Fixed-point theorems

Fixed-point theorems are fundamental results in mathematics that establish conditions under which a function will have a point that maps to itself. In simpler terms, if you have a function \( f \) defined on a certain space, a fixed point \( x \) satisfies the equation \( f(x) = x \). Fixed-point theorems are widely applicable in various areas such as analysis, topology, and applied mathematics.

The Atiyah–Bott fixed-point theorem is a fundamental result in algebraic topology and differential geometry, developed by mathematicians Michael Atiyah and Raoul Bott in the context of the study of fixed points of smooth maps on manifolds.

Bekić's theorem is a result in the field of functional analysis, specifically concerning the properties of certain types of topological vector spaces. The theorem addresses the conditions under which a set of continuous linear functionals on a topological vector space can separate points in the space.

The Borel fixed-point theorem is a result in topology, particularly in the context of more general spaces than just traditional fixed-point theorems. It states that any continuous function from a compact convex set in a finite-dimensional Euclidean space to itself has at least one fixed point.

The Brouwer Fixed-Point Theorem is a fundamental result in topology, specifically in the field of fixed-point theory. It states that any continuous function mapping a compact convex set to itself has at least one fixed point.

The Browder Fixed-Point Theorem is a result in functional analysis and topological fixed-point theory, named after the mathematician Felix Browder. This theorem extends the classical Brouwer Fixed-Point Theorem to more general contexts, particularly in infinite-dimensional spaces.

The Earle–Hamilton fixed-point theorem is a result in the field of topology, particularly in the study of fixed points in continuous functions.

A fixed-point theorem is a fundamental result in various branches of mathematics, particularly in analysis and topology, that asserts the existence of fixed points under certain conditions. A fixed point of a function is a point that is mapped to itself by the function. Formally, if \( f: X \rightarrow X \) is a function on a set \( X \), then a point \( x \in X \) is a fixed point if \( f(x) = x \).

Fixed-point theorems are fundamental results in mathematics that guarantee the existence of points that remain unchanged under certain mappings. While fixed-point theorems are traditionally studied in finite-dimensional spaces (like the well-known Banach and Brouwer Fixed-Point Theorems), their generalization to infinite-dimensional spaces presents some unique challenges and requires different techniques. Here’s an overview of some of the key concepts and results related to fixed-point theorems in infinite-dimensional spaces: ### 1.

The Kleene fixed-point theorem is a fundamental result in theoretical computer science and mathematical logic, particularly in the context of domain theory and functional programming. Named after Stephen Cole Kleene, it provides a framework for understanding the existence of fixed points in certain types of functions. In simple terms, a fixed point of a function \( f \) is a value \( x \) such that \( f(x) = x \).

The Lefschetz Fixed-Point Theorem is a fundamental result in algebraic topology that provides a criterion for determining the existence of fixed points of continuous maps on topological spaces. It is particularly useful when dealing with maps between compact, connected, and oriented manifolds.

The Markov–Kakutani fixed-point theorem is a generalization of the classical Brouwer fixed-point theorem, designed for multi-valued functions (or correspondences). It is important in various areas such as game theory, economics, and optimization.

Nielsen theory, often associated with the work of mathematician and physicist Nielsen, primarily pertains to the field of topological and algebraic invariants in the context of knot theory and three-manifolds. One of the key contributions of Nielsen is his work on the concept of "Nielsen classes," which relate to the classification of covering spaces of surfaces and the study of fundamental groups.

The Ryll-Nardzewski fixed-point theorem is a result in the field of functional analysis, specifically concerning fixed points in nonatomic convex sets in topological vector spaces. It generalizes certain fixed-point results, including the well-known Brouwer fixed-point theorem, to more general settings.

The Schauder fixed-point theorem is a fundamental result in fixed-point theory, particularly in the context of functional analysis and topology. It provides conditions under which a continuous function mapping a convex compact subset of a Banach space (or more generally, in a topological vector space) has at least one fixed point.

The Single-Crossing Condition (SCC) is a concept used primarily in economics, particularly in the context of auction theory, mechanism design, and social choice theory. It refers to a specific property of preference orderings among different agents or individuals regarding a set of alternatives. Under the Single-Crossing Condition, the preference rankings of the individuals (or types) can only cross at most once when plotted against a single dimension of preference.