces
• Kakutani fixed-point theorem
• Kleene fixpoint theorem
• Lefschetz fixed-point theorem
• Nielsen fixed-point theorem
• Knaster–Tarski theorem, which states that any order-preserving function on a complete lattice has a smallest fixed point
• Tychonoff fixed point theorem
• Woods Hole fixed-point theorem

See also

• Fixed-point property
• Fixed-point combinator
• Collage theorem
• Diagonal lemma, also known as the fixed-point lemma, for producing self-referential sentences of first-order logic
• Sperner's lemma
• Bourbaki–Witt theorem
• Injective metric space
• Topological degree theory

Further reading

• Agarwal, Ravi P.; Meehan, Maria; O'Regan, Donal (2001). Fixed Point Theory and Applications. Cambridge University Press. ISBN 0-521-80250-4. 
• Aksoy, Asuman; Khamsi, Mohamed A. (1990). Nonstandard Methods in fixed point theory. Springer Verlag. ISBN 0-387-97364-8. 
• 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.